# The Nodal sets of solutions to parabolic equations

Yiqi Huang  Department of Mathematics, MIT, 77 Massachusetts Avenue,
Cambridge, MA 02139-4307, USA<EMAIL_ADDRESS>and  Wenshuai Jiang  School of
Mathematical Sciences, Zhejiang University, Hangzhou 310058, China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we study the parabolic equations
$\partial_{t}u=\partial_{j}\left(a^{ij}(x,t)\partial_{i}u\right)+b^{j}(x,t)\partial_{j}u+c(x,t)u$
in a domain of $\mathbb{R}^{n}$ under the condition that $a^{ij}$ are
Lipschitz continuous. Consider the nodal set $Z_{t}=\\{x:u(x,t)=0\\}$ at a
time $t$-slice. Simple examples show that the singular set
$\mathcal{S}_{t}=\\{x:u(x,t)=|\nabla_{x}u|(x,t)=0\\}$ may coincide with nodal
set. This makes the methods used in the study of nodal sets for elliptic
equations fail, rendering the parabolic case much more complicated.

The current strongest results in the literature establish the finiteness of
the $(n-1)$-dimensional Hausdorff measure of $Z_{t}$, assuming either $n=1$ by
Angenent [3] or that the coefficients are time-independent and analytic by Lin
[36]. With general coefficients, the codimension-one estimate was obtained
under some doubling assumption by Han-Lin [30] but only for space-time nodal
sets. In the first part, we prove that $\mathcal{H}^{n-1}(Z_{t})<\infty$ in
full generality, i.e. for any dimension, with time-dependent coefficients and
with merely Lipschitz regular leading coefficients $a^{ij}$.

In the second part, we study the evolutionary behavior of nodal sets. When
$n=1$, it is proved by Angenent [3] that the number of nodal points is non-
increasing in time. For the $n$-dimensional case, we construct examples
showing that measure monotonicity fails. In contrast, we prove dimension
monotonicity, i.e., the Hausdorff dimension of the nodal set is non-increasing
in time. This is the first monotonicity property for nodal sets in general
dimensions. All the assumptions here are sharp.

###### Contents

  1. 1 Introduction
    1. 1.1 Finite Measure Theorem
    2. 1.2 Dimension Monotonicity Theorem
    3. 1.3 Outline
  2. 2 Background and Heat Equations
    1. 2.1 Caloric Function and monotone frequency
    2. 2.2 Uniqueness of tangent maps of heat equations
    3. 2.3 Maximum Principle
    4. 2.4 Quantitative Stratification at a time slice
  3. 3 General Parabolic Equations and Almost Monotone Frequency
    1. 3.1 Almost Monotonicity for Localized Frequency
    2. 3.2 Frequency and Doubling Index
    3. 3.3 Equivalence between different normalizations
  4. 4 Quantitative Uniqueness of Tangent Maps
    1. 4.1 Growth estimates under doubling lower bound
    2. 4.2 Gradient estimates for Green’s function expansion
    3. 4.3 Quantitative Uniqueness of tangent maps
  5. 5 Cone-Splitting in a Fixed Time Slice
  6. 6 Neck Structure and Neck Decomposition
    1. 6.1 Neck Regions
    2. 6.2 Neck Decomposition
  7. 7 Proof of the Main Theorems
    1. 7.1 Proof of Theorem 1.14
    2. 7.2 Proof of Theorem 1.4 and Theorem 1.19

##  1\. Introduction

The study of nodal sets of functions has long been a prominent research topic,
particularly concerning the zero sets of solutions to partial differential
equations. In this paper, we systematically investigate the properties of
nodal sets of general parabolic equations. Let us consider a solution $u(x,t)$
to the parabolic equations with time-dependent coefficients

$\partial_{t}u=\partial_{i}(a^{ij}(x,t)\partial_{j}u)+b^{i}(x,t)\partial_{i}u+c(x,t)u,$
(1.1)

in $Q_{2}:=\\{(x,t):|x|<2,-4<t\leq 0\\}\subset\mathbb{R}^{n}\times\mathbb{R}$.
where the coefficients $a^{ij}$ are elliptic and the coefficients $b,c$ are
bounded,

$(1+\lambda)^{-1}\delta^{ij}\leq
a^{ij}\leq(1+\lambda)\delta^{ij},\quad|b^{i}|,|c|\leq\lambda.$ (1.2)

We assume $a^{ij}$ satisfies Lipschitz regularity condition with respect to
the parabolic distance

$\displaystyle|a^{ij}(x,t)-a^{ij}(y,s)|\leq\lambda\cdot
d((x,t),(y,s))\equiv\lambda\cdot(||x-y||^{2}+|s-t|)^{1/2}.$ (1.3)

###  1.1. Finite Measure Theorem

First we study the measure of nodal sets of $u(\cdot,t)$ for each fixed $t$.

Particularly, when $u(x,t)$ and all coefficients are time-independent, $u$
satisfies an elliptic equation, whose nodal set has been extensively studied.
If the coefficients are analytic, Donnelly and Fefferman [20] obtained optimal
upper and lower bound estimates for the Haudorff measure of nodal sets of
eigenfunctions of Laplacian, and Lin [36] provided another proof for the
optimal upper bound. The non-analytic conditions are much more complicated. To
name a few, there are several explicit but not optimal upper bound estimates
obtained by Hardt-Simon [32] and also by Han-Lin [29] with another proof, and
also lower bounds by [13, 54]. Recently, Logunov [41] proved the optimal lower
bound for the measure, thereby resolving Yau’s conjecture, and also obtained
polynomial growth upper bounds [42], see also [19, 21, 28, 31, 33, 35, 38, 40,
43, 48] for related results.

As for the solutions to parabolic equations in domain of $\mathbb{R}^{n}$, the
lack of unique continuation in the time direction makes controlling the local
behavior of solutions much more involved. In [36], Lin proved the measure of
nodal sets at each time slice is finite for the equations with analytic and
time-independent coefficients. Nonetheless, with general non-analytic and
time-dependent coefficients, very few results have been discovered. In the
case $n=1$, Angenent [3] proved that the nodal point set at each time slice is
discrete. For higher dimensions, Han-Lin [30] proved that the nodal set has
finite space-time codimension-one Hausdorff measure under doubling assumption
on the solution. However, at each time slice, controlling the measure of the
nodal sets remains an open problem.

As our first result, we address this case in full generality: for any
dimension, with time-dependent coefficients and with merely Lipschitz regular
leading coefficients $a^{ij}$, we can prove that the $(n-1)$-dimensional
Hausdorff measure is finite at each time slice.

######  Theorem 1.4.

Let $u$ be a solution of (1.1) in $Q_{2}$ satisfying (1.2) and (1.3). For any
$t_{0}>-4$, if $u(\cdot,t_{0})$ is not identically zero, then we have the
following Hausdorff measure estimate

$\displaystyle\mathcal{H}^{n-1}(Z_{t_{0}}\cap
B_{1})<C(n,\lambda,\Lambda)<\infty,$ (1.5)

where $Z_{t_{0}}=\\{x\in B_{2}:u(x,t_{0})=0\\}$ and
$\Lambda\equiv\int_{Q_{2}}u^{2}dxdt/\int_{B_{3/2}\times\\{t_{0}\\}}u^{2}dx$.

######  Remark 1.6.

Note that here the Lipschitz regularity (1.3) is a sharp assumption. If one
only assumes Hölder assumption on $a^{ij}$, i.e.
$|a^{ij}(x,t)-a^{ij}(y,s)|\leq\lambda\cdot|(x,t),(y,s)|^{\alpha}$ for
$\alpha\in(0,1)$, then there exists some nontrivial solution that vanishes on
some open subset, see [45, 46, 52].

######  Remark 1.7.

More generally the same result (1.5) holds for $u$ satisfying
$|\partial_{t}u-\partial_{i}(a^{ij}(x,t)\partial_{j}u)|\leq\lambda(|\nabla
u|+|u|)$ with (1.2) and (1.3). The proof is verbatim with minor modifications.

######  Remark 1.8.

It is proved that the unique continuation property holds for solution $u$ to
(1.1) satisfying (1.2) and (1.3), i.e., if $u(\cdot,t_{0})$ has infinite
vanishing order at $x_{0}\in B_{2}$ for some fixed $t_{0}>-4$, then
$u(\cdot,t_{0})\equiv 0$ at time $t_{0}$, see [10, 22, 25, 37].

######  Remark 1.9.

Following the same lines as in [33] we can improve the Hausdorff estimates
(1.5) to Minkowski estimates ${\rm Vol}(B_{r}(Z_{t_{0}}\cap B_{1}))\leq
C(n,\lambda,\Lambda)r$.

Note that here our estimate is local. Hence one can directly apply it to the
global solutions in $\mathbb{R}^{n}\times\mathbb{R}$. We say a solution
$u(x,t)$ of (1.1) in $\mathbb{R}^{n}\times(-4,0]$ satisfies the backward
uniqueness property if the following holds:

(BUP): If $u(\cdot,0)\equiv 0$ then $u\equiv 0$ in
$\mathbb{R}^{n}\times(-4,0].$

As a direct consequence, Theorem 1.4 gives the following theorem.

######  Theorem 1.10.

Let $u$ be a nonzero solution of (1.1) in $\mathbb{R}^{n}\times(-4,0]$
satisfying (1.2) and (1.3) and (BUP). Then for any $t>-4$ and any
$x\in\mathbb{R}^{n}$ we have

$\displaystyle\mathcal{H}^{n-1}(Z_{t}\cap B_{1}(x))<\infty.$ (1.11)

The backward uniqueness property for parabolic equations has been extensively
investigated, see for instance [15, 18, 24, 37, 39, 53, 55]. From [55], we
know that the BUP holds if $|\nabla_{x}a^{ij}(x,t)|\leq\frac{\lambda}{1+|x|}$
and $|\partial_{t}a^{ij}(x,t)|\leq\lambda$ and $|u(x,t)|\leq Be^{A|x|^{2}}$
for some $A,B>0$. Therefore, Theorem 1.10 recovers the main theorem of
Angenent [3] and generalizes it to higher dimensions. Moreover, while Angenent
[3] proved the number of zero point set is locally finite, Theorem 1.10 gives
an explicit and effective control of the number of zero points.

Actually, our method also works for equations with Hölder continuous leading
coefficients. For any fixed $0<\alpha<1$, assume $a^{ij}$ satisfies Hölder
condition with respect to parabolic distance

$\displaystyle|a^{ij}(x,t)-a^{ij}(y,s)|\leq\lambda\left(||x-y||^{2}+|s-t|\right)^{\alpha/2}.$
(1.12)

Note that in this case the unique continuation property fails, the following
doubling assumption at time $t=0$ must be imposed to exclude the bad behaviors
that the solution vanishes in some open ball. See [30] for similar growth
assumption and see [33] for the elliptic case.

$\displaystyle\sup_{Q_{2r}(x,0)\subset
Q_{2}}\frac{\fint_{Q_{2r}(x,0)}u^{2}}{\fint_{Q_{r}(x,0)}u^{2}}\leq\Lambda,$
(1.13)

where $Q_{r}(x,0)=\\{(y,s):|x-y|<r,-r^{2}<s\leq 0\\}.$

######  Theorem 1.14.

Let $u$ be a solution of (1.1) in $Q_{2}$ satisfying (1.2) (1.12) and (1.13).
Assume $u(\cdot,0)$ is not identically zero, then we have the following
Hausdorff measure estimate

$\displaystyle\mathcal{H}^{n-1}(Z_{0}\cap
B_{1})<C(n,\alpha,\lambda,\Lambda)<\infty,$ (1.15)

where $Z_{0}=\\{x\in B_{2}:u(x,0)=0\\}$.

######  Remark 1.16.

Recall that Han-Lin [30] proved the space-time estimate for nodal set by
assuming the doubling assumption for all subballs, while here we only require
the doubling at time $t=0$ to get the measure estimate at this time slice.

The main body of this paper will be devoted to proving the more general Hölder
case, i.e. Theorem 1.14. Then Theorem 1.4 is a direct consequence once the
doubling property 1.13 is established under the Lipschitz assumption.

###  1.2. Dimension Monotonicity Theorem

Next we study the evolutionary behavior of nodal sets over time. According to
the work of Angenent [3], when $n=1$ and additionally $a_{t}$ is bounded, the
number of zero points of $u(x,t)$, which is a solution to
$u_{t}=a(x,t)u_{xx}+b(x,t)u_{x}+c(x,t)u$ with Dirichlet boundary condition, is
non-increasing in $t$. This result suggests that in principle the complexity
of solutions to parabolic equations decays over time, naturally raising
interest in the behavior of such solutions in higher dimensions. However, as
far as we know, nothing has been established for general dimensions yet.

In this subsection, we continue the study of evolutionary properties of
parabolic solutions in higher dimensions. Unlike the case $n=1$ in [3], in
general, one cannot expect the Hausdorff measure of the nodal set to be non-
increasing. We construct examples demonstrating that the Hausdorff measure of
the nodal sets can increase over time, even for solutions to heat equations
when $n\geq 2$.

######  Example 1.17.

Let $f(s)=\frac{3}{2}s-\frac{1}{2}s^{2}-1$ when $0\leq s\leq 3/2$ and
$f(s)\equiv 1/8$ when $s\geq 3/2$. Then $f$ is $C^{1}$-smooth and $s=1$ is the
only zero point. Consider the heat solution

$u(x,t)=\int_{\mathbb{R}^{2}}f(|y|^{2})\rho_{t}(x,y)dy.$

It is easy to check that $u(x,t)=u(x^{\prime},t)$ for $|x|=|x^{\prime}|$.
Denote $f(r^{2},t)=u(x,t)$ with $|x|=r$. Then $f(s,0)=f(s)$, noting that
$f(1)=0$ and $f^{\prime}(1)=1/2>0$, $\Delta f(|y|^{2})|_{\\{|y|=1\\}}=-2<0$,
we can see that $\partial_{s}f(1,t)>1/4>0$, $\partial_{t}f(1,t)<-1<0$ for $t$
sufficiently small. This implies $f(1,t)<0$ and $f(r^{2}_{t},t)=0$ for some
$r_{t}>1$. Hence the nodal set $Z_{t}=\\{x:|x|=r_{t}\\}$ and
$\mathcal{H}^{1}(Z_{t})=2\pi r_{t}$ is increasing near $0$.

######  Example 1.18.

On the other hand, one can also construct solution with Dirichlet boundary.
Let $f(s)=\frac{3}{2}s-\frac{1}{2}s^{2}-1$ when $0\leq s\leq 3/2$ and
$f(s)\equiv 1/8$ when $3/2\leq s\leq 2$ and $f(s)=\frac{1}{24}(s^{2}-8s+15)$
when $2\leq s\leq 3$. By smoothing $f$ near $s=3/2$ and $s=2$, we get a smooth
$\tilde{f}$. Consider the solution $u(x,t)$ of $(\partial_{t}-\Delta)u=0$ in
$B_{3}(0^{2})\times[0,1]$ with Dirichlet boundary $u=0$ on $\partial
B_{3}\times[0,1]$ and $u(x,0)=\tilde{f}(|x|^{2})$. One can check as above that
near $t=0$ the Hausdorff measure of $Z_{t}=\\{x:u(x,t)=0,|x|<3\\}$ is
increasing. Actually, one can use Dirichlet heat kernel estimate to show that
$u>0$ near the boundary ([44]). Hence we only need to consider the situation
near $|x|=1$.

These examples reveal that the evolutionary behavior of parabolic solutions is
too complicated to expect any monotonicity in the size of nodal sets in higher
dimensions. However, while measure monotonicity appears unrealistic, dimension
monotonicity is a more practical expectation. In our second result, we provide
an affirmative answer to this dimension monotonicity. Similar to the case when
$n=1$, if we assume $a^{ij}$ is Lipschitz in the time direction, i.e.,
$|a^{ij}(x,t)-a^{ij}(x,s)|\leq\lambda|s-t|$, then we can show that the
Hausdorff dimension of nodal sets is non-increasing over time.

######  Theorem 1.19.

Let $u$ be a nonzero solution of (1.1) in $Q_{2}$ with Dirichlet boundary
condition $u=0$ on $\\{|x|=2,-4<t\leq 0\\}$. Assume that the coefficients
satisfy (1.2) and (1.3) and $a^{ij}$ is Lipschitz in time direction. Then the
Hausdorff dimension of the nodal set $Z_{t}:=\\{x:u(x,t)=0,|x|<2\\}$ is non-
increasing,

$\displaystyle\dim Z_{t}\leq\dim Z_{s}\leq n-1$ (1.20)

for any $0\geq t\geq s>-4$.

######  Remark 1.21.

It should be noted that the Lipschitz assumption in the time direction is
sharp, as it ensures the unique continuation property for backward uniqueness
of parabolic equations. According to the counter-examples in [45, 46], if
$a^{ij}$ is only Hölder continuous in time, i.e.
$|a^{ij}(x,t)-a^{ij}(x,s)|\leq\lambda|s-t|^{\alpha}$ for $\alpha\in(0,1)$,
then there exists solution $u$ such that $u(\cdot,T)\equiv 0$ while
$u(\cdot,t)$ is not for $t<T$.

This is the first monotonicty property over time for nodal sets of higher-
dimensional parabolic solutions. Similar to the applications in the $n=1$ case
to curve shortening flows or mean curvature flows, as seen in [2, 4, 5, 12,
16, 17, 26], more geometric applications are expected. Furthermore, as noted
in Remark 1.7, Theorem 1.19 holds in the more general inequality setting. This
broadens its applicability to more nonlinear equations.

###  1.3. Outline

In Section 2, we study standard heat equations, recall Poon’s monotonicity,
and prove some of its consequences. We also establish the quantitative
uniqueness for tangent maps of caloric functions and introduce the concept of
quantitative stratification. In Section 3, we examine general parabolic
equations, introducing the localized frequency function and proving the almost
monotonicity of this function. In Section 4, we prove the quantitative
uniqueness of tangent maps for general parabolic solutions. In Section 5, we
present the cone-splitting theorem for general parabolic solutions. The
quantitative uniqueness and the cone-splitting theorem are then utilized to
establish the neck region theory in Section 6. Finally, in Section 7, we
conclude with the proof of the main theorems.

Acknowledgements The authors would like to thank Prof. Toby Colding and Prof.
Gang Tian for their constant support and their interests of this work. The
authors would like to thank Prof. Kening Lu for bringing the reference [3] to
our attention and raising the question about dimension monotonicity during a
conference at Qingdao in 2023. W. Jiang was supported by National Key Research
and Development Program of China (No. 2022YFA1005501), National Natural
Science Foundation of China (Grant No. 12125105 and 12071425) and the
Fundamental Research Funds for the Central Universities K20220218. Y. Huang
was partially supported by NSF DMS Grant 2104349.

##  2\. Background and Heat Equations

In this section, we will recall and prove some results which will be used in
our proofs. First we will recall some results of caloric functions.

###  2.1. Caloric Function and monotone frequency

In this subsection, we consider the solution to the standard heat equation in
$\mathbb{R}^{n}\times[-T,0]$,

$\partial_{t}u=\Delta u.$

Noting that if there is no any growth control the solution may not be unique
with given initial data. Let us assume some appropriate growth assumption on
$u$, say polynomial growth. First we recall the monotonicity formula for $u$
as in Poon [53]. Fix some point $(x_{0},t_{0})$. Consider

$\begin{split}E_{x_{0},t_{0}}(r)&=2r^{2}\int_{t=t_{0}-r^{2}}|\nabla
u|^{2}G_{x_{0},t_{0}}\\\
H_{x_{0},t_{0}}(r)&=\int_{t=t_{0}-r^{2}}u^{2}G_{x_{0},t_{0}}\end{split}$

where

$G_{x_{0},t_{0}}(x,t)\equiv(4\pi(t_{0}-t))^{-\frac{n}{2}}\text{e}^{-\frac{|x-x_{0}|^{2}}{4(t_{0}-t)}}.$

We define the frequency for $r\geq 0$ to be

$N_{x_{0},t_{0}}(r)=\frac{E_{x_{0},t_{0}}(r)}{H_{x_{0},t_{0}}(r)}.$ (2.1)

It turns out that $N$ is non-increasing in $r$. For the sake of completeness,
here we include the proof, which follows Poon [53].

######  Theorem 2.2 (Poon).

$N_{x_{0},t_{0}}^{\prime}(r)\geq 0$ for any $(x_{0},t_{0})$ and any $r>0$.

###### Proof.

Observe that

$\nabla G_{x_{0},t_{0}}=\frac{x-x_{0}}{2(t-t_{0})}G_{x_{0},t_{0}}\quad\text{
and }\quad\partial_{t}G_{x_{0},t_{0}}=-\Delta G_{x_{0},t_{0}}.$

For simplicity we omit the subscripts. Then integration by parts implies that

$E(r)=-2r^{2}\int_{t=t_{0}-r^{2}}u(\Delta u+\nabla
u\cdot\frac{x-x_{0}}{2(t-t_{0})})G_{x_{0},t_{0}}=-2r^{2}\int_{t=t_{0}-r^{2}}u(u_{t}+\nabla
u\cdot\frac{x-x_{0}}{2(t-t_{0})})G_{x_{0},t_{0}}$

$H^{\prime}(r)=-2r\int_{t=t_{0}-r^{2}}2uu_{t}G_{x_{0},t_{0}}-u^{2}\Delta
G_{x_{0},t_{0}}=-4r\int_{t=t_{0}-r^{2}}u(u_{t}+\nabla
u\cdot\frac{x-x_{0}}{2(t-t_{0})})G_{x_{0},t_{0}}=\frac{2E(r)}{r}.$

$\begin{split}E^{\prime}(r)&=4r\int_{t=t_{0}-r^{2}}|\nabla
u|^{2}G_{x_{0},t_{0}}-4r^{3}\int_{t=t_{0}-r^{2}}2\nabla u\cdot\nabla
u_{t}G_{x_{0},t_{0}}+4r^{3}\int_{t=t_{0}-r^{2}}|\nabla u|^{2}\Delta
G_{x_{0},t_{0}}\\\ &=4r\int_{t=t_{0}-r^{2}}|\nabla
u|^{2}G_{x_{0},t_{0}}-8r^{3}\int_{t=t_{0}-r^{2}}(\nabla u\cdot\nabla
u_{t}+\nabla^{2}u(\nabla u,\frac{x-x_{0}}{2(t-t_{0})}))G_{x_{0},t_{0}}\\\
&=-8r^{3}\int_{t=t_{0}-r^{2}}(\nabla u\cdot\nabla(u_{t}+\nabla
u\cdot\frac{x-x_{0}}{2(t-t_{0})})G_{x_{0},t_{0}}\\\
&=8r^{3}\int_{t=t_{0}-r^{2}}(u_{t}+\nabla
u\cdot\frac{x-x_{0}}{2(t-t_{0})})^{2}G_{x_{0},t_{0}}.\end{split}$

Therefore,

$\begin{split}&\quad E^{\prime}(r)H(r)-E(r)H^{\prime}(r)\\\
&=8r^{3}\Big{(}\int_{t=t_{0}-r^{2}}u^{2}G_{x_{0},t_{0}}\Big{)}\Big{(}\int_{t=t_{0}-r^{2}}\Big{(}u_{t}+\nabla
u\cdot\frac{x-x_{0}}{2(t-t_{0})}\Big{)}^{2}G_{x_{0},t_{0}}\Big{)}-8r^{3}\Big{(}\int_{t=t_{0}-r^{2}}u(u_{t}+\nabla
u\cdot\frac{x-x_{0}}{2(t-t_{0})})G_{x_{0},t_{0}}\Big{)}^{2}.\end{split}$

Hence, by Cauchy-Schwarz inequality we have $N^{\prime}(r)\geq 0$. ∎

Since $H^{\prime}(r)=2E(r)/r$, we have $H^{\prime}(r)/H(r)=2N(r)/r$.
Integrating from $r_{1}$ to $r_{2}$ we have

$\log\frac{H(r_{2})}{H(r_{1})}=2\int_{r_{1}}^{r_{2}}\frac{N(s)}{s}ds.$ (2.3)

If we define the doubling index $D(r)$ as

$D(r)\equiv\log_{4}\frac{H(2r)}{H(r)}.$

Then from (2.3) and the monotonicity of $N(r)$ we can establish the
equivalence between frequency and doubling index

$N(r)\leq D(r)\leq N(2r).$ (2.4)

This is an important global doubling control with respect to frequency. It
could be used to prove the unique continuation for heat equation.

We say $u$ is a homogeneous caloric polynomial of order $k$ centered at
$(x_{0},t_{0})$, if $u$ satisfies the heat equation and $u(\lambda
x+x_{0},\lambda^{2}t+t_{0})=\lambda^{k}u(x+x_{0},t+t_{0})$ for any $t<0$ and
$\lambda>0$.

######  Lemma 2.5.

$N(r)$ is constant if and only if $u$ is a homogeneous caloric polynomial of
order $N(1)$.

###### Proof.

We may assume $(x_{0},t_{0})=(0,0)$ and $u(0,0)=0$. By Cauchy-Schwarz
inequality, $N^{\prime}(r)\equiv 0$ if and only if there exists some $k$ such
that

$2u_{t}\cdot t+\nabla u\cdot x=k\cdot u.$ (2.6)

First we assume (2.6) and prove that $u$ is a homogeneous caloric polynomial
of order $k$. Fix $(x,t)$. Set $F(\lambda)=u(\lambda
x,\lambda^{2}t)-\lambda^{k}u(x,t)$. Note that by (2.6) we have

$\begin{split}F^{\prime}(\lambda)&=\nabla u(\lambda x,\lambda^{2}t)\cdot
x+u_{t}(\lambda x,\lambda^{2}t)\cdot(2\lambda t)-k\lambda^{k-1}u(x,t)\\\
&=\lambda^{-1}(\nabla u(\lambda x,\lambda^{2}t)\cdot(\lambda x)+u_{t}(\lambda
x,\lambda^{2}t)\cdot(2\lambda^{2}t))-k\lambda^{k-1}u(x,t)\\\
&=\lambda^{-1}ku(\lambda x,\lambda^{2}t)-k\lambda^{k-1}u(x,t)\\\
&=\lambda^{-1}kF(\lambda).\end{split}$

Since $F(1)=0$, this implies that $F(\lambda)\equiv 0$. The other direction is
trivial by chain rule. The proof is finished.

The following Liouville type theorem for ancient caloric function is standard.
∎

######  Lemma 2.7.

Let $u$ be a caloric function with polynomial growth of order $\Lambda$, i.e.
$|u(x,t)|\leq C(1+|x|+|t|^{1/2})^{\Lambda}$. Then $u$ is a polynomial of order
$d\leq\Lambda$.

###### Proof.

For any $(x,t)$, denote $R=\left(|x|^{2}+|t|\right)^{1/2}$. For any $r\geq
2R$, by parabolic estimate we get for any $\ell+2k=d$

$\displaystyle|D_{x}^{\ell}D_{t}^{k}u(x,t)|\leq
C(n,d)r^{-\ell-2k}\left(\fint_{Q_{r}}|u(y,s)|^{2}dyds\right)^{1/2}\leq
Cr^{-\ell-2k+\Lambda}.$ (2.8)

If $\ell+2k>\Lambda$, letting $r\to\infty$ we have
$D_{x}^{\ell}D_{t}^{k}u(x,t)=0$. This implies $u$ is a polynomial with order
at most $\Lambda$. ∎

As a corollary, we notice that the frequency can only be pinched near the
integers.

######  Lemma 2.9.

Let $u$ be a caloric function with $N(1)\leq\Lambda$ and with polynomial
growth of order $\Lambda$, i.e. $|u(x,t)|\leq C(1+|x|+|t|^{1/2})^{\Lambda}$.
For any $\epsilon>0$, there exists some $\delta(\epsilon,\Lambda)>0$ such that
if $N(1)-N(1/10)\leq\delta$, then there exists some integer $d$ such that

  1. (1)

$|N(r)-d|\leq\epsilon$ for any $r\in[1/10,1]$.

  2. (2)

There exists some caloric homogeneous polynomial $P$ of order $d$ such that
$|u-P|_{K_{\epsilon^{-1}}}\leq\epsilon$, where $K_{r}=\\{(x,t):|x|\leq
r,1/10+1/r\leq t\leq 1\\}$.

###### Proof.

This is proved by contradiction argument. Let $u_{i}$ be a sequence of caloric
functions with $N^{i}(1)\leq\Lambda$ and $|u_{i}(x,t)|\leq
C_{i}(1+|x|+|t|^{1/2})^{\Lambda}$. And $N^{i}(1)-N^{i}(1/10)\leq 1/i$ with
dist$(N(r),\mathbf{Z})\geq\epsilon_{0}$ for some $r\in[1/10,1]$. We may
normalize it so that $H^{i}(1)=\int_{t=-1}u_{i}^{2}G_{0,0}=1$ for all $i$. By
(2.3) and $N^{i}(1)\leq\Lambda$, we have $H^{i}(1/10)\geq 10^{-\Lambda}$.
Also, we can take $C_{i}=C$ for some fixed $C$ after the normalization.

Consider $K_{R}=\\{(x,t):|x|\leq R,1/10+1/R\leq t\leq 1\\}$. Since
$G_{0,0}(x,t)\geq c(R)$ on $K_{R}$, we have $\int_{K_{R}}u_{i}^{2}\leq C(R)$.
Hence for each $R$ we can extract a subsequence of $u_{i}$ that converges to
$u_{R}$ in $C^{\infty}(K_{R/2})$. By diagonal argument, there exists some
caloric function $u_{\infty}$ such that $u_{i}$ converges to $u_{\infty}$ in
$C^{\infty}_{loc}(K_{\infty})$ by passing to a subsequence. Since $u_{i}$ has
uniform polynomial growth, by dominated convergence theorem we have
$H^{i}(r)\to H^{\infty}(r)$ and $E^{i}(r)\to E^{\infty}(r)$ for any
$r\in[1/10,1]$. Since $H^{i}(r)\geq 10^{-\Lambda}$ for any $r\in[1/10,1]$, we
have

$N^{i}(r)=\frac{E^{i}(r)}{H^{i}(r)}\to N^{\infty}(r).$

Hence by assumption and monotonicity of $N$, we have
$N^{\infty}(1)=N^{\infty}(1/10)$. This implies that $u_{\infty}$ is indeed a
homogeneous caloric polynomial and thus $N^{\infty}=d$ for some integer $d$.
Contradiction arises. ∎

The following Lemma says that away from integers the frequency drops at a
definite rate.

######  Lemma 2.10.

Let $u$ be a caloric polynomial with $N(1)\leq\Lambda$. For any $\epsilon>0$,
there exists some $\delta(\epsilon,\Lambda)>0$ such that if $N(1)\leq
d-\epsilon$, then $N(\delta)\leq d-1+\epsilon$.

###### Proof.

When $\epsilon\geq 1/2$, the conclusion trivially holds by monotonicity of
$N$. Assume $\epsilon<1/2$. Let $\delta_{0}(\epsilon/10,\Lambda)$ be as in
Lemma 2.9.

If $N(1/10)\geq N(1)-\delta_{0}$, then we have
dist$(N(1),\mathbf{Z})\leq\epsilon/10$ by Lemma 2.9. This implies that
$N(1)\leq d-1+\epsilon/10$ and thus $N(1/10)\leq N(1)\leq d-1+\epsilon/10$ by
monotonicity.

Now suppose $N(1/10)\leq N(1)-\delta_{0}$. Applying Lemma 2.9 to the scale
$1/10$, we can conclude that $N(10^{-2})\leq N(1/10)-\delta_{0}$ by the same
arguments. Therefore, after iterating by at most $K$ times with
$K\leq\delta_{0}^{-1}$, we can conclude that $N(10^{-K})\leq
N(1)-K\delta_{0}\leq d-1+\epsilon$. ∎

######  Remark 2.11.

Instead of requiring $u$ to be a polynomial, actually we only need some minor
growth assumption on $u$ for Lemma 2.9 and Lemma 2.10 to hold.

######  Remark 2.12.

It is possible to prove that $\epsilon$ is independent of $\Lambda$ by
delicate analysis like what [48] did for harmonic function.

The following Lemma proves the orthogonality of homogeneous caloric polynomial
of different orders(see also [14]).

######  Lemma 2.13.

Suppose $P$, $Q$ be two homogeneous caloric polynomials with order $d_{1}\neq
d_{2}$. Then we have

$\int_{t=t_{0}}PQG_{0,0}=\int_{t=t_{0}}(\nabla P\cdot\nabla Q)G_{0,0}=0.$

###### Proof.

Note that

$d_{1}\int_{t=t_{0}}PQG_{0,0}=\int_{t=t_{0}}(2\partial_{t}P\cdot t+\nabla
P\cdot x)QG_{0,0}=\int_{t=t_{0}}(2t_{0}\Delta P+\nabla P\cdot x)QG_{0,0}.$

Integration by parts gives

$2t_{0}\int_{t=t_{0}}\Delta PQG_{0,0}=-2t_{0}\int_{t=t_{0}}(\nabla
P\cdot\nabla Q)G_{0,0}-\int_{t=t_{0}}(\nabla P\cdot x)QG_{0,0}.$

Hence we have

$d_{1}\int_{t=t_{0}}PQG_{0,0}=-2t_{0}\int_{t=t_{0}}(\nabla P\cdot\nabla
Q)G_{0,0}.$

On the other hand, by similar arguments we have

$d_{2}\int_{t=t_{0}}PQG_{0,0}=-2t_{0}\int_{t=t_{0}}(\nabla P\cdot\nabla
Q)G_{0,0}.$

Since $d_{1}\neq d_{2}$, the result follows. ∎

######  Remark 2.14.

Let $P(x,t)$ be a homogeneous caloric polynomial of order $d$. Then $P(x,-1)$
is an eigenfunction of the Ornstein-Uhlenbeck operator $\mathcal{L}(u)=\Delta
u-\frac{1}{2}\langle\nabla u,x\rangle$ with eigenvalue $d/2$. Note that
$\mathcal{L}$ is self-adjoint with respect to the weighted volume $\langle
f,g\rangle=\int_{\mathbb{R}^{n}}(fg)\text{e}^{-\frac{|x|^{2}}{4}}$. If $P,Q$
are homogeneous caloric polynomials with different orders $d_{1},d_{2}$, we
have $\langle P(x,-1),Q(x,-1)\rangle\equiv 0$ as they are eigenfunctions of
$\mathcal{L}$ with different eigenvalues.

Next we prove that the frequency of a polynomial is bounded by its order. This
is a direct consequence of the previous orthogonality lemma.

######  Corollary 2.15.

Let $P$ be a caloric polynomial of order $d$. Then we have $N^{P}(r)\leq d$
for all $r>0$.

###### Proof.

Write $P=\sum_{i=0}^{d}P_{i}$, where each $P_{i}$ is a homogeneous caloric
polynomial of order $i$. Then by previous Lemma 2.13

$\begin{split}\int_{t=-r^{2}}|\nabla
P|^{2}G_{0,0}&=\int_{t=-r^{2}}\langle\sum_{i}\nabla P_{i},\sum_{j}\nabla
P_{j}\rangle G_{0,0}=\sum_{i}\int_{t=-r^{2}}|\nabla P_{i}|^{2}G_{0,0}\\\
\int_{t=-r^{2}}P^{2}G_{0,0}&=\int_{t=-r^{2}}(\sum
P_{i})^{2}G_{0,0}=\sum_{i}\int_{t=-r^{2}}P_{i}^{2}G_{0,0}.\end{split}$

By Lemma 2.5 or the calculation in Lemma 2.13 we have
$2r^{2}\int_{t=-r^{2}}|\nabla
P_{i}|^{2}G_{0,0}=i\int_{t=-r^{2}}P_{i}^{2}G_{0,0}$ for any $i\leq d$. This
implies that

$2r^{2}\int_{t=-r^{2}}|\nabla P|^{2}G_{0,0}\leq d\int_{t=-r^{2}}P^{2}G_{0,0}.$

Thus $N^{P}(r)\leq d$. ∎

######  Corollary 2.16.

Let $P$ be a caloric polynomial of order $d$. For any $0<\epsilon<1$, if
$N^{P}(1)\leq\epsilon$ then for any $R>1$

$\displaystyle\sup_{Q_{R}}|P(x,t)-a_{0}|\leq C(n,d,R)|a_{0}|\sqrt{\epsilon}$
(2.17)

for some nonzero constant $a_{0}$.

###### Proof.

Write $P=\sum_{i=0}^{d}a_{i}P_{i}$, where each $P_{i}$ is a homogeneous
caloric polynomial of order $i$ with unit
$\int_{t=-1}|P_{i}(y,s)|^{2}G_{0,0}dyds$. By previous Lemma 2.13, we can
compute that

$\displaystyle\int_{t=-1}|\nabla P|^{2}G_{0,0}$
$\displaystyle=\int_{t=-1}\langle\sum_{i}a_{i}\nabla P_{i},\sum_{j}a_{j}\nabla
P_{j}\rangle G_{0,0}=\sum_{i=1}^{d}\int_{t=-1}a_{i}^{2}|\nabla
P_{i}|^{2}G_{0,0}=\sum_{i=1}^{d}\int_{t=-1}a_{i}^{2}|\nabla
P_{i}|^{2}G_{0,0}.$ (2.18)

Noting that $2\int_{t=-1}|\nabla
P_{i}|^{2}G_{0,0}=i\int_{t=-1}P_{i}^{2}G_{0,0}=i$, we get

$\displaystyle N^{P}(1)=\frac{2\int_{t=-1}|\nabla
P|^{2}G_{0,0}}{\int_{t=-1}|P|^{2}G_{0,0}}=\frac{2\sum_{i=1}^{d}a_{i}^{2}i}{\sum_{i=0}^{d}a_{i}^{2}}\leq\epsilon.$
(2.19)

This implies

$\displaystyle\sum_{i=1}^{d}a_{i}^{2}\leq\frac{\epsilon}{2-\epsilon}a_{0}^{2}.$
(2.20)

This gives the deserved estimate. ∎

###  2.2. Uniqueness of tangent maps of heat equations

In this subsection we prove an effective estimate for tangent map of heat
equation. Let $h$ define on $Q_{2}=B_{2}\times[-4,0]$. For any
$(x_{0},t_{0})\in Q_{1}$, define a localized frequency for $h$ by

$N^{h}_{x_{0},t_{0}}(r)=\frac{E_{x_{0},t_{0}}^{h}(r)}{H^{h}_{x_{0},t_{0}}(r)}=\frac{2r^{2}\int_{Q_{2}\cap\\{t=t_{0}-r^{2}\\}}|\nabla
u|^{2}G_{x_{0},t_{0}}}{\int_{Q_{2}\cap\\{t=t_{0}-r^{2}\\}}u^{2}G_{x_{0},t_{0}}}.$

See a detailed discussion of localized frequency in 3.9 in the next section.

######  Theorem 2.21.

Let $h$ be a solution of heat equation on $Q_{2}$ with doubling assumption
$\sup_{r_{2}\leq r\leq r_{1}}\text{\rm
log}_{4}\frac{\fint_{Q_{2r}(0,0)}h^{2}}{\fint_{Q_{r}(0,0)}h^{2}}\leq\Lambda$ .
For any $\epsilon>0$ there exist $\delta_{0}(n,\Lambda,\epsilon)$ such that if
the localized frequency $|N_{0,0}^{h}(r)-d|\leq\delta\leq\delta_{0}$ for any
$r_{2}\leq r\leq r_{1}\leq 1$ and for some integer $d$. Then there exists a
unique homogeneous caloric polynomial $P_{d}$ of order $d$ such that for any
$r_{2}\leq r\leq\delta r_{1}$ that

$\displaystyle\sup_{Q_{1}}|h_{0,0;r}-P_{d}|\leq\epsilon$ (2.22)

where
$h_{0,0;r}(x,t)=\frac{h(rx,r^{2}t)}{\left(\fint_{Q_{1}}|h(rx,r^{2}t)|^{2}dxdt\right)^{1/2}}$.

###### Proof.

Since $h$ is solution of heat equation, then $h$ is smooth. By Taylor
expansion at $(0,0)$, we can get (see also Lemma 1.2 of Han [27]) for all
$(x,t)\in Q_{1}$

$\displaystyle
h(x,t)=P_{0}(x,t)+P_{1}(x,t)+P_{2}(x,t)+\cdots+P_{d}(x,t)+R(x,t)$ (2.23)

where each $P_{k}$ is a caloric homogeneous polynomial of order $k$ and
$R(x,t)$ satisfies for $(x,t)\in Q_{r}$ with $r\leq 1$ that

$\displaystyle|R(x,t)|\leq
C(n,d)r^{-d-1}|(x,t)|^{d+1}\left(\fint_{Q_{2r}}|h(y,t)|^{2}dydt\right)^{1/2}.$
(2.24)

Furthermore, the coefficients of each $P_{i}$ are uniformly bounded by the
$L^{2}$-norm of $h$ over $Q_{1}$. Since $R(x,t)$ is also a solution of heat
equation, by parabolic estimates, we get for all $(x,t)\in Q_{r}$ with $r\leq
1$ and $\ell+2k\leq d$

$\displaystyle|D_{x}^{\ell}D_{t}^{k}R(x,t)|\leq
C(n,d)r^{-d-1+\ell+2k}|(x,t)|^{d+1-\ell-2k}\left(\fint_{Q_{2r}}|h(y,t)|^{2}dydt\right)^{1/2}.$
(2.25)

We will show that the caloric polynomial $P_{d}$ from (2.23) is the desired
polynomial up to normalization with unit $L^{2}$-norm on $Q_{1}$. To see this,
it suffices to show that for any $\epsilon>0$ if
$\delta\leq\delta(n,\Lambda,\epsilon)$ and $r_{2}\leq r\leq\delta r_{1}$ we
have

$\displaystyle\fint_{Q_{r}}|\sum_{i=0}^{d-1}P_{i}|^{2}\leq\epsilon\fint_{Q_{r}}|h(y,t)|^{2}dydt$
(2.26)

and

$\displaystyle\fint_{Q_{r}}|R(y,t)|^{2}dydt\leq\epsilon\fint_{Q_{r}}|h(y,t)|^{2}dydt$
(2.27)

Let us first prove (2.27). Assume there exists $\epsilon_{0}>0$ and a sequence
of $r_{2,i}\leq r_{i}\leq\delta_{i}r_{1,i}$ and $\delta_{i}\to 0$ and heat
solution $h_{i}(x,t)$ and $R_{i}(x,t)$ satisfying the assumption of the
theorem and (2.23), however,

$\displaystyle\fint_{Q_{r_{i}}}|R_{i}(y,t)|^{2}dydt\geq\epsilon_{0}\fint_{Q_{r_{i}}}|h_{i}(y,t)|^{2}dydt$
(2.28)

Denote
$\hat{h}_{i}(y,s)=\frac{h_{i}(r_{i}y,r_{i}^{2}s)}{\left(\fint_{Q_{1}}|h_{i}(r_{i}y,r_{i}^{2}s)|^{2}dyds\right)^{1/2}}$.
Then

$\displaystyle\fint_{Q_{1}}|\hat{R}_{i}(y,t)|^{2}dydt\geq\epsilon_{0}\fint_{Q_{1}}|\hat{h}_{i}(y,t)|^{2}dydt$
(2.29)

By apriori estimate for parabolic equation and the polynomial growth
assumption of $h_{i}$, we know that $\hat{h}_{i}$ and $\hat{R}_{i}$ will
converge smoothly to ancient solutions of heat equation $\hat{h}_{\infty}$ and
$\hat{R}_{\infty}$ on $\mathbb{R}^{n}$. From the expansion (2.23), we also get

$\displaystyle\hat{h}_{\infty}=\sum_{k=0}^{d}\hat{P}_{k,\infty}+\hat{R}_{\infty}$
(2.30)

where $\hat{P}_{k,\infty}$ is the limit of $\hat{P}_{k,i}$ with $i\to\infty$
(since each $\hat{P}_{k,i}$ has uniformly bounded coefficients which must
converge up to a subsequence). On the other hand, by smooth convergence one
can easily show that the localized frequency converges to the standard
frequency of $\hat{h}_{\infty}$ and equals to $d$ (see also Lemma 3.23 ), then
$\hat{h}_{\infty}$ is a caloric polynomial of order $d$. We will show that
$\hat{R}_{\infty}$ is zero which will lead to a contradiction from the limit
of (2.29). Actually, if $\hat{R}_{\infty}$ is not zero, since
$\hat{h}_{\infty}$ is a homogeneous polynomial of order $d$,
$\hat{R}_{\infty}$ must be a polynomial of order at most $d$ (may not be
homogeneous). Assume $\hat{R}_{\infty}$ contains a non zero term
$x^{\ell}t^{k}$ with $\ell+2k\leq d$. By the smooth convergence of
$\hat{R}_{i}$ we have that

$\displaystyle D_{x}^{\ell}D_{t}^{k}\hat{R}_{i}(0,0)\to
D_{x}^{\ell}D_{t}^{k}\hat{R}_{\infty}(0,0)\neq 0.$ (2.31)

However, by (2.25) we have $D_{x}^{\ell}D_{t}^{k}\hat{R}_{i}(0,0)=0$ for each
$i$. This is a contradiction. Hence we get (2.27). Since
$\hat{R}_{\infty}\equiv 0$ and $\hat{h}_{\infty}$ is a homogeneous polynomial
of order $d$, this implies $\sum_{k=0}^{d-1}\hat{P}_{k,\infty}\equiv 0$. This
gives (2.26). Therefore, by (2.23), (2.26),(2.27), we get for all $r_{2}\leq
r\leq\delta r_{1}$ that

$\displaystyle\fint_{Q_{r}}|h(x,t)-P_{d}(x,t)|^{2}dxdt\leq
2\epsilon\fint_{Q_{r}}|h(x,t)|^{2}dxdt.$ (2.32)

This implies (2.22). We finish the proof. ∎

###  2.3. Maximum Principle

In this subsection, we recall the maximum principle of parabolic equation with
weak coefficients. Let us consider the following equation in
$Q_{2}:=\\{(x,t):|x|<2,-4\leq t\leq 0\\}\subset\mathbb{R}^{n}\times\mathbb{R}$

$\partial_{t}u=\partial_{i}(a^{ij}(x,t)\partial_{j}u)+b^{i}(x,t)\partial_{i}u+c(x,t)u,$
(2.33)

where the coefficients $a^{ij}$ are elliptic and assumed to be in
$C^{\alpha,\alpha/2}({Q_{2}})$ with $\alpha\in(0,1)$, and the coefficients
$b,c$ are bounded,

$(1+\lambda)^{-1}\delta^{ij}\leq
a^{ij}\leq(1+\lambda)\delta^{ij},\quad|b^{i}|,|c|\leq\lambda\text{ and
}|a^{ij}(x,t)-a^{ij}(y,s)|\leq\lambda\left(|x-y|^{2}+|s-t|\right)^{\alpha/2}.$
(2.34)

######  Theorem 2.35 (Maximum Principle).

Let $u$ be a Lipschitz function satisfying (2.33) (2.34) and the Dirichlet
boundary condition $u=0$ on $\\{|x|=2\\}\times[-4,0]$. Suppose $u\leq 0$ at
$t=-4$. Then $u<0$ for $t>-4$ or $u\equiv 0$ for $t\geq-4$.

###### Proof.

Let us consider the positive part of $u$ and define
$u_{+}(x,t)=\max\\{0,u(x,t)\\}$. Hence $u_{+}(x,-4)=0$. To prove the maximum
principle it suffices to show $u_{+}\leq 0$. Noting that $u(x,t)=0$ on
$\partial B_{2}(0)$, multiplying $u_{+}$ to (2.33) and integrating by parts we
get

$\displaystyle\int_{B_{2}}a^{ij}\partial_{j}u_{+}\partial_{i}u_{+}(x,t)dx-\lambda\int_{B_{2}}|\nabla
u_{+}||u_{+}|(x,t)dx-\lambda\int_{B_{2}}|u_{+}|^{2}(x,t)dx\leq-\frac{1}{2}\partial_{t}\int_{B_{2}}|u_{+}|^{2}(x,t)dx.$
(2.36)

By Cauchy inequality we get

$\displaystyle C_{1}(\lambda)\int_{B_{2}}|\nabla u_{+}|^{2}(x,t)dx-
C_{2}(\lambda)\int_{B_{2}}|u_{+}|^{2}(x,t)dx\leq-\partial_{t}\int_{B_{2}}|u_{+}|^{2}(x,t)dx$
(2.37)

Integrating $t$ from $-4$ to $s$ we get

$\displaystyle C_{1}(\lambda)\int_{-4}^{s}\int_{B_{2}}|\nabla
u_{+}|^{2}(x,t)dx-
C_{2}(\lambda)\int_{-4}^{s}\int_{B_{2}}|u_{+}|^{2}(x,t)dx\leq-\int_{B_{2}}|u_{+}|^{2}(x,s)dx+\int_{B_{2}}|u_{+}|^{2}(x,-4)dx.$
(2.38)

Hence for any $s\geq-4$ that

$\displaystyle\int_{B_{2}}|u_{+}|^{2}(x,s)dx\leq
C_{2}(\lambda)\int_{-4}^{s}\int_{B_{2}}|u_{+}|^{2}(x,t)dx.$ (2.39)

Integrating $s$ from $-4$ to $\ell$ we get

$\displaystyle\int_{-4}^{\ell}\int_{B_{2}}|u_{+}|^{2}(x,s)dxds\leq
C_{2}(\lambda)(\ell+4)\int_{-4}^{\ell}\int_{B_{2}}|u_{+}|^{2}(x,s)dxds.$
(2.40)

If $\ell+4<\frac{1}{2C_{2}(\lambda)}$, then we have
$\int_{-4}^{\ell}\int_{B_{2}}|u_{+}|^{2}(x,s)dxds=0$ and thus $u_{+}\equiv 0$
in $B_{2}\times[-4,\ell]$. We can now start at $t=\ell$ and then by induction
to deduce that $u_{+}(x,t)=0$ for all $t$. Hence we have proven the weak
maximum principle $u(x,t)\leq 0$ for all $t$. To see the strong maximum
principle, assume $u(x_{0},t)=0$ for some $t>-4$ and $|x_{0}|<2$, by parabolic
Harnack inequality (see [47, 51]) we can see that $u(x,-4)\equiv 0$ for all
$|x|<2$. In this case, using weak maximum principle to $-u$ and $u$ we get
$u\equiv 0$. ∎

###  2.4. Quantitative Stratification at a time slice

In this subsection, we will mainly introduce the quantitative stratification
which was first proposed in [8, 9].

First we define the symmetry of functions.

######  Definition 2.41.

Consider the continuous function $u:\mathbb{R}^{n}\to\mathbb{R}$.

  * (1)

$u$ is called $0$-symmetric with respect to $x_{0}$ if $u$ is a homogeneous
polynomial, i.e., $u(\lambda x+x_{0})=\lambda^{k}u(x+x_{0})$ for any
$\lambda>0$ and some $k$.

  * (1)

$u$ is called $k$-symmetric with respect to $x_{0}+V$ if $u$ is $0$-symmetric
with respect to $x_{0}$ and further symmetic with respect to some
$k$-dimensional subspace $V$, i.e. $u(x+y)=u(x)$ for any $x\in\mathbb{R}^{n}$
and $y\in V$.

Next we define the quantitative symmetry for $u$.

######  Definition 2.42.

Let $u:B_{2}\to\mathbb{R}$ be a continuous function. For any fixed $x\in
B_{1}$, we define $u$ is $(k,\eta,r,x)$-symmetric if there exists a
$k$-symmetric polynomial $P$ with $\fint_{B_{1}}|P|^{2}=1$ such that

$\sup_{B_{1}}|u_{x,r}(y)-P(y)|\leq\eta.$ (2.43)

where
$u_{x,r}(y)=\frac{u(x+ry)}{\left(\fint_{B_{1}}|u(x+ry)|^{2}\right)^{1/2}}$.

With this we can now give the definition of quantitative stratification

######  Definition 2.44.

Let $u:B_{2}\to\mathbb{R}$ be a continuous function. Given $k,\eta>0,r>0$, the
$(k,\eta)$-singular stratum is defined by

$\mathcal{S}^{k}_{\eta}(u):=\\{x\in B_{1}:u\text{ is not
}(k+1,\eta,s,x)\text{-symmetric for any }s\geq 0\\}.$ (2.45)

The above definitions will be used in during our proofs of the main theorems.

##  3\. General Parabolic Equations and Almost Monotone Frequency

In this section we will prove the almost monotonicity formula for frequency
defined for general parabolic solution $u$. We will consider the solution in a
bounded domain. Since the function is locally defined, we need to localized
the frequency (see Definition 3.9) introduced in [53] (see also (2.1)). We
will see that the localized frequency is approximating to the standard
frequency and the localized frequency is almost monotone (see Theorem 3.30)
which is good enough for the applications. We will also deduce some properties
of this localized frequency based on the almost monotonicity.

From now on we consider the following equation in
$Q_{2}:=\\{(x,t):|x|<2,-4<t\leq 0\\}\subset\mathbb{R}^{n}\times\mathbb{R}$

$\partial_{t}u=\partial_{i}(a^{ij}(x,t)\partial_{j}u)+b^{i}(x,t)\partial_{i}u+c(x,t)u,$
(3.1)

where the coefficients $a^{ij}$ are elliptic and assumed to be in
$C^{\alpha,\alpha/2}({Q_{2}})$ with $\alpha\in(0,1)$, and the coefficients
$b,c$ are bounded,

$(1+\lambda)^{-1}\delta^{ij}\leq
a^{ij}\leq(1+\lambda)\delta^{ij},\quad|b^{i}|,|c|\leq\lambda\text{ and
}|a^{ij}(x,t)-a^{ij}(y,s)|\leq\lambda\left(|x-y|^{2}+|s-t|\right)^{\alpha/2}$
(3.2)

We consider the following growth assumption at $t=0$ for solution $u$ (See
[30] [33]): there exists some positive constant $\Lambda$ such that

$\sup_{Q_{2r}(x,0)\subset
Q_{2}}\text{log}_{4}\frac{\fint_{Q_{2r}(x,0)}u^{2}}{\fint_{Q_{r}(x,0)}u^{2}}\leq\Lambda,$
(3.3)

where $Q_{r}(x,t)=\\{(y,s):|x-y|<r,t-r^{2}<s\leq t\\}.$

First we define the rescaled map of $u$.

######  Definition 3.4.

For $(x,t)\in Q_{1}$ and $\ell\leq 1$ we define

$u_{x,t;\ell}(y,s):=\frac{u(x+\ell
A_{x,t}(y),t+\ell^{2}s)}{\Big{(}\fint_{Q_{1}}(u(x+\ell
A_{x}(y),t+\ell^{2}s))^{2}dyds\Big{)}^{1/2}}.$ (3.5)

Here $A_{x,t}(y)=(\sqrt{a})^{ij}y_{i}e_{j}$ and $(\sqrt{a})^{ij}$ is the
square root of the coefficients matrix $a^{ij}(x,t)$.

######  Remark 3.6.

Write $\tilde{u}=u_{x,t;\ell}$. Then $\tilde{u}$ satisfies the following
rescaled equation

$\partial_{s}\tilde{u}=\partial_{i}(\tilde{a}^{ij}(y,s)\partial_{j}\tilde{u})+\tilde{b}^{i}(y,s)\partial_{i}\tilde{u}+\tilde{c}\tilde{u},$

where

$\begin{split}\tilde{a}(y,s)&=a(x,t)^{-1}\cdot a(x+rA_{x,t}(y),t+\ell^{2}s)\\\
\tilde{b}(y,s)&=\ell\cdot\sqrt{a}(x,t)^{-1}\cdot b(x+\ell
A_{x,t}(y),t+\ell^{2}s)\\\ \tilde{c}(y,s)&=\ell^{2}\cdot c(x+\ell
A_{x,t}(y),t+\ell^{2}s)\end{split}$

Hence we have $\tilde{a}(0,0)=(\delta^{ij})$ and moreover

$|\tilde{a}^{ij}(Y)-\tilde{a}^{ij}(Z)|\leq
C(\lambda,\alpha)\ell^{\alpha}d(Y,Z)^{\alpha},(1+C(\lambda,\alpha)\ell^{\alpha})^{-1}\delta^{ij}\leq\tilde{a}^{ij}\leq(1+C(\lambda,\alpha)){\ell}^{\alpha})\delta^{ij},|\tilde{b}^{i}|\leq(1+\lambda)^{-1/2}{\ell},|c|\leq\lambda{\ell}^{2}.$

Here $d(Y,Z)$ is the parabolic distance define by
$d((x,t),(y,s))=(||x-y||^{2}+|s-t|)^{1/2}$.

Next we generalize the definition of frequency to general parabolic solution
which is only locally defined. Let $u$ be a solution to (3.1) (3.2) defined on
$Q_{2R_{0}}$ with doubling assumption (3.3). For any $(x_{0},t_{0})\in
Q_{R_{0}}$ and $0\leq r\leq R_{0}$, we define

$\displaystyle E^{u,R_{0}}_{x_{0},t_{0}}(r)$
$\displaystyle=2r^{2}\int_{\\{t=t_{0}-r^{2}\\}\cap B_{R_{0}}}|\nabla
u|^{2}G_{x_{0},t_{0}}$ (3.7) $\displaystyle H^{u,R_{0}}_{x_{0},t_{0}}(r)$
$\displaystyle=\int_{\\{t=t_{0}-r^{2}\\}\cap B_{R_{0}}}u^{2}G_{x_{0},t_{0}}.$
(3.8)

######  Definition 3.9.

(Localized Frequency) For any $(x_{0},t_{0})\in Q_{R_{0}}$, $\ell\leq 1$, let
$\tilde{u}=u_{x_{0},t_{0},{\ell}}$. Then we define the localized frequency for
any $r>0$ as

$N_{x_{0},t_{0}}^{u;R_{0}}(r\ell):=N_{0,0}^{\tilde{u},R_{0}/\ell}(r):=\frac{E^{\tilde{u},R_{0}/\ell}_{0,0}(r)}{H^{\tilde{u},R_{0}/\ell}_{0,0}(r)}.$
(3.10)

######  Remark 3.11.

One can easily check that $N_{x_{0},t_{0}}^{u;R_{0}}(r)$ is well defined which
is independent of $\ell$.

######  Remark 3.12.

When $a^{ij}=\delta^{ij}$, then $N^{u;\infty}_{x_{0},t_{0}}$ coincides with
the standard global frequency defined in Poon [53].

######  Remark 3.13.

Throughout this article, we will assume $R_{0}=1$ and omit this superscript.
Further, when $(x_{0},t_{0})=(0,0)$, the subscript would be omitted as well.
For instance, we write $N^{u}(r)\equiv N_{0,0}^{u;1}(r)$ for simplicity. By
rescaling, we have

$N_{x_{0},t_{0}}^{u}(r\ell)=\frac{2\int_{\\{t=-1\\}\cap B_{1/(r\ell)}}|\nabla
u_{x_{0},t_{0};r\ell}|^{2}G_{0,0}}{\int_{\\{t=-1\\}\cap
B_{1/(r\ell)}}u^{2}_{x_{0},t_{0};r\ell}G_{0,0}}=N^{u_{x_{0},t_{0};\ell};1/\ell}(r)=N^{u_{x_{0},t_{0};r\ell};1/(r\ell)}(1).$
(3.14)

###  3.1. Almost Monotonicity for Localized Frequency

In this section, we prove that the localized frequency function is almost non-
decreasing. First we introduce an important approximation Lemma.

######  Lemma 3.15.

(Caloric approximation) Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with
doubling assumption (3.3). Let $\epsilon\in(0,1/10)$. There exists
$\ell_{0}=C(n,\lambda,\alpha,\Lambda,\epsilon)$ such that for any $x\in B_{1}$
and $\ell\leq\ell_{0}$, there exists an ancient caloric polynomial
$h:\mathbb{R}^{n}\times(-\infty,0]\to\mathbb{R}$ of order at most
$C(n,\lambda,\alpha)\Lambda$, such that

$\fint_{Q_{1}}h^{2}=1,\quad\text{ and
}\quad||h-u_{x,0;\ell}||_{C^{1;1}(Q_{1/\epsilon})}\leq\epsilon.$ (3.16)

###### Proof.

We prove the result by contradiction argument. Suppose the result be false.
Consider $u_{x,0;1/i}$ with $i\to\infty$. Since
$\fint_{Q_{1}}u_{x,0;1/i}^{2}=1$, we have
$\fint_{Q_{2^{k}}}u_{x,0;1/i}^{2}\leq C(n,\lambda)^{k\Lambda}$ for any $i$ and
any $k\leq\log_{2}i$ by doubling assumption 1.13. Then by interior estimates,

$\sup_{i}||u_{x,0;1/i}||_{C^{1+\alpha;1+\alpha/2}(Q_{2^{k-1}})}\leq
C(n,\lambda,\alpha)^{k\Lambda}.$

For each $k$, there exists some $h_{k}$ such that
$||u_{x,0;1/i}-h_{k}||_{C^{1;1}(Q_{2^{k-1}})}\to 0$ by passing to a
subsequence. Moreover each $h_{k}$ is a caloric function according to the
estimates of $\tilde{a}^{ij}$. By taking $k\to\infty$, we have $h_{k}\to h$ in
$C^{1;1}$ on compact sets with $h$ satisfying the polynomial growth condition.
Then we can conclude by Lemma 2.7 that $h$ is indeed a caloric polynomial of
order $C(n,\lambda,\alpha)\Lambda$. The contradiction arises. ∎

Next we discuss several applications of this approximation lemma. The first is
the almost monotonicity of the weighted integral in each time slice.

Recall that $H^{h}(r)=\int_{t=-r^{2}}h^{2}G_{0,0}$ is non-decreasing in $r$
for caloric function $h$. By the convergence, we can easily see that the
localized version is also almost non-deceasing in $r$. As it is used
frequently, for simplicity we define $H^{\tilde{u}}$ with
$\tilde{u}=u_{x,0;\ell}$ as

$H_{0,0}^{\tilde{u}}(r)=\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}\tilde{u}^{2}G_{0,0}.$ (3.17)

Sometimes we omit the subscript $(0,0)$ if there is no confusion.

######  Lemma 3.18.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $\epsilon\in(0,1/10)$. There exists
$\ell_{0}=C(n,\lambda,\alpha,\Lambda,\epsilon)$ such that for any $x\in B_{1}$
and $\ell\leq\ell_{0}$, we have the following almost monotonicity for
$H^{\tilde{u}}$ where $\tilde{u}=u_{x,0;\ell}$

$H^{\tilde{u}}(r_{1})\leq H^{\tilde{u}}(r_{2})+\epsilon\quad\text{ for any
}\quad 0\leq r_{1}\leq r_{2}\leq 1.$ (3.19)

######  Remark 3.20.

We can see from the proof below that $H^{\tilde{u}}_{(x_{0},0)}$ is also
almost monotone for any $(x_{0},0)\in Q_{R}$ if
$\ell\leq\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon,R).$ This will be used in
the proof of Lemma 3.51.

###### Proof.

For any $\epsilon^{\prime}>0$, by Lemma 3.15, if
$\ell\leq\ell(n,\Lambda,\lambda,\alpha,\epsilon^{\prime})$, there exists a
caloric polynomial $h$ with order $C(n,\lambda,\alpha)\Lambda$ such that
$\tilde{u}=u_{x,0;\ell}$ satisfies

$||h-u_{x,0;\ell}||_{C^{1;1}(Q_{1/\epsilon^{\prime}})}\leq\epsilon^{\prime}.$
(3.21)

Since $h$ is a polynomial but $G_{0,0}$ has exponential decay, for
$\ell\leq\ell(n,\Lambda,\lambda,\alpha,\epsilon^{\prime})$ sufficiently small
and any $r\in[0,1]$ we have

$\Bigg{|}\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0}-\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}h^{2}G_{0,0}\Bigg{|}\leq\epsilon^{\prime}.$ (3.22)

Also, by the convergence and polynomial growth of $\tilde{u}$ and $h$, for any
$r\in[0,1]$ we have

$\Bigg{|}\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}h^{2}G_{0,0}-\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}\tilde{u}^{2}G_{0,0}\Bigg{|}\leq\int_{\\{t=-r^{2}\\}\cap
B_{1/\epsilon^{\prime}}}|h^{2}-\tilde{u}^{2}|G_{0,0}+\int_{\\{t=-r^{2}\\}\cap
A_{1/\epsilon^{\prime},1/\ell}}|h^{2}-\tilde{u}^{2}|G_{0,0}\leq
C(n,\lambda,\Lambda)\epsilon^{\prime}.$

Recall that $(H^{h,\infty})^{\prime}(r)\geq 0$. Then for any $0\leq r_{1}\leq
r_{2}\leq 1$, we have

$H^{\tilde{u}}(r_{1})\leq H^{h,\infty}(r_{1})+C\epsilon^{\prime}\leq
H^{h,\infty}(r_{2})+2C\epsilon^{\prime}\leq
H^{\tilde{u}}(r_{2})+3C\epsilon^{\prime}.$

The proof is finished by taking
$\epsilon^{\prime}=\epsilon^{\prime}(\epsilon,n,\alpha,\lambda,\Lambda)$ for
any given $\epsilon>0$. ∎

Before the proof of the almost monotonicity for frequency, in the next Lemma
we establish the frequency closeness in the interval away from 0, say
$[\delta,1]$.

######  Lemma 3.23.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $\epsilon,\delta\in(0,1/10)$. There exists
$\ell_{0}=C(\lambda,\alpha,\Lambda,\epsilon,\delta)$ such that for any $x\in
B_{1}$ and $\ell\leq\ell_{0}$, there exists an ancient caloric polynomial
$h:\mathbb{R}^{n}\times(-\infty,0]\to\mathbb{R}$, of order at most
$C(n,\lambda,\alpha)\Lambda$, such that for any $r\in[\delta,1]$

$|N^{\tilde{u};1/\ell}(r)-N^{h;\infty}(r)|\leq\epsilon,$ (3.24)

where $\tilde{u}=u_{x,0;\ell}$.

###### Proof.

For any $\epsilon^{\prime}>0$, by Lemma 3.15, if
$\ell\leq\ell(n,\Lambda,\lambda,\alpha,\epsilon^{\prime})$, there exists a
caloric polynomial $h$ with order $C(n,\lambda,\alpha)\Lambda$ such that
$\tilde{u}=u_{x,0;\ell}$ satisfies

$\fint_{Q_{1}}h^{2}=1,\quad\text{ and
}\quad||h-u_{x,0;\ell}||_{C^{1,1}(Q_{1/\epsilon^{\prime}})}\leq\epsilon^{\prime}.$
(3.25)

Thus, given $\delta>0$, by the monotonicity of
$H^{h}(r)=\int_{t=-r^{2}}h^{2}G_{0,0}$ for caloric function $h$ we can get for
any $r\in[\delta,1]$ that

$\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0}\geq c(\lambda,\alpha,\Lambda,\delta)>0.$
(3.26)

Since $h$ is a polynomial but $G_{0,0}$ has exponential decay, for
$\ell\leq\ell(n,\Lambda,\lambda,\alpha,\epsilon^{\prime})$ sufficiently small
and any $r\in[0,1]$ we have

$\begin{split}\Bigg{|}\int_{\\{t=-r^{2}\\}}|\nabla
h|^{2}G_{0,0}&-\int_{\\{t=-r^{2}\\}\cap B_{1/\ell}}|\nabla
h|^{2}G_{0,0}\Bigg{|}\leq\epsilon^{\prime}\\\
\Bigg{|}\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0}&-\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}h^{2}G_{0,0}\Bigg{|}\leq\epsilon^{\prime}.\end{split}$ (3.27)

Also, by the approximating estimate and polynomial growth of $\tilde{u}$ and
$h$, for any $r\in[0,1]$ we have

$\begin{split}\Bigg{|}\int_{\\{t=-r^{2}\\}\cap B_{1/\ell}}|\nabla
h|^{2}G_{0,0}&-\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}|\nabla\tilde{u}|^{2}G_{0,0}\Bigg{|}\leq\Psi(\epsilon^{\prime}|~{}n,\lambda,\alpha,\Lambda)\\\
\Bigg{|}\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}h^{2}G_{0,0}&-\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}\tilde{u}^{2}G_{0,0}\Bigg{|}\leq\Psi(\epsilon^{\prime}|~{}n,\lambda,\alpha,\Lambda),\end{split}$
(3.28)

The notation $\Psi(\epsilon^{\prime}|~{}n,\lambda,\alpha,\Lambda)$ satisfies
$\lim_{\epsilon^{\prime}\to
0}\Psi(\epsilon^{\prime}|~{}n,\lambda,\alpha,\Lambda)=0$ for any fixed
$n,\lambda,\alpha,\Lambda$.

Moreover, since $h$ is a caloric polynomial of order at most
$C(n,\lambda,\alpha)\Lambda$, by Corollary 2.15 we have for $r\leq 1$ that

$N^{h,\infty}(r)=\frac{2r^{2}\int_{\\{t=-r^{2}\\}}|\nabla
h|^{2}G_{0,0}}{\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0}}\leq
C(n,\lambda,\alpha)\Lambda.$ (3.29)

Combining all these inequalities (3.26) (3.27) (3.28) (3.29), we can prove for
any $r\in[\delta,1]$ that

$\begin{split}&\quad\frac{1}{2}|N^{h;\infty}(r)-N^{\tilde{u};1/{\ell}}(r)|\\\
&=\Bigg{|}\frac{r^{2}\int_{\\{t=-r^{2}\\}\cap
B_{1/{\ell}}}|\nabla\tilde{u}|^{2}G_{0,0}}{\int_{\\{t=-1\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{0,0}}-\frac{r^{2}\int_{\\{t=-r^{2}\\}}|\nabla
h|^{2}G_{0,0}}{\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0}}\Bigg{|}\\\ \\\
&=r^{2}\Bigg{|}\frac{(\int_{\\{t=-r^{2}\\}\cap
B_{1/{\ell}}}|\nabla\tilde{u}|^{2}G_{0,0})(\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0})-(\int_{\\{t=-r^{2}\\}}|\nabla
h|^{2}G_{0,0})(\int_{\\{t=-r^{2}\\}\cap
B_{1/{\ell}}}{\tilde{u}^{2}G_{0,0})}}{{(\int_{\\{t=-r^{2}\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{0,0}})(\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0})}\Bigg{|}\\\
\\\ &\leq r^{2}\Bigg{|}\frac{\int_{\\{t=-r^{2}\\}\cap
B_{1/{\ell}}}|\nabla\tilde{u}|^{2}G_{0,0}-\int_{\\{t=-r^{2}\\}}|\nabla
h|^{2}G_{0,0}}{\int_{\\{t=-r^{2}\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{0,0}}\Bigg{|}+r^{2}\Bigg{|}\frac{(\int_{\\{t=-r^{2}\\}}|\nabla
h|^{2}G_{0,0})(\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0}-\int_{\\{t=-r^{2}\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{0,0})}{(\int_{\\{t=-r^{2}\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{0,0})(\int_{\\{t=-r^{2}\\}}h^{2}G_{0,0})}\Bigg{|}\\\
\\\
&\leq\Psi(\epsilon^{\prime}|~{}n,\delta,\lambda,\alpha,\Lambda).\end{split}$

Therefore for any $\epsilon>0$ by fixing $\epsilon^{\prime}$ we have

$|N^{h;\infty}(r)-N^{\tilde{u};1/\ell}(r)|\leq\Psi(\epsilon^{\prime}|~{}n,\delta,\lambda,\alpha,\Lambda)\leq\epsilon,$

which finishes the whole proof. ∎

Now we are ready to prove the almost monotonicity formula for $u$.

######  Theorem 3.30.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $\epsilon\in(0,1/10)$. There exists
$\ell_{0}=C(\lambda,\alpha,\Lambda,\epsilon)$ such that the following almost
monotonicity formula holds: for any $x\in B_{1}$ and any $r_{1}\leq
r_{2}\leq\ell_{0}$,

$N^{u}_{x,0}(r_{1})\leq N^{u}_{x,0}(r_{2})+\epsilon.$ (3.31)

Moreover, if $N^{u}_{x,0}(r)\leq\epsilon$ with $r\leq\ell_{0}$, then
$\sup_{Q_{r}(x,0)}|u(y,s)-a_{0}|\leq C(n,\Lambda)\sqrt{\epsilon}|a_{0}|$ for
some nonzero constant $a_{0}$.

###### Proof.

For simplicity, we assume $(x,t)=(0,0)$ and we write $u_{0,0;\ell}\equiv
u_{\ell}$. Also we write $N^{u}(r)\equiv N^{u}_{0,0}(r)$.

Take any $\epsilon\in(0,1/10)$. Fix $\delta$ as in Lemma 2.10 corresponding to
$\epsilon/20$. Then by Lemma 3.23, there exists some
$\ell_{0}(\lambda,\alpha,\Lambda,\epsilon)$ such that for any
$\ell\leq\ell_{0}$, there exists some $h_{\ell}$ such that for any
$r\in[\delta,1]$

$|N^{u}(\ell
r)-N^{h_{\ell};\infty}(r)|=|N^{u_{\ell};1/\ell}(r)-N^{h_{\ell};\infty}(r)|\leq\epsilon/100.$
(3.32)

Along with the monotonicity of $N^{h_{\ell};\infty}$, (3.32) imply that
$N^{u}(r)\leq N^{u}(\ell_{0})+\epsilon$ for any
$r\in[\delta\ell_{0},\ell_{0}]$. Hence, to prove the theorem, it suffices to
prove that $N^{u}(r\ell_{0})\leq N^{u}(\ell_{0})+\epsilon$ for any
$r\leq\delta$. We choose $d$ to be the minimal integer such that
$|N^{u}(\ell_{0})-d|\leq 1/2.$ Hence we have $d-1/2<N^{u}(\ell_{0})\leq
d+1/2$.

Claim:  If $N^{u}(\ell_{0})\leq d+\epsilon/10$, then $N^{u}(r\ell_{0})\leq
d+\epsilon/2$ for any $r\leq 1$.

We first suppose the claim be true and prove the theorem. It suffices to
consider the following three cases.

Case 1:  Suppose $N^{u}(\ell_{0})\in[d+\epsilon/10,d+1/2]$. Applying Lemma
2.10 to $h_{\ell_{0}}$ and by (3.32), we have $N^{u}(\delta\ell_{0})\leq
d+\epsilon/10$. According to the Claim, for any $r\leq 1$, we have
$N^{u}(r\delta r_{0})\leq d+\epsilon/2\leq N^{u}(\ell_{0})+\epsilon$. This
finishes the proof of Case 1.

Case 2:  Suppose $N^{u}(\ell_{0})\in[d-\epsilon/10,d+\epsilon/10]$. By the
claim, we have $N^{u}(r\ell_{0})\leq d+\epsilon/2<N^{u}(\ell_{0})+\epsilon$
for any $r\leq 1$.

Case 3:  Suppose $N^{u}(\ell_{0})\in[d-1/2,d-\epsilon/10]$. Applying Lemma
2.10 to $h_{\ell_{0}}$ and by (3.32), we have $N^{u}(\delta\ell_{0})\leq
d-1+\epsilon/10$. By the claim we have $N^{u}(r\delta\ell_{0})\leq
d-1+\epsilon/2<N^{u}(\ell_{0})$ for any $r\leq 1$. This finishes the proof.

In the following we will prove the claim. Since $N^{u}(\ell_{0})\leq
d+\epsilon/10$, we can define

$r_{1}\equiv\inf\\{s\geq 0:N^{u}(r\ell_{0})\leq d+\epsilon/3\text{ for any
}r\in[s,1].\\}$ (3.33)

Noting that frequency closeness (3.32) and monotonicity of $N^{h;\infty}$ we
have that $r_{1}\leq\delta$. We assume $r_{1}>0$. Otherwise the claim holds
trivially. Then we apply Lemma 3.23 at the scale $r_{1}\ell_{0}$ to obtain
$h_{r_{1}\ell_{0}}$ such that for any $r\in[\delta,1]$ we have

$|N^{u}(rr_{1}\ell_{0})-N^{h_{r_{1}\ell_{0}};\infty}(r)|\leq\epsilon/100.$
(3.34)

Together with the monotonicity of $N^{h_{r_{1}\ell_{0}};\infty}$, this implies
that $N^{u}(r\ell_{0})\leq N^{u}(r_{1}\ell_{0})+\epsilon/50<d+\epsilon/2$ for
any $r\in[\delta r_{1},r_{1}]$. Since $N^{h_{r_{1}\ell_{0}};\infty}(1)\leq
d+3\epsilon/8<d+1-\epsilon/20$, applying Lemma 2.10 to $h_{r_{1}\ell_{0}}$ and
by (3.34) we have $N^{u}(\delta r_{1}\ell_{0})\leq d+\epsilon/10$.

Now we can finish the proof by induction. Indeed, for each $k$ we could define

$r_{k+1}\equiv\inf\\{s\geq 0:N^{u}(r\ell_{0})\leq d+\epsilon/3\text{ for any
}r\in[s,r_{k}].\\}$ (3.35)

and by the same arguments as above we can prove that

  1. (1)

$r_{k+1}\leq\delta r_{k}$.

  2. (2)

$N^{u}(r\ell_{0})\leq d+\epsilon/2$ for any $r\in[\delta r_{k+1},1]$.

  3. (3)

$N^{u}(\delta r_{k+1}\ell_{0})\leq d+\epsilon/10$.

This finishes the proof of almost monotonicity. Noting Corollary 2.16, Lemma
3.23 and Lemma 3.15, one can easily deduce the estimate with
$N^{u}_{x,t}(r)\leq\epsilon$ by contradiction. ∎

As a corollary, we prove that the frequency of $u$ is only pinched near
integers and drops at a definite rate when it is away from integers.

######  Lemma 3.36.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $\epsilon\in(0,1/10)$ and $x\in B_{1}$. There exist some
$\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ and some
$\delta(n,\lambda,\alpha,\Lambda,\epsilon)$ such that:

  1. (1)

If $|N^{u}_{x,0}(r_{1})-N^{u}_{x,0}(r_{2})|\leq\delta$ for some $10r_{1}\leq
r_{2}\leq\ell_{0}$, then there exists some integer $d$ such that
$|N^{u}_{x,0}(r)-d|\leq\epsilon$ for any $r\in[r_{2},r_{1}]$.

  2. (2)

If $N^{u}_{x,0}(r)\leq d-\epsilon$ for some $r\leq\ell_{0}$, then
$N^{u}_{x,0}(\tilde{r})\leq d-1+\epsilon$ for any $\tilde{r}\leq\delta r$.

###### Proof.

The first conclusion follows directly from Lemma 2.9 and Lemma 3.23. The
second one can be proved using the same arguments as in Lemma 2.10 combined
with the almost monotonicity theorem 3.30. ∎

Another consequence of the almost monotonicity is the following finitely many
non-pinch result. This will be used in the proof of neck region decomposition
in section 6.

######  Corollary 3.37.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). For any $\epsilon,\delta>0$ and $x\in B_{1}$ if
$r\leq\ell_{0}(\lambda,\alpha,\Lambda,\epsilon,\delta)$ and
$r_{i}=r\epsilon^{i}$, then there exists at most
$K(n,\delta,\lambda,\alpha,\Lambda)$ many $\\{i\geq 0\\}$ such that

$\displaystyle|N^{u}_{x,0}(r_{i})-N^{u}_{x,0}(r_{i+1})|\geq\delta.$ (3.38)

###### Proof.

By Theorem 3.30 if $r\leq\ell_{0}(n,\lambda,\alpha,\Lambda,\delta)$ we have
for all $r_{i}=r\epsilon^{i}$ that

$\displaystyle N^{u}_{x,0}(r_{i+1})\leq N^{u}_{x,0}(r_{i})+\delta^{2}$ (3.39)

Denote $I=\\{i\geq
0:|N^{u}_{x,0}(r_{i})-N^{u}_{x,0}(r_{i+1})|\geq\delta\\}=\\{0\leq
i_{1}<i_{2}<i_{3}<\cdots\\}$. Thus

$\displaystyle N^{u}(x,0)(r_{i_{1}+1})\leq N^{u}_{x,0}(r_{i_{1}})-\delta$
(3.40)

By the almost monotonicity of the localized frequency we get

$\displaystyle N^{u}_{x,0}(r_{i_{2}})\leq
N^{u}_{x,0}(r_{i_{1}+1})+\delta^{2}\leq
N^{u}_{x,0}(r_{i_{1}})-\delta+\delta^{2}$ (3.41)

Since $|N^{u}_{x,0}(r_{i_{2}})-N^{u}_{x,0}(r_{i_{2}+1})|\geq\delta$, we get

$\displaystyle N^{u}_{x,0}(r_{i_{3}})\leq
N^{u}_{x,0}(r_{i_{2}+1})+\delta^{2}\leq
N^{u}_{x,0}(r_{i_{2}})-\delta+\delta^{2}\leq
N^{u}_{x,0}(r_{i_{1}})-2(\delta-\delta^{2}).$ (3.42)

By iteration we get

$\displaystyle N^{u}_{x,0}(r_{i_{k}})\leq
N^{u}_{x,0}(r_{i_{1}})-(k-1)(\delta-\delta^{2}).$ (3.43)

Noting $0\leq N_{x,0}^{u}(r)\leq C(n,\lambda,\Lambda,\alpha)$, we conclude
that $\\#I\leq K(n,\delta,\lambda,\alpha,\Lambda)$. Hence we complete the
proof. ∎

###  3.2. Frequency and Doubling Index

In this subsection, we establish the equivalence of the localized frequency
and the doubling index. Thus the doubling index is also almost monotone.
Recall that we define $H^{\tilde{u}}$ with $\tilde{u}=u_{x,0;\ell}$ as

$H^{\tilde{u}}(r)=\int_{\\{t=-r^{2}\\}\cap B_{1/\ell}}\tilde{u}^{2}G_{0,0}.$
(3.44)

######  Definition 3.45.

We define the localized doubling index $D^{\tilde{u}}(r)$ with
$\tilde{u}=u_{x,0;\ell}$ as

$D_{0,0}^{\tilde{u}}(r)\equiv\log_{4}\frac{H^{\tilde{u}}(2r)}{H^{\tilde{u}}(r)}.$
(3.46)

Sometimes we omit the subscript $(0,0)$ and write it as $D^{\tilde{u}}(r)$ if
there is no confusion. Like the frequency function $N$, (3.14), the doubling
index also has similar scaling property

$D^{u_{x,0;\ell}}(r)=D^{u_{x,0;r\ell}}(1).$ (3.47)

Next we can conclude that the doubling index is also almost monotone.

######  Theorem 3.48.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $\epsilon\in(0,1/10)$. There exists
$\ell_{0}=C(\lambda,\alpha,\Lambda,\epsilon)$ such that for any
$\ell\leq\ell_{0}$ we have for any $r\leq 1$ that

$N^{\tilde{u},1/\ell}(r)-\epsilon\leq D^{\tilde{u}}(r)\leq
N^{\tilde{u},1/\ell}(2r)+\epsilon.$ (3.49)

Moreover, the following almost monotonicity formula holds: for any $x\in
B_{1}$ and any $r_{1}\leq r_{2}\leq 1/2$,

$D^{\tilde{u}}(r_{1})\leq D^{\tilde{u}}(r_{2})+\epsilon.$ (3.50)

If $D^{\tilde{u}}(r)\leq\epsilon$ for some $r\leq 1/2$ then there exists a
nonzero constant $a_{0}$ such that $\sup_{Q_{r}}|\tilde{u}-a_{0}|\leq
C(n,\Lambda)\sqrt{\epsilon}|a_{0}|.$

###### Proof.

For $r=1$, we can prove the theorem by applying the contradiction argument and
(2.4). The general case follows from the scaling property of $N$, see (3.14),
and $D$, see (3.47). Then the almost monotonicity follows directly from
Theorem 3.48. ∎

###  3.3. Equivalence between different normalizations

Recall that in the definition of tangent function we normalize such the
function has unit $L^{2}$-norm over $Q_{1}$. However, sometimes we need to use
other normalization. In the following lemma, we will establish the equivalence
between several different normalizations.

######  Lemma 3.51.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). If $\ell\leq\ell_{0}(n,\lambda,\alpha,\Lambda)$ is chosen small, then
for any $x\in B_{1}$ and $\tilde{u}=u_{x,0;\ell}$ the following holds

$\displaystyle\fint_{Q_{1}}\tilde{u}^{2}\leq
C_{1}(n,\lambda,\alpha,\Lambda)\int_{\\{t=-1\\}\cap
B_{1/\ell}}\tilde{u}^{2}G_{0,0}\leq
C_{2}(n,\lambda,\alpha,\Lambda)\fint_{Q_{1}}\tilde{u}^{2}$ (3.52)

###### Proof.

Noting that $\fint_{Q_{1}}\tilde{u}^{2}=1$, we only need to show that
$\int_{\\{t=-1\\}\cap B_{1/\ell}}\tilde{u}^{2}G_{0,0}$ has uniform upper and
lower bounds.

To see the upper bound of $\int_{\\{t=-1\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{0,0}$, by the doubling assumption and parabolic
estimate, we have point-wise estimate for $R\geq 1$ that

$\displaystyle\sup_{Q_{R}}|\tilde{u}|\leq
C(n,\lambda,\alpha,\Lambda)R^{C(n,\Lambda)}.$ (3.53)

Since $G_{0,0}$ is exponentially decay, this implies

$\displaystyle\int_{\\{t=-1\\}\cap B_{1/{\ell}}}\tilde{u}^{2}G_{0,0}\leq
C(n,\lambda,\alpha,\Lambda).$ (3.54)

To see the lower bound of $L_{0}:=\int_{\\{t=-1\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{0,0}$, for any $x_{0}\in B_{1}$ we have for any
$R>1$ that

$\displaystyle\int_{\\{t=-1\\}\cap B_{1/{\ell}}}\tilde{u}^{2}G_{x_{0},0}$
$\displaystyle=\int_{\\{t=-1\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}(4\pi)^{-n/2}e^{-\frac{|x-x_{0}|^{2}}{4}}$ (3.55)
$\displaystyle\leq\int_{\\{t=-1\\}\cap B_{1/{\ell}}\cap
B_{R}(x_{0})}\tilde{u}^{2}(4\pi)^{-n/2}e^{-\frac{|x-x_{0}|^{2}}{4}}+\int_{\\{t=-1\\}\cap
B_{1/{\ell}}\setminus
B_{R}(x_{0})}\tilde{u}^{2}(4\pi)^{-n/2}e^{-\frac{|x-x_{0}|^{2}}{4}}$ (3.56)
$\displaystyle\leq
C(n,R)L_{0}+C(n,\lambda,\alpha,\Lambda)R^{C(n,\Lambda)}e^{-R^{2}/10}$ (3.57)

By the monotonicity of $H$ in Lemma 3.18, for any $\epsilon>0$ if
${\ell}\leq{\ell}_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ we have for any
$s\leq 1$ that

$\displaystyle\int_{\\{t=-s\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{x_{0},0}\leq\int_{\\{t=-1\\}\cap
B_{1/{\ell}}}\tilde{u}^{2}G_{x_{0},0}+\epsilon\leq
C(n,R)L_{0}+C(n,\lambda,\alpha,\Lambda)R^{C(n,\Lambda)}e^{-R^{2}/10}+\epsilon.$
(3.58)

In particular,

$\displaystyle\int_{\\{t=-s\\}\cap B_{\sqrt{s}}(x_{0})}\tilde{u}^{2}\leq
C(n)s^{n/2}\left(C(n,R)L_{0}+C(n,\lambda,\alpha,\Lambda)R^{C(n,\Lambda)}e^{-R^{2}/10}+\epsilon\right).$
(3.59)

Since the above holds for any $x_{0}\in B_{1}$, using covering we conclude
that for any $0\leq s\leq 1$

$\displaystyle\int_{\\{t=-s\\}\cap B_{1}}\tilde{u}^{2}\leq
C(n)\left(C(n,R)L_{0}+C(n,\lambda,\alpha,\Lambda)R^{C(n,\Lambda)}e^{-R^{2}/10}+\epsilon\right).$
(3.60)

This implies

$\displaystyle 1=\fint_{Q_{1}}\tilde{u}^{2}\leq
C(n)\left(C(n,R)L_{0}+C(n,\lambda,\alpha,\Lambda)R^{C(n,\Lambda)}e^{-R^{2}/10}+\epsilon\right).$
(3.61)

By fixing $R,\epsilon$ thus fixing
$\ell_{0}=\ell_{0}(\lambda,\Lambda,n,\alpha)>0$ we get that $L_{0}\geq
C(n,\Lambda,\lambda,\alpha)>0$. Hence we finish the whole proof. ∎

##  4\. Quantitative Uniqueness of Tangent Maps

In this section we aim to prove the quantitative version of the uniqueness of
tangent maps for parabolic solutions. Based on the almost monotonicity of the
localized frequency and parabolic estimate we can get quantitative estimate on
each time slice. This is the key point to get nodal set estimate on time
slice. The main theorem in this section is the following quantitative
uniqueness of tangent maps. Some arguments are motivated by [1, 27, 33, 48].

######  Theorem 4.1.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption (3.3)
and $x\in B_{1}$. For any $\epsilon>0$ there exist
$\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ and
$\delta(n,\lambda,\alpha,\Lambda,\epsilon)$ such that if
$|N^{u}_{x,0}(r_{1})-N^{u}_{x,0}(r_{2})|\leq\delta$ for
$10\delta^{-1}r_{2}\leq r_{1}\leq\ell_{0}$, then there exists some integer $d$
and some homogeneous caloric polynomial $P_{d}$ with normalization
$\fint_{Q_{1}}|P_{d}|^{2}=1$ such that for any $r\in[r_{2},\delta r_{1}]$

$\sup_{Q_{1}}|u_{x,0;r}-P_{d}|\leq\epsilon.$ (4.2)

Recall that we say a function $u:B_{1}\to\mathbb{R}$ is
$(k,\eta,r,x)$-symmetric if there exists a $k$-symmetric homogeneous
polynomial $P$ with $\fint_{B_{1}}|P|^{2}=1$ such that
$\sup_{B_{1}}|u_{x,r}(y)-P(y)|\leq\eta,$ where
$u_{x,r}(y)=\frac{u(x+ry)}{\left(\fint_{B_{1}}|u(x+ry)|^{2}\right)^{1/2}}$. We
say $u$ is uniformly $(k,\eta,x)$-symmetric in $[r_{2},r_{1}]$ if there exists
a $k$-symmetric homogeneous polynomial $P$ with $\fint_{B_{1}}|P|^{2}=1$ such
that for any $r\in[r_{2},r_{1}]$

$\sup_{B_{1}}|u_{x,r}-P|\leq\eta.$ (4.3)

By (4.2), we know that at time slice $s=0$,
$\sup_{B_{1}}|u_{x,0;r}-P_{d}|(y,0)\leq\epsilon$. By Lemma 3.51, we have
$C(n,d)^{-1}\leq\fint_{B_{1}}|P_{d}|^{2}(x,0)dx\leq C(n,d)$. Furthermore, by
the definition of homogeneous polynomial, $P_{d}(x,0)$ is also a homogeneous
polynomial in $x$. Hence as a consequence of Theorem 4.1 we get

######  Theorem 4.4.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). For any $\epsilon>0$ and $x\in B_{1}$ there exist
$\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ and
$\delta(n,\lambda,\alpha,\Lambda,\epsilon)$ such that if
$|{N}^{u}_{x,0}(r_{1})-{N}^{u}_{x,0}(r_{2})|\leq\delta$ for
$10\delta^{-1}r_{2}\leq r_{1}\leq\ell_{0}$, then there exists some integer $d$
and some homogeneous polynomial $\hat{P}_{d}:\mathbb{R}^{n}\to\mathbb{R}$ with
normalization $\fint_{B_{1}}|\hat{P}_{d}|^{2}=1$ such that for any
$r\in[r_{2},\delta r_{1}]$

$\sup_{y\in B_{1}}|\hat{u}_{x,r}(y)-\hat{P}_{d}(y)|\leq\epsilon.$ (4.5)

where
$\hat{u}_{x,r}(y)=\frac{u(x+ry,0)}{\Big{(}\fint_{B_{1}}|u(x+ry,0)|^{2}dy\Big{)}^{1/2}}$.
In particular, $u(\cdot,0)$ is uniformly $(0,\epsilon,x)$-symmetric in
$[r_{2},\delta r_{1}]$.

We should remark that the homogeneous polynomial $\hat{P}_{d}(y)$ may not
satisfy any equation. However, this is good enough to deduce the quantitative
estimates as solution of elliptic PDEs in [33].

Theorem 4.1 is part of the result of Theorem 4.39. Hence it suffices to prove
Theorem 4.39 in the following subsection. The proofs depends on several
crucial estimates.

###  4.1. Growth estimates under doubling lower bound

In this subsection we will prove a growth estimate for $u_{x,0;r}$ which is
one key ingredient in the proof of quantitative uniqueness .

Let $x\in B_{1}$ and choose $\ell<1$ be small. Write $\tilde{u}=u_{x,0;\ell}$.
Recall that $\tilde{u}$ satisfies the equation

$(\partial_{t}-\Delta)\tilde{u}=\partial_{i}(\tilde{a}^{ij}-\delta^{ij})\partial_{j}\tilde{u}+\tilde{b}^{i}\partial_{i}\tilde{u}+\tilde{c}\tilde{u},$

where we have $\tilde{a}(0,0)=(\delta^{ij})$ and

$|\tilde{a}^{ij}(Y)-\tilde{a}^{ij}(Z)|\leq
C(\lambda,\alpha)\ell^{\alpha}d(Y,Z)^{\alpha},(1+C(\lambda,\alpha)\ell^{\alpha})^{-1}\delta^{ij}\leq\tilde{a}^{ij}\leq(1+C(\lambda,\alpha))\ell^{\alpha})\delta^{ij},|\tilde{b}^{i}|\leq(1+\lambda)^{-1/2}\ell,|c|\leq\lambda\ell^{2}.$

Recall that

$H^{\tilde{u}}(r)=\int_{\\{t=-r^{2}\\}\cap
B_{1/\ell}}\tilde{u}^{2}G_{0,0}\quad\text{ and }\quad
D_{0,0}^{\tilde{u}}(r)\equiv\log_{4}\frac{H^{\tilde{u}}(2r)}{H^{\tilde{u}}(r)}.$

For any $\eta>1$, write the $\eta$-parabolic metric
$d^{\eta}((x,t),(y,s))\equiv\max\\{\eta^{-1}||x-y||,\sqrt{|t-s|}\\}$. Define
the backward $\eta$-parabolic ball as
$Q^{\eta}_{r}(x_{0},t_{0})\equiv\\{(x,t):||x-x_{0}||<\eta
r,0<t_{0}-t<r^{2}\\}$. Let us first prove the following growth estimate for
$\tilde{u}$.

######  Lemma 4.6.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). For fixed $x\in B_{1}$ let $\tilde{u}=u_{x,0;\ell}$. Assume
$D_{0,0}^{\tilde{u}}(r)\geq\gamma>0$ for any $r_{0}\leq r\leq 1$. Then if
$\ell\leq\ell_{0}(n,\lambda,\alpha,\Lambda)$ we have the following estimates
for all $r_{0}\leq r\leq 1$ and all $1\leq\eta<\frac{1}{10\ell}$ that

  1. (1)

$\sup_{Q_{r}^{\eta}}|\tilde{u}|\leq
C(n,\lambda,\alpha,\Lambda,\eta)~{}r^{\gamma}.$ (4.7)

  2. (2)

$\sup_{Q_{r}^{\eta}}|\nabla\tilde{u}|\leq
C(n,\lambda,\alpha,\Lambda,\eta)~{}r^{\gamma-1}.$ (4.8)

###### Proof.

We will first prove the estimate for $r=r_{0}$. We will see that the proof
works for general $r>r_{0}$. Since $\fint_{Q_{1}}\tilde{u}^{2}=1$, by Lemma
3.51, we have $\int_{\\{t=-1\\}\cap B_{1/\ell}}\tilde{u}^{2}G_{0,0}\leq
C(n,\lambda,\alpha,\Lambda)$ if $\ell\leq\ell_{0}(n,\lambda,\alpha,\Lambda)$.
Since $D_{0,0}^{\tilde{u}}(r)\geq\gamma$ for any $r\in[r_{0},1]$, then
$H_{0,0}^{\tilde{u}}(r)\leq Cr^{2\gamma}$ for $r\in[r_{0},1]$. By Almost
Monotonicity Theorem 3.48 and taking
$\epsilon_{0}=\epsilon_{0}(n,\Lambda,\eta)>0$, for any $y\in B_{\eta
r_{0}}(0)$, we have $D_{(y,0)}^{\tilde{u}}(r)\geq\epsilon_{0}$ for all $r\geq
r_{1}\geq 0$ with some $r_{1}=r_{1}(y,0)\geq 0$, and $\sup_{Q_{2\eta
r_{1}}(y,0)}|\tilde{u}-a_{0}|\leq|a_{0}|/100$ for some nonzero constant
$a_{0}$. Let us now fix $y\in B_{\eta r_{0}}$. By Lemma 3.51 and doubling
assumption of $\tilde{u}$, we can deduce that $H_{y,0}^{\tilde{u}}(r_{0})\leq
C(n,\lambda,\Lambda,\eta)H_{0,0}^{\tilde{u}}(r_{0})\leq Cr_{0}^{2\gamma}.$ If
$r_{1}(y,0)<r_{0}$, we have for any $r_{1}\leq r\leq r_{0}$ that

$\int_{B_{r}(y)\times\\{t=-r^{2}\\}}\tilde{u}^{2}\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{0}^{2\gamma-2\epsilon_{0}}r^{2\epsilon_{0}+n}\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{0}^{2\gamma}r^{n}$ (4.9)

and for all $r\leq r_{1}$

$\displaystyle\int_{B_{r}(y)\times\\{t=-r^{2}\\}}\tilde{u}^{2}\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{0}^{2\gamma-2\epsilon_{0}}r_{1}^{2\epsilon_{0}}r^{n}\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{0}^{2\gamma}r^{n}.$ (4.10)

Thus if $r_{1}(y,0)<r_{0}$ for all $y\in B_{\eta r_{0}}$ then the above holds
for all $y\in B_{\eta r_{0}}(0)$. By covering we have for all $0\leq r\leq
r_{0}$ that

$\displaystyle\int_{B_{\eta r_{0}}(0)\times\\{t=-r^{2}\\}}\tilde{u}^{2}\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{0}^{2\gamma+n}$ (4.11)

This implies

$\displaystyle\int_{B_{\eta r_{0}}(0)\times[-r_{0}^{2},0]}\tilde{u}^{2}\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{0}^{2\gamma+n+2}.$ (4.12)

On the other hand, if $r_{1}(y,0)\geq r_{0}$ for some $y\in B_{\eta r_{0}}$,
noting that $H_{y,0}^{\tilde{u}}(r_{0})\leq
C(n,\lambda,\Lambda,\eta)H_{0,0}^{\tilde{u}}(r_{0})$ and $\sup_{Q_{2\eta
r_{1}}(y,0)}|\tilde{u}-a_{0}|\leq|a_{0}|/100$, we get

$\displaystyle\int_{B_{\eta
r_{0}}(0)\times[-r_{0}^{2},0]}\tilde{u}^{2}\leq\int_{Q_{\eta
r_{0}}(0,0)}\tilde{u}^{2}\leq\int_{Q_{2\eta r_{0}}(y,0)}\tilde{u}^{2}\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{0}^{2\gamma+n+2}.$ (4.13)

Replacing $r_{0}$ by any $1>r>r_{0}$ and using the above argument, we can get

$\displaystyle\int_{B_{\eta r}(0)\times[-r^{2},0]}\tilde{u}^{2}\leq
C(n,\lambda,\alpha,\Lambda,\eta)r^{2\gamma+n+2}.$ (4.14)

The proof is now finished by standard parabolic estimates. ∎

###  4.2. Gradient estimates for Green’s function expansion

Next we study the expansion of the Green’s function of the heat operator.
Recall that for $s\leq t$,

$G(x-y,t-s)=G(x,t;y,s)=\frac{1}{(4\pi(t-s))^{n/2}}\exp({-\frac{|x-y|^{2}}{4(t-s)}})$

And $G(x,t;y,s)\equiv 0$ when $s>t$. It is direct that we have the following
gradient estimates: for any $\mu,l\in\mathds{N}$, there exist constants
$C_{1}(n,\mu,l)$, $c_{2}(n,\mu,l)$ that for $t>0$

$|D_{x}^{\mu}D_{t}^{l}G(x,t)|\leq\frac{C_{1}}{|(x,t)|^{n+\mu+2l}}\exp(-\frac{|x|^{2}}{c_{2}t}).$
(4.15)

By Taylor’s theorem and simple rearrangement, for any $(y,s)\neq(0,0)$, we can
write the expansion at $(x,t)=(0,0)$ in $Q_{R}$ with $R<|(y,s)|$ as

$G(x,t;y,s)=\sum_{k=0}^{d}P_{y,s}^{k}(x,t)+R^{d}_{y,s}(x,t)$ (4.16)

where $P_{y,s}^{k}(x,t)$ are homogeneous caloric polynomials of degree $k$
with

$P_{y,s}^{k}(x,t)=\sum_{|\mu|+2l=k}D^{\mu}_{x}D_{t}^{l}G(-y,-s)\frac{x^{\mu}t^{l}}{\mu!l!}.$
(4.17)

and the remainder term

$R^{d}_{y,s}(x,t)=\sum_{k=0}^{d}\sum_{\begin{subarray}{c}|\mu|+l=k\\\
|\mu|+2l>d\end{subarray}}D^{\mu}_{x}D_{t}^{l}G(-y,-s)\frac{x^{\mu}t^{l}}{\mu!l!}+\sum_{|\mu|+l=d+1}D^{\mu}_{x}D_{t}^{l}G(\kappa\cdot
x-y,\theta\cdot t-s)\frac{x^{\mu}t^{l}}{\mu!l!}$ (4.18)

where $\kappa(x,t;y,s),\theta(x,t;y,s)$ are constants in $(0,1)$.

We have the following estimates on the expansions which are crucial in the
proof of uniqueness of tangent maps in the next section.

######  Lemma 4.19.

There exist constants $C$ depending on $n,d$ such that the following estimates
hold:
For any $s<0$

$\displaystyle|P^{k}_{y,s}(x,t)|\leq
C|(y,s)|^{-n-k}|(x,t)|^{k},\quad|\nabla_{y}P^{k}_{y,s}(x,t)|\leq
C|(y,s)|^{-n-k-1}|(x,t)|^{k}.$ (4.20)

Moreover, if $|(y,s)|_{\eta}\geq 2|(x,t)|_{\eta}$, then (4.16) holds with

$\displaystyle|R^{d}_{y,s}(x,t)|$ $\displaystyle\leq
C\sum_{j=d+1}^{2(d+1)}|(y,s)|^{-n-j}|(x,t)|^{j},$ (4.21)
$\displaystyle|\nabla_{y}R^{d}_{y,s}(x,t)|$ $\displaystyle\leq
C\sum_{j=d+1}^{2(d+1)}|(y,s)|^{-n-j-1}|(x,t)|^{j},$ (4.22)

where $|(x,t)|_{\eta}=\max\\{\eta^{-1}|x|,|t|\\}.$

###### Proof.

The first two inequalities come from gradient estimate of Green’s function
4.15 directly. We prove the estimate for $R^{d}_{y,s}(x,t)$. And the estimate
for the last inequality comes from gradient estimate 4.15 and the Taylor
expansion 4.16.

By gradient estimate (4.15), we have

$\displaystyle\quad\Big{|}\sum_{k=0}^{d}\sum_{\begin{subarray}{c}|\mu|+l=k\\\
|\mu|+2l>d\end{subarray}}D^{\mu}_{x}D_{t}^{l}G(-y,-s)\frac{x^{\mu}t^{l}}{\mu!l!}\Big{|}\leq
C\sum_{k=0}^{d}\sum_{\begin{subarray}{c}|\mu|+l=k\\\
|\mu|+2l>d\end{subarray}}|(y,s)|^{-(n+\mu+2l)}|(x,t)|^{\mu+2l}\exp(-\frac{|y|^{2}}{c_{2}s})\leq
C\sum_{j=d+1}^{2d}|(y,s)|^{-n-j}|(x,t)|^{j},$

where the second inequality follows from the substitution $j=\mu+2l$.

For the second term of (4.18), observe that when $|(y,s)|_{\eta}\geq
2|(x,t)|_{\eta}$, we have

$2|(\kappa\cdot x-y,\theta\cdot t-s)|_{\eta}\geq|(y,s)|_{\eta}.$

Then by gradient estimate (4.15) we have

$\displaystyle\big{|}\sum_{|\mu|+l=d+1}D^{\mu}_{x}D_{t}^{l}G(\kappa\cdot
x-y,\theta\cdot t-s)\frac{x^{\mu}t^{l}}{\mu!l!}\big{|}\leq
C\sum_{|\mu|+l=d+1}|(\kappa\cdot x-y,\theta\cdot
t-s)|^{-n-\mu-2l}|(x,t)|^{\mu+2l}\leq
C\sum_{j=d+1}^{2(d+1)}|(y,s)|^{-n-j}|(x,t)|^{j}.$

This finishes the proof. ∎

###  4.3. Quantitative Uniqueness of tangent maps

In this subsection we will prove the quantitative uniqueness of tangent maps.
The following Theorem 4.23 gives a uniqueness of tangent map with respect to a
caloric function $h$. Based on the doubling estimate of $h$ in Theorem 4.23
and the quantitative uniqueness of heat equation in Theorem 2.21 we can prove
the desired quantitative uniqueness Theorem 4.39.

######  Theorem 4.23.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $x\in B_{1}$ and $\epsilon\leq\alpha/10$. Then for any $\eta>0$,
there exists some $\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon,\eta)$ such that
if $|N^{u}_{x,0}(r_{1})-N^{u}_{x,0}(r_{2})|\leq\alpha/4$ for $10r_{2}\leq
r_{1}\leq\ell_{0}$, then there exists some caloric function $h$ defined on
$Q_{1}^{\eta}$ such that for any $r\in[r_{2}/r_{1},1]$ we have

$\sup_{(y,s)\in
Q_{r}^{\eta}}|\tilde{u}(y,s)-h(y,s)|\leq\epsilon^{2}(\fint_{Q_{r}}\tilde{u}^{2})^{1/2},$
(4.24)

where $\tilde{u}=u_{x,0;r_{1}}$. Moreover, if
$\epsilon\leq\epsilon_{0}(n,\lambda,\alpha,\Lambda)$ then $h$ satisfies the
doubling assumption at $(0,0)$ for all radius $r_{2}/r_{1}\leq r\leq 1$, i.e.

$\log_{4}\frac{\fint_{Q_{2r}}h^{2}}{\fint_{Q_{r}}h^{2}}\leq
C(n,\lambda)\Lambda.$ (4.25)

###### Proof.

Let $\tilde{r}\equiv r_{2}/r_{1}$ for simplicity. Since
$|N^{u}_{x,0}(r_{1})-N^{u}_{x,0}(r_{2})|\leq\alpha/4$ for $10r_{2}\leq
r_{1}\leq\ell_{0}$, by Theorem 3.30 and 3.48, we have
$D^{\tilde{u}}(s)\in[\gamma,\gamma+\alpha/2]$ for any $s\in[\tilde{r},1/2]$
and for some $\gamma$ if $\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ small.
Now we choose the largest integer $d$ such that $d<\gamma+\alpha$. Hence $d$
satisfies

$\gamma+\alpha-d>0\quad\text{ and }\quad\gamma+\alpha-(d+1)\leq 0.$

With this $d$, we construct the function $\phi$ as follows: For any $(x,t)\in
B_{\eta}\times[-1,0]$, we define

$\phi(x,t)=\int_{Q^{\eta}_{1}}(G(x,t;y,s)-G(0,0;y,s))f(y,s)dyds-\sum_{k=1}^{d}\int_{Q^{\eta}_{1}\setminus
Q^{\eta}_{\tilde{r}}}P^{k}_{y,s}(x,t)f(y,s)dyds$ (4.26)

where $P^{k}_{y,s}(x,t)$ is defined as in last section, see Lemma 4.19. Here
$f=(\partial_{t}-\Delta)\tilde{u}$. Note that $f$ may not be well-defined
pointwisely due to the low regularity of $u$. However we will address this
issue using integral by parts in the proof. We will prove the uniform bound
for $\phi$ provided that only the Hölder continuity of $a^{ij}$ is assumed.
For simplicity, we assume the coefficients $b^{i}=c=0$ without loss of
generality, as the terms involving $b^{i}$ or $c$ would have growth of higher
order.

Set $r=|(x,t)|_{\eta}\equiv\max\\{\eta^{-1}|x|,\sqrt{|t|}\\}$. We have the
following claim about the growth estimates for $\phi(x,t)$:

Claim : If $r\geq\tilde{r}$, we have

$|\phi(x,t)|\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{1}^{\alpha}|(x,t)|^{d+\alpha/2}.$ (4.27)

If $r\leq\tilde{r}$, we have

$|\phi(x,t)|\leq
C(n,\lambda,\alpha,\Lambda,\eta)r_{1}^{\alpha}\tilde{r}^{d+\alpha/2}.$ (4.28)

The claim implies $\phi$ is well-defined and finite. Noting that
$(\partial_{t}-\Delta)\phi(x,t)=f(x,t)$ we have the function
$h\equiv\tilde{u}-\phi$ is the caloric function we want defined in
$Q^{\eta}_{1}$. In the following we prove the first estimate and the second
one can proved in a similar way.

Suppose $(x,t)\in Q^{\eta}_{1}$ and $r=|(x,t)|_{\eta}$. If $r\geq\tilde{r}$,
we can split the integral into the following three parts

$\begin{split}I_{1}&=\int_{Q^{\eta}_{2r}}(G(x,t;y,s)-G(0,0;y,s))f(y,s)dyds\\\
I_{2}&=-\sum_{k=1}^{d}\int_{Q^{\eta}_{2r}\setminus
Q^{\eta}_{\tilde{r}}}P^{k}_{y,s}(x,t)f(y,s)dyds\\\
I_{3}&=\int_{Q^{\eta}_{1}\setminus
Q^{\eta}_{2r}}\Big{(}G(x,t;y,s)-\sum_{k=0}^{d}P^{k}_{y,s}(x,t)\Big{)}f(y,s)dyds.\end{split}$
(4.29)

If $r<\tilde{r}$, we can split the integral as above by replacing $r$ by
$\tilde{r}$. We will only consider the case $r\geq\tilde{r}$. One will see
that the same argument implies the estimate in the claim for $r<\tilde{r}$.

Note that

$f(y,s)=\partial_{s}\tilde{u}-\Delta\tilde{u}=(\partial_{s}\tilde{u}-\Delta\tilde{u})-(\partial_{s}\tilde{u}-\partial_{i}(\tilde{a}^{ij}\partial_{j}\tilde{u}))=\partial_{i}\Big{(}(\tilde{a}^{ij}-\delta^{ij})\partial_{j}\tilde{u}\Big{)}.$

Using integration by parts, we can write

$\displaystyle I_{1}$
$\displaystyle=\int_{Q^{\eta}_{2r}}G(x,t;y,s)f(y,s)dyds-\int_{Q^{\eta}_{2r}}G(0,0;y,s)f(y,s)dyds$
$\displaystyle=-\int_{Q^{\eta}_{2r}}\partial_{y_{i}}G(x,t;y,s)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)dyds+\int_{\partial
B_{2\eta
r}\times[-(2r)^{2},0]}G(x,t;y,s)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)\cdot
n_{i}dyds$
$\displaystyle+\int_{Q^{\eta}_{2r}}\partial_{y_{i}}G(0,0;y,s)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)dyds+\int_{\partial
B_{2\eta
r}\times[-(2r)^{2},0]}G(0,0;y,s)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)\cdot
n_{i}dyds$

where $n_{i}$ is $i$-th component of the spatial outer normal.

Note that

$\displaystyle\quad\int_{Q^{\eta}_{2r}}\partial_{y_{i}}G(x,t;y,s)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{i}}\tilde{u}(y,s)dyds$
$\displaystyle\leq
Cr_{1}^{\alpha}\int_{Q^{\eta}_{2r}}|\nabla_{y}G(x,t;y,s)||(y,s)|^{\alpha}|\nabla\tilde{u}(y,s)|dyds$
$\displaystyle\leq
Cr_{1}^{\alpha}r^{\gamma-1+\alpha}\int_{Q^{\eta}_{2r}}|(x-y,t-s)|_{\eta}^{-n-1}dyds$
$\displaystyle\leq
Cr_{1}^{\alpha}r^{\gamma-1+\alpha}\int_{Q^{\eta}_{3r}}|(y,s)|_{\eta}^{-n-1}dyds\leq
Cr_{1}^{\alpha}r^{\gamma+\alpha}.$

Next we deal with the boundary term.

$\begin{split}&\quad\int_{\partial B_{2\eta
r}\times[-(2r)^{2},0]}G(x,t;y,s)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)\cdot
n_{i}dyds\\\ &\leq Cr_{1}^{\alpha}r^{\gamma+\alpha-1}\int_{\partial B_{2\eta
r}\times[-(2r)^{2},0]}G(x,t;y,s)dyds\leq
Cr_{1}^{\alpha}r^{\gamma+\alpha}.\end{split}$ (4.30)

The remaining two terms of $I_{1}$ could be estimated in the same way.
Therefore we prove that $|I_{1}|\leq Cr_{1}^{\alpha}r^{\gamma+\alpha}.$ Next
we deal with $I_{2}$. Integration by parts gives

$\displaystyle I_{2}=$
$\displaystyle\sum_{k=1}^{d}\int_{Q^{\eta}_{2r}\setminus
Q^{\eta}_{\tilde{r}}}\partial_{y_{i}}(P^{k}_{y,s}(x,t))(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)dyds$
$\displaystyle-\sum_{k=1}^{d}\int_{\partial B_{2\eta
r}\times[-(2r)^{2},0]}(P^{k}_{y,s}(x,t))(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)\cdot
n_{i}dyds$ $\displaystyle+\sum_{k=1}^{d}\int_{\partial
B_{\eta\tilde{r}}\times[-\tilde{r}^{2},0]}(P^{k}_{y,s}(x,t))(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)\cdot
n_{i}dyds.$

For the first term, we have

$\displaystyle\quad\sum_{k=1}^{d}$ $\displaystyle\int_{Q^{\eta}_{2r}\setminus
Q^{\eta}_{\tilde{r}}}\partial_{y_{i}}(P^{k}_{y,s}(x,t))(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)dyds$
$\displaystyle\leq Cr_{1}^{\alpha}\sum_{k=1}^{d}\int_{Q^{\eta}_{2r}\setminus
Q^{\eta}_{\tilde{r}}}|(y,s)|^{\alpha}|\nabla_{y}(P^{k}_{y,s}(x,t)||\nabla\tilde{u}(y,s)|dyds$
$\displaystyle\leq Cr_{1}^{\alpha}\sum_{k=1}^{d}\sum_{i\geq
0,2^{-i+1}r\geq\tilde{r}}\int_{Q^{\eta}_{2^{1-i}r}\setminus
Q^{\eta}_{2^{-i}r}}|(y,s)|^{\alpha}|\nabla_{y}(P^{k}_{y,s}(x,t)||\nabla\tilde{u}(y,s)|dyds$
$\displaystyle\leq Cr_{1}^{\alpha}\sum_{k=1}^{d}r^{k}\sum_{i\geq
0,2^{-i+1}r\geq\tilde{r}}\int_{Q^{\eta}_{2^{1-i}r}\setminus
Q^{\eta}_{2^{-i}r}}|(y,s)|^{-n-k-1+\alpha}|(y,s)|^{\gamma-1}dyds$
$\displaystyle\leq
Cr_{1}^{\alpha}\sum_{k=1}^{d}r^{k}\sum_{i=0}^{\infty}(\frac{r}{2^{i}})^{\gamma+\alpha-k}\leq
Cr_{1}^{\alpha}r^{\gamma+\alpha}\sum_{k=1}^{d}\sum_{i=0}^{\infty}(\frac{1}{2^{\gamma+\alpha-k}})^{i}\leq
Cr_{1}^{\alpha}r^{\gamma+\alpha},$

where the last inequality follows from the fact that $\gamma+\alpha-d>0$. The
two boundary terms could be estimates similar as in (4.30).

Next we deal with $I_{3}$. We have

$\displaystyle I_{3}=$ $\displaystyle-\int_{Q^{\eta}_{1}\setminus
Q^{\eta}_{2r}}\partial_{y_{i}}R^{d}_{y,s}(x,t)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)dyds$
$\displaystyle+\int_{\partial
B_{\eta}\times[-1,0]}R^{d}_{y,s}(x,t)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)\cdot
n_{i}dyds$ $\displaystyle-\int_{\partial B_{2\eta
r}\times[-(2r)^{2},0]}R^{d}_{y,s}(x,t)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)\cdot
n_{i}dyds$

For the first term, by Lemma 4.19,

$\displaystyle\quad\int_{Q^{\eta}_{1}\setminus
Q^{\eta}_{2r}}\partial_{y_{i}}R^{d}_{y,s}(x,t)(\tilde{a}^{ij}(y,s)-\delta^{ij})\partial_{y_{j}}\tilde{u}(y,s)dyds$
$\displaystyle\leq Cr_{1}^{\alpha}\int_{Q^{\eta}_{1}\setminus
Q^{\eta}_{2r}}|\nabla_{y}R^{d}_{y,s}(x,t)||(y,s)|^{\alpha}|\nabla\tilde{u}(y,s)|dyds$
$\displaystyle\leq
Cr_{1}^{\alpha}\sum_{i=1}^{M}\int_{Q^{\eta}_{2^{i+1}r}\setminus
Q^{\eta}_{2^{i}r}}|\nabla_{y}(R^{k}_{y,s}(x,t)||(y,s)|^{\alpha}||\nabla\tilde{u}(y,s)|dyds$
$\displaystyle\leq
Cr_{1}^{\alpha}\sum_{j=d+1}^{2(d+1)}r^{j}\sum_{i=1}^{M}\int_{Q^{\eta}_{2^{i+1}r}\setminus
Q^{\eta}_{2^{i}r}}|(y,s)|^{-n-j-1}\cdot|(y,s)|^{\alpha}|\cdot|(y,s)|^{\gamma-1}dyds$
$\displaystyle\leq
Cr_{1}^{\alpha}r^{\gamma+\alpha}\sum_{j=d+1}^{2(d+1)}\sum_{i=1}^{M}(2^{\gamma+\alpha-j})^{i},$

where $M$ is an integer such that $2^{M}r\leq 1<2^{M+1}r$.

Recall that $\gamma+\alpha-(d+1)\leq 0$. If $\gamma+\alpha<d+1$, we have

$\sum_{j=d+1}^{2(d+1)}\sum_{i=1}^{M}(2^{\gamma+\alpha-j})^{i}\leq
C(\alpha,\Lambda).$

If $\gamma+\alpha=d+1$, then we have

$\displaystyle\sum_{j=d+1}^{2(d+1)}\sum_{i=1}^{M}(2^{\gamma+\alpha-j})^{i}=M+\sum_{j=d+2}^{2(d+1)}\sum_{i=1}^{\infty}(2^{d+1-j})^{i}\leq\log_{2}(1/r)+C(\alpha,\Lambda).$

To handle the boundary terms is similar. Therefore, we have the following
estimate for $I_{3}$,

$I_{3}\leq Cr_{1}^{\alpha}r^{\gamma+\frac{\alpha}{2}}.$

Therefore, we have proved that

$|\phi(x,t)|\leq Cr_{1}^{\alpha}r^{\gamma+\alpha/2}.$ (4.31)

For $|(x,t)|_{\eta}=r\leq\tilde{r}$ the same argument as above by replacing
$r$ by $\tilde{r}$ we get

$|\phi(x,t)|\leq Cr_{1}^{\alpha}\tilde{r}^{\gamma+\alpha/2}.$ (4.32)

Hence we finish the proof of the claim.

Noting that $Q_{r}^{\eta}=\\{(x,t):|(x,t)|_{\eta}\leq r\\}$, as a direct
consequence of the claim, we get for any $r\in[\tilde{r},1]$ that

$\displaystyle\sup_{Q_{r}^{\eta}}|\phi(x,t)|\leq
Cr_{1}^{\alpha}r^{\gamma+\alpha/2}.$ (4.33)

On the other hand, Since $D^{\tilde{u}}(r)\leq\gamma+\alpha/2$ for
$r\in[\tilde{r},1/2]$ and by Lemma 3.51, we have

$H^{\tilde{u}}(r)\geq Cr^{2\gamma+\alpha}H^{\tilde{u}}(1)\geq
Cr^{2\gamma+\alpha}.$

If we pick $\ell_{0}(n,\alpha,\lambda,\Lambda,\eta,\epsilon)$ small enough and
note that $r_{1}\leq\ell_{0}$, then for any $r\in[\tilde{r},1]$ we have

$\sup_{Q_{r}^{\eta}}|\phi(x,t)|\leq Cr_{1}^{\alpha}r^{\gamma+\alpha/2}\leq
Cr_{1}^{\alpha}(H^{\tilde{u}}(r))^{1/2}\leq\epsilon^{2}(H^{\tilde{u}}(r))^{1/2}.$
(4.34)

Also, by Lemma 3.51 we have

$\sup_{Q_{r}^{\eta}}|\phi(x,t)|\leq
Cr_{1}^{\alpha}(H^{\tilde{u}}(r))^{1/2}\leq\epsilon^{2}(\fint_{Q_{r}}\tilde{u}^{2})^{1/2}.$

Therefore, for any $r\in[\tilde{r},1]$ we have

$\displaystyle\frac{\fint_{Q_{2r}}h^{2}}{\fint_{Q_{r}}h^{2}}\leq
100\frac{\fint_{Q_{2r}}\tilde{u}^{2}+\fint_{Q_{2r}}\phi^{2}}{\fint_{Q_{r}}\tilde{u}^{2}-\fint_{Q_{r}}\phi^{2}}\leq
100\frac{(1+C\epsilon^{2})\fint_{Q_{2r}}\tilde{u}^{2}}{(1-C\epsilon^{2})\fint_{Q_{r}}\tilde{u}^{2}}.$

This proves the doubling bound for $h$. ∎

Next we will prove the frequency closeness between $\tilde{u}$ and the
approximated caloric function $h$. Note that the definition of $H^{\tilde{u}}$
requires the integral over the ball $B_{1/r_{1}}$. However we only have the
approximation over the ball $B_{\eta}$. The following two lemmas show that
there is no essential difference between these two integrals.

######  Lemma 4.35.

Let $P$ be a caloric polynomial of order at most $d$. Then for any $\epsilon$,
there exists some $\eta_{0}(n,d,\epsilon)>0$ such that for any $\eta>\eta_{0}$
and $s>0$

$\int_{\\{\mathbb{R}^{n}\setminus B_{\eta
s}\\}\cap\\{t=-s^{2}\\}}P^{2}G_{0,0}\leq\epsilon\int_{\\{t=-s^{2}\\}}P^{2}G_{0,0}.$

###### Proof.

First we suppose $P$ be a caloric homogeneous polynomial of order $d$. By
change of variable formula, we have

$\int_{\\{\mathbb{R}^{n}\setminus B_{\eta
s}\\}\cap\\{t=-s^{2}\\}}P^{2}G_{0,0}=\frac{s^{2d}}{(4\pi)^{n/2}}\int_{\\{\mathbb{R}^{n}\setminus
B_{\eta}\\}\cap\\{t=-1\\}}P^{2}(x,-1)\exp(-\frac{|x|^{2}}{4})dx.$

Similarly, we have

$\int_{\\{t=-s^{2}\\}}P^{2}G_{0,0}=\frac{s^{2d}}{(4\pi)^{n/2}}\int_{\\{t=-1\\}}P^{2}(x,-1)\exp(-\frac{|x|^{2}}{4})dx.$

By normalization we may assume

$\int_{\\{t=-1\\}}P^{2}(x,-1)\exp(-\frac{|x|^{2}}{4})dx=1.$

By lemma 3.51 and standard estimate, we have

$|P(x,-1)|\leq C(n,d)(|x|^{d}+1).$

Therefore, by taking $\eta$ large enough we have

$\int_{\\{\mathbb{R}^{n}\setminus
B_{\eta}\\}\cap\\{t=-1\\}}P^{2}(x,-1)\exp(-\frac{|x|^{2}}{4})dx\leq
C^{\prime}(n,d)\int_{\eta}^{\infty}r^{d+n-1}\exp(-\frac{r^{2}}{4})dr\leq\epsilon.$

This finishes the proof for caloric homogeneous polynomials.

For general case, we can write $P=\sum_{i=0}^{d}P_{i}$ with each $P_{i}$
homogeneous polynomial of order $i$. Then by lemma 2.13, we have

$\int_{\\{t=-s^{2}\\}}P^{2}G_{0,0}=\sum_{i}\int_{\\{t=-s^{2}\\}}P_{i}^{2}G_{0,0}.$

And Hölder inequality gives

$\int_{\\{\mathbb{R}^{n}\setminus B_{\eta
s}\\}\cap\\{t=-s^{2}\\}}P^{2}G_{0,0}\leq(d+1)\sum_{i}\int_{\\{\mathbb{R}^{n}\setminus
B_{\eta s}\\}\cap\\{t=-s^{2}\\}}P_{i}^{2}G_{0,0}.$

The proof is now finished using homogeneous case. ∎

The polynomial case will be used in the proof of the following general case.

######  Lemma 4.36.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $x\in B_{1}$. Then for any $\epsilon>0$ there exists some
$\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ and
$\eta_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ such that for any
$\ell\leq\ell_{0}$ and $\eta\geq\eta_{0}$ with $\tilde{u}=u_{x,0;\ell}$ and
any $0\leq r\leq 1$

$\int_{\\{B_{1/\ell}\setminus B_{\eta
r}\\}\cap\\{t=-r^{2}\\}}\tilde{u}^{2}G_{0,0}\leq\epsilon\int_{\\{B_{1/\ell}\\}\cap\\{t=-r^{2}\\}}\tilde{u}^{2}G_{0,0}\equiv\epsilon
H^{\tilde{u}}(r).$

In particular, for $r\in(0,1]$ we have

$\Big{|}\frac{\int_{\\{B_{\eta
r}\\}\cap\\{t=-r^{2}\\}}\tilde{u}^{2}G_{0,0}}{H^{\tilde{u}}(r)}-1\Big{|}\leq\epsilon.$

###### Proof.

We prove this by contradiction. Suppose the result be false with some
$\epsilon_{0}>0$. Then we can find a sequence $\tilde{u}_{i}\equiv
u_{x,0;1/i}$ with $i\to\infty$ and each $\tilde{u}_{i}$ has polynomial growth
of order $C(n,\lambda,\alpha)\Lambda$. By Lemma 3.15, there exists some
caloric polynomial $P$ of order $C(n,\lambda,\alpha)\Lambda$ such that
$||\tilde{u}_{i}-P||_{C^{1;1}(\Omega)}\to 0$ in any compact $\Omega\subset
Q_{1/i}$.

Take $\eta(n,\lambda,\alpha,\Lambda,\epsilon_{0})$ as in Lemma 4.35 with order
$d=C(n,\lambda,\alpha)\Lambda$. Then we have

$\int_{\\{\mathbb{R}^{n}\setminus B_{\eta
s}\\}\cap\\{t=-s^{2}\\}}P^{2}G_{0,0}\leq\frac{\epsilon_{0}}{2}\int_{\\{t=-s^{2}\\}}P^{2}G_{0,0}.$

Since $\tilde{u}_{i}$ and $P$ has polynomial growth, then as $i\to\infty$ we
have

$\int_{\\{B_{i}\\}\cap\\{t=-s^{2}\\}}\tilde{u}^{2}_{i}G_{0,0}\to\int_{\\{t=-s^{2}\\}}P^{2}G_{0,0}\quad\text{
and }\quad\int_{\\{B_{\eta
s}\\}\cap\\{t=-s^{2}\\}}\tilde{u}^{2}_{i}G_{0,0}\to\int_{\\{B_{\eta
s}\\}\cap\\{t=-s^{2}\\}}P^{2}G_{0,0}.$

This proves that for large $i$

$\int_{\\{B_{i}\setminus B_{\eta
s}\\}\cap\\{t=-s^{2}\\}}\tilde{u}_{i}^{2}G_{0,0}\leq\epsilon_{0}\int_{\\{t=-s^{2}\\}}\tilde{u}_{i}^{2}G_{0,0}.$

The contradiction arises. ∎

With the lemma above, we can prove that the closeness of frequency between
$\tilde{u}$ and $h$ in Theorem 4.23.

######  Proposition 4.37.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $x\in B_{1}$ and $\epsilon\leq\alpha/10$. There exists some
$\eta(n,\lambda,\alpha,\Lambda,\epsilon)$ and
$\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ such that if
$|N^{u}_{x,0}(r_{1})-N^{u}_{x,0}(r_{2})|\leq\alpha/4$ for $10r_{2}\leq
r_{1}\leq\ell_{0}$, let $h$ be the approximated caloric function as in Theorem
4.23. Then we have the following frequency closeness

$|N^{\tilde{u},1/r_{1}}(s)-N^{h,\eta}(s)|\leq\epsilon.$ (4.38)

###### Proof.

By Theorem 4.23, for any $r\in[\tilde{r},1]$ we have

$\sup_{Q_{r}^{\eta}}|\tilde{u}(y,s)-h(y,s)|\leq\epsilon^{2}(H^{\tilde{u}}(r))^{1/2}.$

Hence for any $r\in[\tilde{r},1]$, we have

$\int_{B_{\eta
r}\cap\\{t=-r^{2}\\}}|\tilde{u}-h|^{2}G_{0,0}\leq\epsilon^{4}H^{\tilde{u}}(r)$

and

$\int_{B_{\eta r}\cap\\{t=-r^{2}\\}}|\tilde{u}+h|^{2}G_{0,0}\leq
10H^{\tilde{u}}(r).$

Therefore we have

$\int_{B_{\eta
r}\cap\\{t=-r^{2}\\}}|\tilde{u}^{2}-h^{2}|G_{0,0}\leq\int_{\\{B_{\eta
r}\\}\cap\\{t=-r^{2}\\}}\frac{|\tilde{u}-h|^{2}}{\epsilon^{2}}G_{0,0}+\int_{B_{\eta
r}\cap\\{t=-r^{2}\\}}\frac{\epsilon^{2}|\tilde{u}+h|^{2}}{4}G_{0,0}\leq
4\epsilon^{2}H^{\tilde{u}}(r).$

This proves that

$\int_{B_{\eta
r}\cap\\{t=-r^{2}\\}}\tilde{u}^{2}G_{0,0}-4\epsilon^{2}H^{\tilde{u}}(r)\leq\int_{B_{\eta
r}\cap\\{t=-r^{2}\\}}h^{2}G_{0,0}\leq\int_{B_{\eta
r}\cap\\{t=-r^{2}\\}}\tilde{u}^{2}G_{0,0}+4\epsilon H^{\tilde{u}}(r).$

Now we take $\eta(n,\lambda,\alpha,\Lambda,\epsilon)$ as in Lemma 4.36. We
have

$(1-5\epsilon^{2})H^{\tilde{u}}(s)\leq\int_{B_{\eta
s}\cap\\{t=-s^{2}\\}}h^{2}G_{0,0}\leq(1+5\epsilon^{2})H^{\tilde{u}}(s).$

One can obtain the same estimates for gradient terms $|\nabla h|$. This proves
the closeness of frequency. ∎

Combining with Theorem 2.21, we prove that $u$ is uniformly almost symmetric
when frequency is pinched.

######  Theorem 4.39.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $x\in B_{1}$ and $\epsilon\leq c(n,\lambda,\alpha,\Lambda)$. Then
there exist some $\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon)$ and
$\delta(n,\lambda,\alpha,\Lambda,\epsilon)$ such that if
$|N^{u}_{x,0}(r_{1})-N^{u}_{x,0}(r_{2})|\leq\delta$ for
$10\delta^{-1}r_{2}\leq r_{1}\leq\ell_{0}$, there exists some integer $d$ such
that:

  1. (1)

For any $r\in[r_{2},r_{1}]$ we have $|N^{u}_{x,0}(r)-d|\leq\epsilon$.

  2. (2)

There exists some homogeneous caloric polynomial $P_{d}$ with normalization
$\fint_{Q_{1}}|P_{d}|^{2}=1$ such that for any $r\in[r_{2},\delta r_{1}]$

$\sup_{Q_{1}}|u_{x,0;r}-P_{d}|\leq\epsilon.$

###### Proof.

The first conclusion comes from Lemma 3.36. We prove the second conclusion.
Let $\tilde{u}=u_{x,0;r_{1}}$. Denote $\hat{r}=r/r_{1}$. Let $h$ be the
approximated caloric function as in Theorem 4.23. By Proposition 4.37, we can
apply Theorem 2.21 to find a homogeneous caloric polynomial $P_{d}$ of order
$d$ such that for any $r\in[2r_{2},\delta r_{1}]$ such that

$\displaystyle\sup_{Q_{1}}|h_{0,0;\hat{r}}-P_{d}|\leq\epsilon.$ (4.40)

On the other hand, by the scaling property of rescaled maps we have

$u_{x,0;r}(y,s)=\frac{\tilde{u}(\hat{r}z,\hat{r}^{2}s)}{(\fint_{Q_{\hat{r}}}\tilde{u}^{2})^{1/2}}\quad\text{
and }\quad
h_{0,0;r}(y,s)=\frac{h(\hat{r}y,\hat{r}^{2}s)}{(\fint_{Q_{\hat{r}}}h^{2})^{1/2}}.$

By Theorem 4.23, we have

$\Big{|}\fint_{Q_{\hat{r}}}\tilde{u}^{2}-\fint_{Q_{\hat{r}}}h^{2}\Big{|}\leq
C\epsilon^{2}\fint_{Q_{\hat{r}}}\tilde{u}^{2}.$

Hence

$\displaystyle\quad\sup_{Q_{1}}|u_{x,0;r}-h_{0,0;\hat{r}}|=\sup_{Q_{\hat{r}}}|\frac{\tilde{u}}{(\fint_{Q_{\hat{r}}}\tilde{u}^{2})^{1/2}}-\frac{h}{(\fint_{Q_{\hat{r}}}h^{2})^{1/2}}|$
$\displaystyle\leq\sup_{Q_{\hat{r}}}|\frac{\tilde{u}}{(\fint_{Q_{\hat{r}}}\tilde{u}^{2})^{1/2}}-\frac{h}{(\fint_{Q_{\hat{r}}}\tilde{u}^{2})^{1/2}}|+|\frac{h}{(\fint_{Q_{\hat{r}}}\tilde{u}^{2})^{1/2}}-\frac{h}{(\fint_{Q_{\hat{r}}}h^{2})^{1/2}}|$
$\displaystyle\leq C\epsilon^{2}.$

By triangle inequality, now we have

$\sup_{Q_{1}}|u_{x,0;r}-P_{d}|\leq\sup_{Q_{1}}|u_{x,0;r}-h_{0,0;\hat{r}}|+|h_{0,0;\hat{r}}-P_{d}|\leq\epsilon.$

The proof is now finished. ∎

##  5\. Cone-Splitting in a Fixed Time Slice

Recall that we say a continuous function $u$ is uniformly
$(k,\eta,x)$-symmetric in $[r_{2},r_{1}]$ if there exists a $k$-symmetric
homogeneous polynomial $P$ with $\fint_{B_{1}}|P|^{2}=1$ such that for any
$r\in[r_{2},r_{1}]$

$\sup_{B_{1}}|u_{x,r}-P|\leq\eta.$ (5.1)

The main purpose in this section is the following cone-splitting principle in
the time $t$-slice.

######  Theorem 5.2 (Quantitative Cone-Splitting).

Fix $\tau<c(n,\lambda)$. Let $\eta>0$ be given. Let $u$ be a solution to (3.1)
(3.2) on $Q_{2}$ with doubling assumption (3.3). Let $x\in B_{1}$. Then there
exist some $r_{0}(n,\lambda,\alpha,\Lambda,\eta)$ and
$\epsilon(n,\lambda,\alpha,\Lambda,\eta)$ such that if

  1. (1)

$u(\cdot,0)$ is uniformly $(k,\epsilon,x)$-symmetric in $[r_{2},r_{1}]$ with
respect to $k$-dimensional subspace $V$ with $10r_{2}\leq r_{1}\leq r_{0}$,

  2. (2)

$u(\cdot,0)$ is $(0,\epsilon,r,y)$-symmetric for some $y\in
B_{c(n,\lambda)r}(x)\setminus B_{\tau r}(x+V)$ and some $r\in[r_{2},r_{1}]$,

then $u(\cdot,0)$ is uniformly $(k+1,\eta,x)$-symmetric in $[r_{2},r_{1}]$.

###### Proof.

Denote $f(y)=u(y,0)$ By the condition (1), there exists a $k$-symmetric
homogeneous polynomial $P$ with $\fint_{B_{1}}|P|^{2}=1$ such that for all
$r\in[r_{2},r_{1}]$

$\displaystyle\sup_{B_{1}}|f_{x,r}-P|\leq\epsilon,$ (5.3)

and by condition (2), there exists a homogeneous polynomial $\tilde{P}$ with
$\fint_{B_{1}}|\tilde{P}|^{2}=1$ such that

$\displaystyle\sup_{B_{1}}|f_{y,r}-\tilde{P}|\leq\epsilon,$ (5.4)

where $y\in B_{c(n,\lambda)r}(x)\setminus B_{\tau r}(x+V)$ and some
$r\in[r_{2},r_{1}]$. To prove the theorem, it suffices to show that there
exist a $(k+1)$-symmetric homogeneous polynomial $\hat{P}$ with
$\fint_{B_{1}}|\hat{P}|^{2}=1$ such that if
$\epsilon\leq\epsilon(n,\lambda,\alpha,\Lambda,\eta)$ we have

$\displaystyle\sup_{B_{1}}|P-\hat{P}|\leq\eta.$ (5.5)

To see this, by scaling, without loss of generality we assume (2) holds with
$r=1$. By (5.3) and (5.4) we have for some $a,b>0$ that

$\displaystyle\sup_{z\in B_{1}(x)\cap
B_{1}(y)}|aP(z-x)-b\tilde{P}(z-y)|\leq\epsilon(a+b).$ (5.6)

In particular, for $\tilde{a}=a/(a+b)$ and $\tilde{b}=b/(a+b)$ we have

$\displaystyle\sup_{z\in B_{1}(x)\cap
B_{1}(y)}|\tilde{a}P(z-x)-\tilde{b}\tilde{P}(z-y)|\leq\epsilon$ (5.7)

and $\tilde{a}+\tilde{b}=1$. By contradiction argument, to get (5.5) it
suffices to prove $P$ is $(k+1)$-symmetric when

$\displaystyle\sup_{z\in B_{1}(x)\cap
B_{1}(y)}|\tilde{a}P(z-x)-\tilde{b}\tilde{P}(z-y)|=0$ (5.8)

and $y-x\notin V$ and $0<d(x,y)<1$. Since $P$ and $\tilde{P}$ are polynomial,
by (5.8) we get that

$\displaystyle\tilde{a}P(z-x)=\tilde{b}\tilde{P}(z-y)$ (5.9)

Noting that $\tilde{a}+\tilde{b}=1$, we have $\tilde{a},\tilde{b}>0.$
Furthermore, by $0$-homogeneous of $P$ and $\tilde{P}$ it is easy to show that
$P(z+(y-x))=P(z)$ for all $z$ which means $P$ is $(k+1)$-symmetric with
respect to $\tilde{V}={\rm span}\\{V,y-x\\}$. Hence we finish the whole proof.
∎

Next we define the set of points where the frequency of the parabolic function
$u$ is pinched as

$\mathcal{V}^{u}_{\delta,d,r}(x,0)\equiv\\{y\in
B_{r}(x):|N_{y,0}^{u}(s)-d|\leq\delta\text{ for any
}s\in[10^{-2}r,\delta^{-1}r]\\}$ (5.10)

######  Definition 5.11 ($(k,\tau)$-independent).

We say a subset $S\subset B_{r}\subset\mathbb{R}^{n}$ is
$(k,\tau)$-independent in $B_{r}$ if for any affine $(k-1)$-plane
$L\subset\mathbb{R}^{n}$ there exists some point $x\in S$ such that
$d(x,L)\geq\tau r$.

The following is the generalization of Theorem 4.39.

######  Proposition 5.12.

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $x\in B_{1}$ and
$\epsilon\leq\epsilon_{0}(n,\lambda,\alpha,\Lambda)$, $\tau>0$. Then there
exist some $\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon,\tau)$ and
$\delta(n,\lambda,\alpha,\Lambda,\epsilon,\tau)$ such that if
$\mathcal{V}^{u}_{\delta,d,r}(x,0)$ is $(k,\tau)$-independent in $B_{r}(x)$
for some $r\leq\ell_{0}$, then $u(\cdot,0)$ is uniformly
$(k,\epsilon,y)$-symmetric in $[r/10,10r]$ for any
$y\in\mathcal{V}^{u}_{\delta,d,r}(x,0)$.

###### Proof.

This follows from Theorem 4.4 and Quantitative Cone-splitting Theorem 5.2. ∎

##  6\. Neck Structure and Neck Decomposition

In this section, we will introduce the neck region and prove the neck
decomposition. This was first introduced in [34, 49] to study the structure of
Einstein manifolds and Yang-Mills equations. Such kind of technique turns out
to be very useful to many interesting topics, see for instance [6, 7, 50]. The
main purpose in this section is the following theorem. Based on Proposition
5.12, the proof of this result is quite standard now(see also [33]).

######  Theorem 6.1 (Neck Decomposition).

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). For each $\eta>0$ and
$\delta\leq\delta_{0}(n,\lambda,\alpha,\Lambda,\eta)$ we have

$\begin{split}&B_{1}\subset\bigcup_{a}(\mathcal{N}^{a}\cap
B_{r_{a}}(x_{a})\big{)}\cup\bigcup_{b}B_{r_{b}}(x_{b})\cup\Big{(}S_{0}\cup\bigcup_{a}\mathcal{C}_{0,a}\Big{)}\end{split}$
(6.2)

such that

  * •

$u(\cdot,0)$ is $(k+1,\eta,2r_{b},x_{b})$-symmetric.

  * •

$\mathcal{N}^{a}\subset B_{r_{a}}(x_{a})$ is $(d_{a},k,\delta,\eta)$-neck
region at $0$-slice for some integer $d_{a}\leq C(n)\Lambda.$

  * •

$\sum_{a}r^{k}_{a}+\sum_{b}r^{k}_{b}+\mathcal{H}^{k}\Big{(}S_{0}\cup\bigcup_{a}\mathcal{C}_{0,a}\Big{)}\leq
C(n,\lambda,\alpha,\Lambda,\epsilon,\eta).$

  * •

$S_{0}\cup\bigcup_{a}\mathcal{C}_{0,a}$ is $k$-rectifiable and
$\mathcal{H}^{k}(S_{0})=0$.

Moreover, for any $\epsilon>0$, if
$\eta<\eta(n,\lambda,\alpha,\Lambda,\epsilon)$ and
$\delta<\delta(n,\lambda,\alpha,\Lambda,\eta,\epsilon)$, the quantitative
stratum $\mathcal{S}^{k}_{\epsilon}$ satisfies

$\mathcal{S}^{k}_{\epsilon}\cap
B_{1}\subset\Big{(}S_{0}\cup\bigcup_{a}\mathcal{C}_{0,a}\Big{)}.$ (6.3)

###  6.1. Neck Regions

We construct the neck regions at $t$-slice. This is essentially equivalent to
the neck regions in Euclidean space as defined in [33]. Throughout this
section, the time slice $t=0$ is fixed.

######  Definition 6.4.

(Neck Region) Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling
assumption (3.3). Fix $t=0$. Let $\mathcal{C}\subset B_{r}$ a closed subset
and $r_{x}:\mathcal{C}\to\mathbb{R}_{\geq 0}$ a radius function such that the
closed balls {$\bar{B}_{r_{x}/5}(x)$} are disjoint. The subset
$\mathcal{N}=B_{r}\setminus\bar{B}_{r_{x}}(\mathcal{C})$ is called a
$(d,k,\delta,\eta)$-neck region at $0$-slice  if

  1. (1)

For any $x\in\mathcal{C}$, we have $|N_{x,0}(s)-d|\leq\delta$ for any
$s\in[r_{x},\delta^{-1}r]$ with $r_{x}\leq 10^{-2}r$.

  2. (2)

For any $x\in\mathcal{C}$, $u(\cdot,0)$ is uniformly $(k,\delta,x)$-symmetric
in $[r_{x},10r]$ with respect to $V_{x}$ but $u$ is not
$(k+1,\eta,s,x)$-symmetric for any $s\in[r_{x},10r]$.

  3. (3)

For any $B_{2s}(x)\subset B_{2r}$ with $s\geq r_{x}$, we have
$x+V_{x}\subset\cup_{y\in\mathcal{C}\cap B_{s}(x)}B_{10^{-10}(s+r_{y})}(y)$.

######  Remark 6.5.

In the neck region definition we can assume $V_{x}$ is unique for all scale
$s\in[r_{x},10r]$, the reason is the quantitative uniqueness of tangent map in
Theorem 4.4.

######  Remark 6.6.

By Corollary 3.37 and Cone-splitting Theorem 5.2, for any $y\in\mathcal{N}$
with $d(y,V_{x})=s$ for some $V_{x}$, we have that $B_{\hat{s}}(y)$ is
$(k+1,\eta)$-symmetric if $\delta\leq\delta(\eta,n)$ for some
$c(\eta)s\leq\hat{s}\leq s$. In particular, $y\notin\mathcal{S}^{k}_{\eta}$

Denote $\mathcal{C}_{+}=\\{x\in\mathcal{C}:r_{x}>0\\}$ and
$\mathcal{C}_{0}=\\{x\in\mathcal{C}:r_{x}=0\\}$. The following neck structure
theorem follows directly by the quantitative uniqueness of tangent map.

######  Theorem 6.7 (Neck Structure Theorem).

Let $\mathcal{N}=B_{2r}\setminus\bar{B}_{r_{x}}(\mathcal{C})$ be a
$(d,k,\delta,\eta)$-neck region at $0$-slice. For any $\epsilon>0$ if
$\delta\leq\delta(n,\eta,k,\epsilon,\Lambda,\lambda,\alpha)$ then

  1. (1)

Fix $x_{0}\in\mathcal{C}$ and let $V=V_{x_{0}}$. Then the projection map
$\pi:\mathcal{C}\to V$ is a bilipschitz map,

$\displaystyle(1-\epsilon)||x-y||\leq||\pi(x)-\pi(y)||\leq||x-y||$ (6.8)

  2. (2)

The measure bound holds

$\sum_{x\in\mathcal{C}_{+}}r_{x}^{k}+\mathcal{H}^{k}({\mathcal{C}_{0}})\leq
C(n)r^{k}$

###### Proof.

To prove (1), noting that $\pi$ is a projection, the upper bound
$||\pi(x)-\pi(y)||\leq||x-y||$ holds trivially. To see the lower bound, for
any given $x,y\in\mathcal{C}$ assume $s=d(x,y)$. Denote
$\pi_{x}:\mathcal{C}\to V_{x}$ is the projection to $V_{x}$. Since
$u(\cdot,0)$ is $(k,\delta,s,x)$-symmetric with respect to $V_{x}$ and $u$ is
$(k,\delta,s,y)$-symmetric, but $u$ is not $(k+1,\eta,2s,x)$-symmetric, for
any $\epsilon$ by Theorem 5.2 if $\delta\leq\delta(\epsilon,\eta)$ we have
that $x-y\in B_{\epsilon s}(V_{x})$. In particular we get

$\displaystyle||\pi_{x}(x-y)||\geq(1-\epsilon^{2})||y-x||.$ (6.9)

Similar argument implies $d_{H}(V_{x}\cap B_{1},V_{x_{0}}\cap
B_{1})\leq\epsilon^{2}$ if $\delta\leq\delta(\epsilon,\eta)$ Hence we have

$\displaystyle||\pi_{x}(x-y)-\pi_{x_{0}}(x-y)||\leq\epsilon^{2}||x-y||.$
(6.10)

Combining the above estimates we get $||\pi(x-y)||\geq(1-\epsilon)||x-y||.$
Hence this proves (1).

To prove (2), noting that $\\{B_{r_{x}/5}(x):x\in\mathcal{C}\\}$ are pairwise
disjoint, by (1) then $\\{B_{r_{x}/10}(\pi(x)):x\in\mathcal{C}\\}$ are
pairwise disjoint. Thus

$\displaystyle\sum_{x\in\mathcal{C}_{+}}r_{x}^{k}\leq
C(n)\sum_{x\in\mathcal{C}_{+}}{\rm Vol}_{V}(B_{r_{x}/10}(\pi(x))\leq
C(n)r^{k},$ (6.11)

where ${\rm Vol}_{V}$ is the $k$-Hausdorff measure in $V$. Since $\pi$ is
bilipschitz, we have $\mathcal{H}^{k}(\mathcal{C}_{0})\leq
C(n)\mathcal{H}^{k}(\pi(\mathcal{C}_{0}))\leq C(n)r^{k}.$ Thus we get the
upper bound estimate.∎

###  6.2. Neck Decomposition

In this subsection we prove the decomposition theorem.

For each $x\in B_{1}$, we will decompose $B_{r}(x)$ into the balls in the
following categories.

  1. (a)

A ball $B_{r_{a}}(x_{a})$ is associated with a $(d,k,\epsilon,\tau,\eta)$-neck
region at $0$-slice
$\mathcal{N}_{a}=B_{2r_{a}}(x_{a})\setminus\bar{B}_{r_{a,x}}(\mathcal{C}_{a})$.

  2. (b)

A ball $B_{r_{b}}(x_{b})$ is such that $u$ is
$(k+1,2\eta,2r_{b},x_{b})$-symmetric.

  3. (c)

A ball $B_{r_{c}}(x_{c})$ is not a $b$-ball and
$\mathcal{V}_{\delta,d,r_{c}}(x_{c},0)$ is $(k,\tau)$-independent in
$B_{r_{c}}(x_{c})$.

  4. (d)

A ball $B_{r_{d}}(x_{d})$ is such that
$\mathcal{V}_{\delta,d,r_{d}}(x_{d},0)\neq\emptyset$ and is not
$(k,\tau)$-independent in $B_{r_{d}}(x_{d})$.

  5. (e)

A ball $B_{r_{e}}(x_{e})$ is such that
$\mathcal{V}_{\delta,d,r_{e}}(x_{e},0)=\emptyset$.

Next we prove the decomposition for $c$-balls.

######  Proposition 6.12.

(Covering of $c$-balls) Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with
doubling assumption (3.3). Let $\eta,\tau>0$. Suppose
$\delta\leq\delta_{0}(n,\lambda,\alpha,\Lambda,\eta,\tau)$ and
$r\leq\ell_{0}(n,\lambda,\alpha,\Lambda,\epsilon,\eta,\tau)$. Suppose
$\sup_{x\in B_{r}}N_{x,0}(s)\leq d+\delta$ for any $s\leq
C(n,\lambda)\delta^{-1}r$ and for some integer $d$. Furthermore we assume that
$\mathcal{V}_{\delta,d,r}(0,0)$ is $(k,\tau)$-independent in $B_{r}$ and that
$B_{2r}$ is not $(k+1,2\eta)$-symmetric. Then we can decompose

$B_{r}\subset\big{(}\mathcal{C}_{0}\cup\mathcal{N})\cup\bigcup_{b\in
B}B_{r_{b}}(x_{b})\cup\bigcup_{d\in D}B_{r_{d}}(x_{d})\cup\bigcup_{e\in
E}B_{r_{e}}(x_{e}),$ (6.13)

where $\mathcal{N}$ is a $(d,k,\epsilon,\eta)$-neck region at $0$-slice.
Furthermore, we have the estimates

$\sum_{b\in B}r_{b}^{k}+\sum_{d\in D}r_{d}^{k}+\sum_{b\in
E}r_{e}^{k}+\mathcal{H}^{k}(\mathcal{C}_{0})\leq C(n)r^{k}.$ (6.14)

###### Proof.

Since $\mathcal{V}_{\delta,d,r}$ is $(k,\tau)$-independent in $B_{r}$, for any
$\epsilon>0$ if $\delta$ and $\ell_{0}$ are sufficiently small, by Proposition
5.12 we have $u$ is $(k,\epsilon,y)$-symmetric in $[r/10,10r]$ with respect to
$k$-plane $V_{y}$ for any $y\in\mathcal{V}_{\delta,d,r}$. We also have
$\mathcal{V}_{\delta,d,r}\subset B_{\epsilon r}(y+V_{y})$ by noting that
$B_{2r}$ is not $(k+1,2\eta)$-symmetric. Fix
$y_{0}\in\mathcal{V}_{\delta,d,r}$, consider a Vitali subcovering of
$\bigcup_{x\in y_{0}+V_{y_{0}}\cap B_{r}}B_{r/10^{2}}(x)$. Denote by
$\\{B_{r/10^{2}}(x_{i}^{1}),i=1,\cdots,L_{1}\\}$. Then
$\mathcal{N}^{1}:=B_{r}\setminus\cup_{i=1}^{L_{1}}B_{r/10^{2}}(x_{i}^{1})$ is
$(d,k,\epsilon,\eta)$-neck region. Note that each ball must be one of the $b$,
$c$, $d$ or $e$-type, we have a covering

$B_{r}\subset\mathcal{N}^{1}\cup\bigcup_{b\in
B^{1}}B_{r_{b}}(x_{b})\cup\bigcup_{c\in
C^{1}}B_{r_{c}}(x_{c})\cup\bigcup_{d\in
D^{1}}B_{r_{d}}(x_{d})\cup\bigcup_{e\in E^{1}}B_{r_{e}}(x_{e}).$ (6.15)

Now for each $c$-ball in $C^{1}$ group, we have $r_{c}=10^{-2}r$. We apply the
same decomposition to obtain the refined covering

$B_{r}\subset\mathcal{N}^{2}\cup\bigcup_{b\in
B^{2}}B_{r_{b}}(x_{b})\cup\bigcup_{c\in
C^{2}}B_{r_{c}}(x_{c})\cup\bigcup_{d\in
D^{2}}B_{r_{d}}(x_{d})\cup\bigcup_{e\in E^{2}}B_{r_{e}}(x_{e}).$ (6.16)

with $r_{c}=10^{-4}r$.

By iterating this decomposition, finally we will obtain a covering

$B_{r}\subset\big{(}\mathcal{C}_{0}\cup\mathcal{N})\cup\bigcup_{b\in
B}B_{r_{b}}(x_{b})\cup\bigcup_{d\in D}B_{r_{d}}(x_{d})\cup\bigcup_{e\in
E}B_{r_{e}}(x_{e}),$ (6.17)

where $\mathcal{C}_{0}$ is the limit of $\\{x_{c},c\in C^{i}\\}$ when
$i\to\infty$. We need to prove that $\mathcal{N}$ is indeed a neck region. The
property 1 comes from our construction trivially. Property 2 and 3 follow from
the Quantitative Cone-splitting Theorem 5.2.

By Neck structure Theorem 6.7, we have

$\sum_{b\in B}r_{b}^{k}+\sum_{d\in D}r_{d}^{k}+\sum_{b\in
E}r_{e}^{k}+\mathcal{H}^{k}(\mathcal{C}_{0})\leq C(n)r^{k}.$ (6.18)

The proof is now finished. ∎

######  Remark 6.19.

The point here is that the approximated plane $V_{x}$ does not depend on the
scale $r$. This essentially comes from the uniqueness of tangent maps 4.39 and
significantly simplifies the construction of neck regions out of $c$-balls.
Similar uniqueness has been used in the elliptic case [33]. See [7] for
general constructions.

The covering for $d$-balls is standard. For completeness we include the proof
here. See also [7] or [33].

######  Proposition 6.20 (Covering of $d$-balls).

Let $u$ be a solution to (3.1) (3.2) on $Q_{2}$ with doubling assumption
(3.3). Let $\delta\leq\delta_{0}=C(n,\lambda,\alpha,\Lambda)$ and
$r\leq\ell_{0}(n,\lambda,\alpha,\Lambda,\delta)$. Suppose $\sup_{x\in
B_{r}}N_{x,0}(s)\leq d+\delta$ for any $s\leq C(n,\lambda)\delta^{-1}r$ and
for some integer $d$. Furthermore we assume that
$\mathcal{V}_{\delta,d,r}(0,0)\neq\emptyset$ and is not $(k,\tau)$-independent
in $B_{r}$ with $\tau<\tau_{0}(n,\lambda)$. Then we have the following
decomposition

$B_{r}\subset\bigcup_{b\in B}B_{r_{b}}(x_{b})\cup\bigcup_{c\in
C}B_{r_{c}}(x_{c})\cup\bigcup_{e\in E}B_{r_{e}}(x_{e})\cup S_{d},$ (6.21)

where each $B_{r_{b}}$ is a $b$-ball, $B_{r_{c}}(x_{c})$ is a $c$-ball and
$B_{r_{e}}(x_{e})$ is an $e$-ball. Furthermore, we have the following measure
estimates

$\begin{split}&\sum_{b\in B}r_{b}^{k}+\sum_{e\in E}r_{e}^{k}\leq
C(n,\tau)r^{k}.\\\ &\sum_{c\in C}r_{c}^{k}\leq C(n)\tau r^{k}.\\\
&\mathcal{H}^{k}(S_{d})=0.\end{split}$ (6.22)

###### Proof.

First we choose a Vitali covering of $B_{r}$ with uniform radius $10^{-1}\tau
r$

$B_{r}\subset\bigcup_{b\in B^{1}}B_{10^{-1}\tau r}(x^{1}_{b})\cup\bigcup_{c\in
C^{1}}B_{10^{-1}\tau r}(x^{1}_{c})\cup\bigcup_{d\in D^{1}}B_{10^{-1}\tau
r}(x^{1}_{d})\cup\bigcup_{e\in E^{1}}B_{10^{-1}\tau r}(x^{1}_{e}).$

and if $x^{1}$ and $y^{1}$ are any distinct centers, we require

$B_{10^{-2}\tau r}(x^{1})\cap B_{10^{-2}\tau r}(y^{1})=\emptyset.$

The disjointness implies that

$\sum_{b\in B^{1}}r_{b}^{k}+\sum_{e\in E^{1}}r_{e}^{k}\leq
C_{1}(n,\tau)r^{k}.$ (6.23)

Since $\mathcal{V}_{\delta,d,r}(0)\neq\emptyset$ and is not
$(k,\tau)$-independent in $B_{r}$, there exists some $V^{k-1}$ such that
$\mathcal{V}_{\delta,d,r}(0)\subset B_{\tau r}(V)$. Since
$\mathcal{V}_{\delta,d,r}(0)\cap C^{1}\neq\emptyset$ and
$\mathcal{V}_{\delta,d,r}(0)\cap D^{1}\neq\emptyset$, by the disjointness we
have

$\sum_{c\in C^{1}}r_{c}^{k}+\sum_{d\in D^{1}}r_{d}^{k}\leq
C_{2}(n)\tau^{-k+1}(\tau r)^{k}=C_{2}(n)\tau r^{k}.$ (6.24)

Next for each $d$-ball we can repeat the same Vitali covering argument, to
refine the covering as

$\begin{split}&B_{r}\subset\bigcup_{b\in
B^{2}}B_{r_{b}}(x^{2}_{b})\cup\bigcup_{c\in
C^{2}}B_{r_{c}}(x^{2}_{c})\cup\bigcup_{d\in
D^{2}}B_{10^{-2}\tau^{2}r}(x^{2}_{d})\cup\bigcup_{e\in
E^{2}}B_{r_{e}}(x^{2}_{e}).\\\ &\sum_{b\in B^{2}}r_{b}^{k}+\sum_{e\in
E^{2}}r_{e}^{k}\leq C_{1}(n,\tau)(1+C_{2}(n)\tau)r^{k}.\\\ &\sum_{c\in
C^{2}}r_{c}^{k}\leq C_{2}(n)\tau(1+C_{2}(n)\tau)r^{k}.\\\ &\sum_{d\in
D^{2}}r_{d}^{k}=\sum_{d\in
D^{2}}(\frac{\tau}{10})^{2k}r^{k}\leq(C_{2}(n)\tau)^{2}r^{k}.\end{split}$
(6.25)

After $M$ times refinement, we obtain

$\begin{split}&B_{r}\subset\bigcup_{b\in
B^{M}}B_{r_{b}}(x^{M}_{b})\cup\bigcup_{c\in
C^{M}}B_{r_{c}}(x^{M}_{c})\cup\bigcup_{d\in
D^{M}}B_{(10^{-1}\tau)^{M}r}(x^{M}_{d})\cup\bigcup_{e\in
E^{M}}B_{r_{e}}(x^{M}_{e}).\\\ &\sum_{b\in B^{M}}r_{b}^{k}+\sum_{e\in
E^{M}}r_{e}^{k}\leq C_{1}(n,\tau)\sum_{i=0}^{M-1}(C_{2}(n)\tau)^{i}r^{k}.\\\
&\sum_{c\in C^{M}}r_{c}^{k}\leq
C_{2}(n)\tau\sum_{i=0}^{M-1}(C_{2}(n)\tau)^{i}r^{k}.\\\ &\sum_{d\in
D^{M}}(\frac{\tau}{10})^{kN}\leq(C_{2}(n)\tau)^{M}.\end{split}$ (6.26)

Let $C_{2}(n)\tau<1/10$, then $\\{x_{d}^{M}\\}\to S_{d}$ in the Hausdorff
sense with $\mathcal{H}^{k}(S_{d})=0$ and the proof is finished. ∎

By applying the covering of $c$-balls and $d$-balls, we can prove the Neck
decomposition theorem 6.1.

###### Proof of Theorem 6.1.

By Lemma 3.51, we have $\sup_{x\in B_{1}}N_{x,0}(1)\leq
C_{0}(n,\lambda,\alpha,\Lambda)$ for some integer $C_{0}$. We can cover
$B_{1}$ using Vitali covering with radius $r=c(n,\lambda)\delta$. Then we can
apply Proposition 6.12 and 6.20 to decompose $c$-balls and $d$-balls
iteratively. Then we obtain the following covering

$B_{1}\subset\bigcup_{a}\big{(}\mathcal{C}_{0,a}\cup\mathcal{N}_{a}\cap
B_{a}(x_{a})\big{)}\cup\bigcup_{b}B_{r_{b}}(x_{b})\cup\bigcup_{e}B_{r_{e}}(x_{e})\cup
S_{d},$ (6.27)

where $\mathcal{H}^{k}(S_{d})=0$ and
$\sum_{a}r^{k}_{a}+\sum_{b}r^{k}_{b}+\sum_{e}r^{k}_{e}+\mathcal{H}^{k}(\bigcup_{a}\mathcal{C}_{0,a})\leq
C(n,\lambda,\alpha,\Lambda,\delta)$.

Next we deal with $e$-balls. According to the definition of $e$-ball, we know
$\mathcal{V}_{\delta,d,r_{e}}(x_{e},0)=\emptyset$. In other words, $\sup_{x\in
B_{e}}N_{x,0}(s)\leq C_{0}(n,\lambda,\alpha,\Lambda)-\delta$ for some $s\geq
10^{-2}r_{e}$. By Lemma 3.36 we get $\sup_{x\in B_{e}}N_{x,0}(s)\leq
C_{0}(n,\lambda,\alpha,\Lambda)-1+\delta$ for $s\geq
c(n,\lambda,\alpha,\Lambda,\delta)r_{e}$. And we can cover $B_{r_{e}}$ with
Vitali covering with this radius $c(n,\lambda,\alpha,\Lambda,\delta)r_{e}$.
Then apply the arguments above again to refine the covering lemmas of
$c$-balls and $d$-balls and continue the process for finitely many (at most
$C_{0}(n,\lambda,\alpha,\Lambda))$ times. Finally we obtain the result.

The second conclusion comes from the Quantitative Cone-splitting theorem 5.2.
Actually, if $x\in\mathcal{N}^{a}$ or $x\in B_{r_{b}}(x_{b})$, by Corollary
3.37 and Theorem 5.2 we have $u$ is $(k+1,\epsilon,r^{\prime},x)$-symmetric
for some small $r^{\prime}$ provided $\eta$ and $\delta$ is sufficiently
small. This proves that $\mathcal{S}^{k}_{\epsilon}\cap
B_{1}\subset\Big{(}S_{0}\cup\bigcup_{a}\mathcal{C}_{0,a}\Big{)}$. ∎

##  7\. Proof of the Main Theorems

In this section we will prove our main theorems.

###  7.1. Proof of Theorem 1.14

For any $\epsilon\leq 1/10$ denote
$\mathcal{S}^{n-1}_{\epsilon}=\mathcal{S}^{n-1}_{\epsilon}(u(\cdot,0))$. We
will see that $Z_{0}\cap B_{1}\subset\mathcal{S}^{n-1}_{\epsilon}\cap B_{1}$
where $Z_{0}=\\{x:u(x,0)=0\\}$. To see this, it suffices to show that
$u(y,0)\neq 0$ for any $y\notin\mathcal{S}_{\epsilon}^{n-1}$.Let
$y\notin\mathcal{S}_{\epsilon}^{n-1}$ we have for some $s>0$ that $u(\cdot,0)$
is $(n,\epsilon,s,y)$-symmetric. This means that
$f(x)=\frac{u(y+sx,0)}{\left(\fint_{B_{1}}|u(y+sx,0)|^{2}dx\right)^{1/2}}$ is
$\epsilon$-close to $1$. Hence $|f(0)-1|\leq\epsilon$ and thus $u(y,0)\neq 0$
if $\epsilon<1/2$. Hence we have $Z_{0}\cap
B_{1}\subset\mathcal{S}^{n-1}_{\epsilon}\cap B_{1}$. The Hausdorff estimate
follows now directly since $\mathcal{H}^{n-1}(\mathcal{S}_{\epsilon}^{n-1}\cap
B_{1})\leq C<\infty$ by Theorem 6.1. Hence we complete the proof. ∎

###  7.2. Proof of Theorem 1.4 and Theorem 1.19

The following doubling estimate is taken from [23].

######  Lemma 7.1 (Theorem 3 in [23]).

Let $u$ be a nonzero solution of (1.1) in $Q_{2}$ satisfying (1.2) and (1.3).
Then, there exists $N(n,\lambda)$ such that the following holds when
$0<r\leq(N\log(N\Theta))^{-1/2}$ that

$\displaystyle\int_{Q_{2r}}u^{2}dxdt\leq
e^{N\log(N\Theta)\log(N\log(N\Theta))}\int_{Q_{r}}u^{2}dxdt,$ (7.2)

where $\Theta=\frac{\int_{Q_{2}}u^{2}dxdt}{\int_{B_{1/2}}u^{2}(x,0)dx}$ and
$Q_{r}=Q_{r}(0,0)=\\{(x,t):|x|<r,-r^{2}<t\leq 0\\}$.

As a direct consequence we can get the following doubling estimate

######  Corollary 7.3.

Let $u$ be a solution of (1.1) in $Q_{2}$ satisfying (1.2) and (1.3). If $u$
is nonzero at $t=0$, then there exists $\Lambda(n,\lambda,\Theta)$ and
$r_{0}(n,\lambda,\Theta)$ such that for any $x\in B_{1}$ and $0<r\leq r_{0}$
that

$\displaystyle\int_{Q_{2r}(x,0)}u^{2}dxdt\leq\Lambda\int_{Q_{r}(x,0)}u^{2}dxdt.$
(7.4)

where $\Theta=\frac{\int_{Q_{2}}u^{2}dxdt}{\int_{B_{3/2}}u^{2}(x,0)dx}$.

###### Proof.

If $u(\cdot,0)$ is not vanishing in any open subset of $B_{1}$, we can
directly apply Lemma 7.1 by replacing center $0$ with given $x\in B_{1}$. If
$u(\cdot,0)$ is vanishing in an open subset $U\subset B_{1}$, we will show
that $u(\cdot,0)\equiv 0$ which contradicts to the assumption that $u$ is
nonzero at $t=0$. To see this, since $u(\cdot,0)$ is nonzero, there exists
some $x_{0}\in U$ and $\hat{r}>0$ such that $u(\cdot,0)$ is nonzero in
$B_{\hat{r}}(x_{0})$ and $B_{2\hat{r}}(x_{0})\subset B_{2}$. Applying Lemma
7.1 we get doubling estimate of $u$ at center $(x_{0},0)$. We need to consider
the following two cases. First, if $\int_{Q_{r}(x_{0},0)}u^{2}dxdt=0$ for some
small $r\leq\hat{r}$, then $u$ vanishes to infinite order at $(x_{0},0)$ in
space-time sense as in [22]. Therefore, by [22] we have that $u(\cdot,0)\equiv
0$. On the other hand, if $\int_{Q_{r}(x_{0},0)}u^{2}dxdt\neq 0$ for all
$r\leq\hat{r}$, consider the tangent map of $u$ at $(x_{0},0)$. One can argue
as Lemma 3.15 to get that the tangent map is a nonzero caloric polynomial.
However $u(\cdot,0)$ is zero near $x_{0}$ which implies that the tangent map
is zero at $t=0$. By the backward uniqueness of heat equation(see [53, 55]),
this is a contradiction since tangent map is a nonzero caloric polynomial.
Thus we finish the proof. ∎

###### Proof of Theorem 1.4.

Theorem 1.4 follows directly by Corollary 7.3 and Theorem 1.14. ∎

Let us now turn to the proof of Theorem 1.19. The following result is
standard.

######  Lemma 7.5.

Let $u$ be a continuous function in $B_{1}\subset\mathbb{R}^{n}$. Assume both
$\\{x:u(x)>0\\}$ and $\\{x:u(x)<0\\}$ are not empty. Then

$\displaystyle\dim Z_{u}\geq n-1,$ (7.6)

where $Z_{u}=\\{x:u(x)=0\\}.$

###### Proof.

Assume $x_{0},y_{0}\in B_{1}$ satisfy $u(x_{0})>0$ and $u(y_{0})<0$. Since
$u\in C^{0}$, then there exists $r_{0}>0$ such that
$\inf_{B_{r_{0}}(x_{0})}u(x)>0$ and $\sup_{B_{r_{0}}(y_{0})}u(x)<0$. Consider
two hyperplanes $H_{1}$ and $H_{2}$ satisfying

  * •

$H_{1}$ and $H_{2}$ are perpendicular to $v:=x_{0}-y_{0}$

  * •

$x_{0}\in H_{1}$ and $y_{0}\in H_{2}$.

Denote line $\ell_{x,y}$ to be the line segment which is parallel to $v$ and
$\ell_{x,y}\cap H_{1}=\\{x\\}$ and $\ell_{x,y}\cap H_{2}=\\{y\\}$. For any
$x\in H_{1}\cap B_{r_{0}}(x_{0})$ there exists a unique $y_{x}\in H_{2}\cap
B_{r_{0}}(y_{0})$ such that $\ell_{x,y_{x}}//v$. Noting that $u|_{H_{1}\cap
B_{r_{0}}(x_{0})}>0$ and $u|_{H_{1}\cap B_{r_{0}}(y_{0})}<0$ and $u$ is
continuous, there must exist $z_{x}\in\ell_{x,y_{x}}$ such that $u(z_{x})=0$.
Define a map $\tau:H_{1}\cap B_{r_{0}}(x_{0})\to Z_{u}$ by $\tau(x)=z_{x}$.
Then $|\tau(x)-\tau(y)|\geq|x-y|$. Hence by the definition of Hausdorff
measure, we have

$\mathcal{H}^{n-1}(\\{z:u(z)=0,z\in\ell_{x,y_{x}},x\in H_{1}\cap
B_{r_{0}}(x_{0})\\})\geq\mathcal{H}^{n-1}(H_{1}\cap B_{r_{0}}(x_{0}))\geq
c(n)r_{0}^{n-1}>0.$

In particular, $\dim Z_{u}\geq n-1.$ ∎

Now we are ready to prove Theorem 1.19.

###### Proof of Theorem 1.19.

By the standard backward uniqueness (see [15, 22, 37, 55]), we know that $u$
is not vanishing at any $t>-4$ otherwise $u\equiv 0$ in $Q_{2}$. Therefore by
Theorem 1.4 we know that $\dim Z_{t}\leq n-1$ for any $t>-4$ (see also [11]).
Therefore, we only need to consider the following two cases.

Case 1: Assume $s=-4$ and $t>-4$. If $\dim Z_{-4}\geq n-1$, then $\dim
Z_{t}\leq n-1\leq\dim Z_{-4}$. If $\dim Z_{-4}<n-1$, then by Lemma 7.5 we can
assume $u(x,-4)\geq 0$. By maximum principle Theorem 2.35 we get $u(x,t)>0$
for any $t>-4$. In particular $Z_{t}=\emptyset$. Hence $\dim Z_{t}\leq\dim
Z_{s}$.

Case 2: Assume $-4<s<t\leq 0$. We have $\dim Z_{t}\leq n-1$ and $\dim
Z_{s}\leq n-1$. If $\dim Z_{s}=n-1$ then we are done. If $\dim Z_{s}<n-1$, the
same argument as above we get $Z_{t}=\emptyset$ and we are also done. Thus we
complete the proof. ∎

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