# Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers

Jacob Bernstein  Department of Mathematics, Johns Hopkins University, 3400 N.
Charles Street, Baltimore, MD 21218<EMAIL_ADDRESS>
###### Abstract.

We show the (normalized) Li-Yau conformal volume of a self-shrinker of mean
curvature flow in Euclidean space bounds its Colding-Minicozzi entropy from
below. This bound is independent of codimension and sharp on planes. As an
application we verify a conjecture of Colding-Minicozzi about the entropy of
closed self-shrinkers of arbitrary codimension for self-shrinkers that are
topologically two-dimensional real projective planes. As part of the proof we
introduce two auxiliary functionals which we call _stable conformal volume_
and _virtual entropy_ which should be of independent interest.

The author was partially supported by the NSF Grant DMS-2203132. The author
would like to thank Jeffrey Case for a stimulating discussion that lead to
this work and Letian Chen for helpful comments on an earlier draft.

##  1\. Introduction

In [32], Li-Yau introduced a quantity associated to branched conformal
immersions into $\mathbb{S}^{N}$ they called the ($N$-)_conformal volume_ –
this quantity is also called _visual volume_ by Gromov [28]. Among other
applications, it has proven to be of significant importance both in the study
of the Willmore energy and spectral properties of surfaces. There is a vast
literature on this quantity – we refer in particular to [14] for helpful
background discussion. In this paper, we relate the conformal volume to
another important geometric quantity introduced in [22] called _Colding-
Minicozzi entropy_ or _entropy_ for short, which has proven important in the
study of mean curvature flow.

Consider the inverse stereographic projection map

(1.1)
$\mathcal{S}:\mathbb{R}^{N}\to\mathbb{S}^{N}\subset\mathbb{R}^{N+1},\mathbf{x}\mapsto\left(\frac{\mathbf{x}}{1+\frac{1}{4}|\mathbf{x}|^{2}},\frac{1-\frac{1}{4}|\mathbf{x}|^{2}}{1+\frac{1}{4}|\mathbf{x}|^{2}}\right).$

For an $n$-dimensional submanifold, $\Sigma\subset\mathbb{R}^{N}$ and
embedding $j:M^{n}\to\Sigma\subset\mathbb{R}^{N}$ define

$\lambda_{LY}^{n}[\Sigma]=V_{c}(N,\mathcal{S}\circ j)$

where $V_{c}(N,\mathcal{S}\circ j)$ is the $N$-conformal volume of the
embedding $\mathcal{S}\circ j:M\to\mathcal{S}(\Sigma)\subset\mathbb{S}^{N}$
introduced in [32]. A straightforward computation yields,

$\lambda_{LY}^{n}[\Sigma]=\sup_{\rho>0,\mathbf{x}_{0}\in\mathbb{R}^{N}}\int_{\Sigma}W^{n}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}(\mathbf{x}).$

where the weights $W^{n}_{\rho}$ are given explicitly by

$W^{n}_{\rho}(\mathbf{x})=\frac{\rho^{n}}{(1+\frac{1}{4}\rho^{2}|\mathbf{x}|^{2})^{n}}.$

These are the volume element on $\Sigma$ induced by homothetic scalings of
$\mathcal{S}^{*}g_{\mathbb{S}}$, i.e., the pull-back of the round metric on
the sphere. Denote the _normalized_ conformal volume by

$\bar{\lambda}_{LY}^{n}[\Sigma]=|\mathbb{S}^{n}|^{-1}\lambda_{LY}^{n}[\Sigma].$

When $\Sigma$ is an affine $n$-plane, $\mathbb{R}^{n}$ or round $n$-sphere,
$\mathbb{S}^{n}$, $\bar{\lambda}_{LY}^{n}[\Sigma]=1$.

Following [22], for a $n$-dimensional submanifold
$\Sigma\subset\mathbb{R}^{N}$, the entropy of $\Sigma$ is the value

$\lambda_{CM}^{n}[\Sigma]=\sup_{\rho>0,\mathbf{x}_{0}\in\mathbb{R}^{N}}(4\pi\rho)^{-\frac{n}{2}}\int_{\Sigma}e^{-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4\rho}}d\mathcal{H}^{n}.$

This quantity is monotone non-increasing along mean curvature flows and has
found many applications both in the study of mean curvature flow and as a
measure of complexity, especially of closed hypersurfaces, e.g., [9, 10, 11,
12, 13, 18, 19, 40]. For affine $n$-planes, $\mathbb{R}^{n}$, the value is
one, but is strictly larger for all other proper submanifolds – see [17, 6].
While there has been extensive study of the entropy of hypersurfaces,
comparatively little is known about it in higher codimension – see however
recent work of Colding-Minicozzi [24, 23].

One readily observes – see Proposition 4.4 – that

$(4\pi\rho)^{-\frac{n}{2}}e^{-\frac{|\mathbf{x}|^{2}}{4\rho}}\leq\left(\frac{n}{4\pi}\right)^{\frac{n}{2}}W^{n}_{(n\rho)^{-\frac{1}{2}}}(\mathbf{x})$

which leads to the non-sharp bound

$\lambda_{CM}^{n}[\Sigma]\leq\left(\frac{n}{4\pi}\right)^{\frac{n}{2}}|\mathbb{S}^{n}|\bar{\lambda}_{LY}^{n}[\Sigma].$

Note that asymptotically the constant behaves like

$\left(\frac{n}{4\pi}\right)^{\frac{n}{2}}|\mathbb{S}^{n}|=\sqrt{2}\left(\frac{e}{2}\right)^{\frac{n}{2}}+l.o.t,n\to\infty.$

which grows rapidly as $n$ grows.

We introduce two auxiliary measures of complexity of submanifolds and then
relate the new quantities with the conformal volume and entropy. These
relationships are particularly striking when the submanifold,
$\Sigma\subset\mathbb{R}^{N}$, is a _self-shrinker_. That is, satisfies

$\mathbf{H}_{\Sigma}+\frac{\mathbf{x}^{\perp}}{2}=\mathbf{0}$

where $\mathbf{H}_{\Sigma}$ is the mean curvature vector and
$\mathbf{x}^{\perp}$ is the normal component of the position vector. Such
submanifolds are important in the study of mean curvature flow, i.e., in the
geometric evolution equation

$t\in
I\mapsto\Sigma_{t}\subset\mathbb{R}^{N},\left(\frac{\partial\mathbf{x}}{\partial
t}\right)^{\perp}=\mathbf{H}_{\Sigma_{t}}.$

Here $I$ is an interval of time and $t\mapsto\Sigma_{t}$ is a smooth evolution
of $n$-dimensional submanifolds. This is because for a self-shrinker $\Sigma$,

$t\in(-\infty,0)\mapsto\sqrt{-t}\Sigma$

is an evolution by mean curvature flow that provides a model of singularity
formation of the flow. As $\lambda_{CM}^{n}$ is invariant under rescaling, it
is constant along this flow.

######  Theorem 1.1.

Let $\Sigma\subset\mathbb{R}^{N}$ be a $n$-dimensional properly embedded
submanifold that is, up to translation and dilation, a self-shrinker. One has

$1\leq\bar{\lambda}_{LY}^{n}[\Sigma]\leq\lambda_{CM}^{n}[\Sigma].$

The inequality of Theorem 1.1 is sharp in the sense that equality holds
throughout on affine $n$-planes and by [17] this double equality characterizes
these planes. When $n\geq 2$ planes and round spheres are characterized by
equality in the first inequality by [14], though for round spheres the second
inequality is strict. While they are not self-shrinkers, certain double
logarithmic spirals provide examples with equality in the second inequality
but not the first – see Appendix A.

In [24, pg. 119], Colding-Minicozzi made the following conjecture:

######  Conjecture 1.2.

If $\Sigma\subset\mathbb{R}^{N}$ is a closed $n$-dimensional self-shrinker,
then

$\sqrt{2}=\lim_{n\to\infty}\left(\frac{2n}{4\pi
e}\right)^{\frac{n}{2}}|\mathbb{S}^{n}|<\left(\frac{2n}{4\pi
e}\right)^{\frac{n}{2}}|\mathbb{S}^{n}|=\lambda_{CM}^{n}[\mathbb{S}^{n}]\leq\lambda_{CM}^{n}[\Sigma].$

When $n=N+1$, i.e., $\Sigma$ is a hypersurface, this was shown in [21] and
when $n=1$ it follows from the classification of self-shrinking curves [24,
Lemma 6.16]. It was shown in [10] that the only non-flat proper self-shrinking
surfaces in $\mathbb{R}^{3}$ with entropy below that of the shrinking
cylinder, i.e., less than
$\lambda_{CM}^{2}[\mathbb{S}^{1}\times\mathbb{R}]=\lambda_{CM}^{1}[\mathbb{S}^{1}]\approx
1.52$, are the flat plane which has entropy 1, the shrinking sphere which has
entropy $\lambda_{CM}^{2}[\mathbb{S}^{2}]\approx 1.47$ and the shrinking
cylinder itself. A strengthening of Conjecture (1.2) is that this phenomena
continues to hold for surfaces of any codimension [24, Conjecture 8.1].

In [32], Li-Yau observed that two-dimensional real projective planes have
conformal volume at least $6\pi$. Thus, Theorem 1.1 gives is a partial
confirmation of Conjecture 1.2:

######  Corollary 1.3.

If $\Sigma\subset\mathbb{R}^{N}$ is a self-shrinker that is topologically
$\mathbb{RP}^{2}$, then

$1.47\approx\lambda_{CM}^{2}[\mathbb{S}^{2}]<\frac{3}{2}\leq\lambda_{CM}^{2}[\Sigma].$

There is a minimal $\mathbb{RP}^{2}$ embedded into
$\mathbb{S}^{4}\subset\mathbb{R}^{5}$, which we denote by
$\Sigma_{\mathbb{RP}^{2}}$ and is sometimes called the Veronese surface, which
has conformal volume $6\pi$. One may check that $2\Sigma_{\mathbb{RP}^{2}}$ is
a self-shrinker and compute that

$\bar{\lambda}_{LY}^{2}[\Sigma_{\mathbb{RP}^{2}}]=\frac{3}{2}<2.207\approx\frac{6}{e}=\frac{3}{2}\lambda_{CM}^{2}[\mathbb{S}^{2}]=\lambda^{2}_{CM}[\Sigma_{\mathbb{RP}^{2}}].$

In the case of tori, the conformal area has been estimated in terms of the
complex structure by Montiel and Ros [36]. Moreover, they gave exact values
for a large class of tori; this was refined by Bryant in [15].

######  Corollary 1.4.

Let $\Sigma\subset\mathbb{R}^{N}$ be a self-shrinking torus conformally
parametrized by the lattice $\mathbb{C}/\Lambda_{\omega}$ where $\omega=x+iy$
and $0\leq x\leq\frac{1}{2}$, $\sqrt{1-x^{2}}\leq y$.

One has

$\frac{\pi
y}{x^{2}+y^{2}-x+1}\leq\bar{\lambda}_{LY}^{2}[\Sigma]\leq\lambda_{CM}^{2}[\Sigma].$

######  Remark 1.5.

This bound comes from the bound established in [36] on (unnormalized)
conformal volume of the lattice

$\frac{4\pi^{2}y}{x^{2}+y^{2}-x+1}\leq
V_{c}(\mathbb{C}/\Lambda_{\tau},[g_{\mathbb{R}}]).$

This inequality becomes strict for large $y$. However, as conjectured and
partially shown in [36] and fully established in [15], it is an equality in
the restricted domain satisfying

$(x-\frac{1}{2})^{2}+y^{2}\leq\frac{9}{4}.$

In this region, the minimum is achieved by the rectangular torus with
$\omega=i\sqrt{2}$ and the maximum is achieved by the hexagonal torus with
$\omega=\frac{1}{2}+i\frac{\sqrt{3}}{2}$ and the values satisfy

$\lambda_{CM}^{2}[\mathbb{S}^{2}]\approx
1.47<1.48\approx\frac{\sqrt{2}\pi}{3}\leq\frac{\pi
y}{x^{2}+y^{2}-x+1}\leq\frac{\sqrt{3}\pi}{3}\approx 1.814.$

###  1.1. Virtual entropy

In order to establish the link between the Li-Yau conformal volume and the
Colding-Minicozzi entropy, it will be helpful to introduce an alternative
notion of entropy. This is inspired by work of Magni-Mantegazza [34], who,
prior to Colding-Minicozzi’s paper on the subject, introduce a quantity based
off of Hamilton’s matrix Harnack inequality [29] and its application to a
generalized notion of Huisken monotonicity [30]. We revisit the approach of
[34]. Using a simple observation about the matrix Harnack inequality in
Euclidean space, we are led to a more flexible concept which should have other
applications beyond the proof of Theorem 1.1 .

Let us first briefly recall some observations and ideas drawn from [30, 34].
For $t>0$ and $u_{0}\in C^{0}\cap L^{1}(\mathbb{R}^{N})$, consider the
convolution by the heat kernel,

(1.2)
$\displaystyle(\mathcal{H}_{t}u_{0})(\mathbf{x})=\int_{\mathbb{R}^{N}}u_{0}(\mathbf{y})\mathcal{H}(t,\mathbf{x},\mathbf{y})d\mathcal{L}^{N}(\mathbf{y});$
$\displaystyle\mathcal{H}(t,\mathbf{x},\mathbf{y})=(4\pi
t)^{-\frac{N}{2}}e^{-\frac{|\mathbf{x}-\mathbf{y}|^{2}}{4t}}.$

For such $u_{0}$, $U(t,\cdot)=\mathcal{H}_{t}u_{0}$ is a solution of the heat
equation continuous on $[0,\infty)\times\mathbb{R}^{N}$. For $t>0$, consider
the collection time $t$ evolutions of positive mass one initial data,

$\mathcal{H}^{+}_{1}(t)=\left\\{\mathcal{H}_{t}u_{0},u_{0}\in
C^{0}(\mathbb{R}^{N}),u_{0}>0,\int_{\mathbb{R}^{N}}u_{0}d\mathcal{L}^{N}=1\right\\}\subset
C^{\infty}\cap L^{1}(\mathbb{R}^{N}),$

For any $\delta>0$, the Gaussian profiles satisfy

$\mathcal{G}_{1}(t)=\left\\{\mathcal{H}(t,\mathbf{x},\mathbf{x}_{0})=(4\pi
t)^{-\frac{N}{2}}e^{-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4t}}:\mathbf{x}_{0}\in\mathbb{R}^{N}\right\\}\subset\mathcal{H}^{+}_{1}(t+\delta).$

For $t>0$, any $U\in\mathcal{H}_{1}^{+}(t)\cup\mathcal{G}_{1}(t)$, satisfies
Hamilton’s matrix Harnack inequality:

(1.3) $\nabla^{2}_{\mathbb{R}}\log U+\frac{1}{2t}g_{\mathbb{R}}\geq 0.$

When traced this is the Li-Yau Harnack inequality. Both inequalities hold in
more general settings – see [29, 33]. For the convenience of the reader, an
elementary proof of this fact in Euclidean space and for positive solutions
given by convolving with the heat kernel as in (1.2) is provided in Lemma 2.7.

Hamilton observed in [30], that (1.3) could allowed one to generalize the
Huisken monotonicity formula [31] – see also [27]. It was this observation
that was used in [34] and which we will also exploit. Following Magni-
Mantegazza, for $t>0$ and $\Sigma\subset\mathbb{R}^{N}$ an $n$-dimensional
submanifold set

$\sigma_{MM}[\Sigma,t]=\sup_{U\in\mathcal{H}^{+}_{1}(t)}(4\pi
t)^{\frac{N-n}{2}}\int_{\Sigma}Ud\mathcal{H}^{n}.$

It is shown in [34] that it suffices to take the sup over
$U\in\mathcal{G}_{1}(t)$ and so

$\sigma_{MM}[\Sigma,t]=\sup_{U\in\mathcal{G}_{1}(t)}(4\pi
t)^{\frac{N-n}{2}}\int_{\Sigma}Ud\mathcal{H}^{n}\leq\lambda_{CM}^{n}[\Sigma]=\sup_{t>0}\sigma_{MM}[\Sigma,t].$

That is, the Magni-Mantegezza functionals lead to the Colding-Minicozzi
entropy.

We modify the approach of Magni-Mantegazza by focusing on the condition (1.3).
To that end, for a positive function $U\in C^{2}(\mathbb{R}^{N})$, set

$\tau(U)=\sup\left\\{\tau\geq 0:2\tau\nabla^{2}_{\mathbb{R}}\log
U+g_{\mathbb{R}}\geq 0\right\\}$

which we call the _virtual time_ of $U$. Note that $\tau(U)\geq 0$, but can be
equal to $0$. When $U\in\mathcal{H}^{+}_{1}(t)\cup\mathcal{G}_{1}(t)$, $t>0$,
(1.3) holds and so

$\tau(U)\geq t>0$

and one has equality for $U\in\mathcal{G}_{1}(t)$. That is, for positive
solutions of the heat equation the virtual time estimates the real time from
above. When $u_{0}\in C^{2}\cap L^{1}(\mathbb{R}^{N})$ has
$\tau(u_{0})=\tau_{0}>0$ and $U(t,\cdot)=\mathcal{H}_{t}u_{0}$ is heat flow
with initial data $u_{0}$ the proof of matrix Harnack inequality from [29]
suggests that

(1.4) $\tau(U(t,\cdot))\geq t+\tau_{0}.$

That is, the virtual time of a positive solution of the heat equation grows at
least linearly in the evolution time. This is slightly subtle due to the non-
compactness of $\mathbb{R}^{N}$ and is proved in Proposition 2.7 using ideas
from [16].

For $T>0$, set

$\mathcal{VT}_{1}^{+}(T)=\left\\{u\in C^{2}\cap
L^{1}(\mathbb{R}^{N}):u>0,\tau(u)\geq
T,\int_{\mathbb{R}^{N}}ud\mathcal{L}^{N}=1\right\\}.$

Clearly, for $0<T_{1}<T_{2}$,
$\mathcal{VT}_{1}^{+}(T_{2})\subset\mathcal{VT}_{1}^{+}(T_{1})$. As observed
above

(1.5)
$\mathcal{H}_{1}^{+}(T)\cup\mathcal{G}_{1}(T)\subset\mathcal{VT}_{1}^{+}(T).$

The inclusion is strict as the elements of $\mathcal{H}_{1}^{+}(T)$ must also
have strong regularity properties not required of elements in
$\mathcal{VT}_{1}^{+}(t)$. Moreover,
$\mathcal{G}_{1}(t)\subset\mathcal{VT}_{1}^{+}(T)$ for $t\geq T$ and
$\mathcal{G}_{1}(t)\cap\mathcal{VT}_{1}^{+}(T)=\emptyset$ for $0<t<T$ Standard
facts about heat flow and (1.4) also imply,

$\mathcal{H}_{t}(\mathcal{VT}_{1}^{+}(T))\subset\mathcal{VT}_{1}^{+}(T+t).$

Given an $n$-dimensional submanifold $\Sigma\subset\mathbb{R}^{N}$ we define
its _virtual entropy_ to be

$\lambda_{V}^{n}[\Sigma]=\sup_{T>0}\sup_{U\in\mathcal{VT}_{1}^{+}(T)}\left((4\pi
T)^{\frac{N-n}{2}}\int_{\Sigma}Ud\mathcal{H}^{n}\right).$

As in [30, 34], this quantity is monotone along reasonable mean curvature
flows.

######  Theorem 1.6.

If $t\in[0,T)\mapsto\Sigma_{t}\subset\mathbb{R}^{N}$ is a mean curvature flow
of $n$-dimensional proper submanifolds with
$\lambda_{V}^{n}[\Sigma_{0}]<\infty$, then

$t\mapsto\lambda_{V}^{n}[\Sigma_{t}]$

is monotone non-increasing in $t$.

It follows immediately from (1.5) that

$\lambda_{CM}^{n}[\Sigma]\leq\lambda_{V}^{n}[\Sigma].$

The long-time behavior of suitable solutions of the heat equation is towards
the Gaussian – see, e.g., [37]. Using Theorem 1.6, this suggests that when
$\Sigma$ is a self-shrinker,

(1.6) $\lambda_{CM}^{n}[\Sigma]=\lambda_{V}^{n}[\Sigma].$

This is proved in Theorem 3.6. It is unclear to the author how general this
equality is.

###  1.2. Stable conformal volume

Finally, we introduce a functional obtained by “stabilizing” the (normalized)
Li-Yau conformal volume in a manner reminiscent of the way stable homotopy
groups are introduced. This is distinct from the stabilization procedure of
[32] which involved conformally embedding in higher and higher dimensional
spheres. This quantity is related to both the Colding-Minicozzi and virtual
entropies.

Given a $n$-dimensional submanifold $\Sigma\subset\mathbb{R}^{N}$, its _stable
conformal volume_ is defined as

$\bar{\lambda}^{\infty}_{LY}[\Sigma]=\liminf_{m\to\infty}\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}].$

Crucially, this sequence is monotonically increasing in the stabilization
parameter:

######  Proposition 1.7.

If $\Sigma\subset\mathbb{R}^{N}$ is a $n$-dimensional proper submanifold, then

$\bar{\lambda}_{LY}^{n}[\Sigma]\leq\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}]\leq\bar{\lambda}^{\infty}_{LY}[\Sigma].$

In particular,
$m\mapsto\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}]$ is increasing
in $m$ and so

$\bar{\lambda}^{\infty}_{LY}[\Sigma]=\lim_{m\to\infty}\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}].$

In contrast with process in [32], this method of stabilization breaks the full
conformal invariance – though the functionals remain invariant under Euclidean
translations and dilations. For instance, it follows from [14] that

$\bar{\lambda}_{LY}^{n+2}[\Sigma\times\mathbb{R}^{2}]=1$

if and only if $\Sigma$ is an affine $n$-plane. In particular, even though
$\bar{\lambda}_{LY}^{n}[\mathbb{R}^{n}]=\bar{\lambda}_{LY}^{n}[\mathbb{S}^{n}]=1$,

$1=\bar{\lambda}_{LY}^{n}[\mathbb{S}^{n}]<\bar{\lambda}_{LY}^{n+2}[\mathbb{S}^{n}\times\mathbb{R}^{2}].$

Rather remarkably, the stable conformal volume lies between the Colding-
Minicozzi entropy and the virtual entropy.

######  Theorem 1.8.

For any $n$-dimensional proper submanifold, $\Sigma\subset\mathbb{R}^{N}$,

$\lambda_{CM}^{n}[\Sigma]\leq\bar{\lambda}_{LY}^{\infty}[\Sigma]\leq\lambda_{V}^{n}[\Sigma].$

That is, the Colding-Minicozzi entropy of $\Sigma$ provides a lower bound for
its stable conformal volume while the virtual entropy provides an upper bound.

These inequalities are sharp as, when $\Sigma$ is a self-shrinker, appealing
to (1.6) yields

$\lambda_{CM}^{n}[\Sigma]=\bar{\lambda}_{LY}^{\infty}[\Sigma]=\lambda_{V}^{n}[\Sigma].$

Theorem 1.1 follows immediately from this, Proposition 1.7 and Theorem 1.8.

##  2\. Properties of elements of $\mathcal{VT}_{1}^{+}(T)$ and their heat
flows

In [29], Hamilton used a (tensor) maximum principle to prove the matrix
Harnack inequality (1.3) for positive solutions of the heat equation. We use
the same idea to establish a linear in (real) time growth of the measure of
convexity, i.e., the $t$ in (1.3). In [29] the underlying manifold is closed
and so, as we work in Euclidean space, we instead adapt an argument of [16].
While [29] treats quite general positive solutions of the heat equation, in
order to simplify exposition we focus on the subclass of solutions given by
(1.2) with initial data in $\mathcal{VT}_{1}^{+}(T)$, $T>0$. As such, we
establish some basic properties of elements of $\mathcal{VT}_{1}^{+}(T)$ and
of their evolution by (1.2).

We first give an elementary proof that (1.3) holds for elements of
$\mathcal{H}_{1}^{+}(t)$.

######  Lemma 2.1.

For $u_{0}\in C^{0}\cap L^{1}(\mathbb{R}^{N})$ with $u_{0}\geq 0$ and
$\int_{\mathbb{R}^{N}}u_{0}d\mathcal{L}^{N}=1$, the function
$U(t,\cdot)=\mathcal{H}_{t}u_{0}\in\mathcal{H}^{+}_{1}(t)$ given by (1.2)
satisfies, for $t>0$,

$\nabla_{\mathbb{R}}^{2}\log U(t,\mathbf{x})\geq-\frac{1}{2t}g_{\mathbb{R}}.$

###### Proof.

The hypotheses on $u_{0}$ ensure that, for $t>0$,

$U>0\mbox{ and }\int_{\mathbb{R}^{N}}U(t,\mathbf{x})d\mathcal{L}^{N}=1.$

As $u_{0}$ is integrable and, for $t>0$,
$|\nabla^{i}_{\mathbb{R},\mathbf{x}}\mathcal{H}(t,\mathbf{x},\mathbf{y})|\leq
C_{N,i}t^{-\frac{N+2i}{2}}$, one may differentiate under the integral sign and
obtain for any parallel unit vector field $\mathbf{v}$ that,

$\displaystyle\nabla_{\mathbb{R}}^{2}$ $\displaystyle
U(t,\mathbf{x})(\mathbf{v},\mathbf{v})=\int_{\mathbb{R}^{N}}u_{0}(\mathbf{y})\nabla^{2}_{\mathbb{R},\mathbf{x}}\mathcal{H}(t,\mathbf{x},\mathbf{y})(\mathbf{v},\mathbf{v})d\mathcal{L}^{N}(\mathbf{y})$
$\displaystyle=\int_{\mathbb{R}^{N}}u_{0}(\mathbf{y})\left(\frac{(\mathbf{v}\cdot\nabla_{\mathbb{R},\mathbf{x}}\mathcal{H}(t,\mathbf{x},\mathbf{y}))^{2}}{\mathcal{H}(t,\mathbf{x},\mathbf{y})^{2}}-\frac{|\mathbf{v}|^{2}}{2t}\right)\mathcal{H}(t,\mathbf{x},\mathbf{y})d\mathcal{L}^{N}(\mathbf{y})$
$\displaystyle=-\frac{U(t,\mathbf{x})}{2t}|\mathbf{v}|^{2}+\int_{\mathbb{R}^{N}}\left(u_{0}^{\frac{1}{2}}(\mathbf{y})\frac{\mathbf{v}\cdot\nabla_{\mathbb{R},\mathbf{x}}\mathcal{H}(t,\mathbf{x},\mathbf{y})}{\mathcal{H}(t,\mathbf{x},\mathbf{y})}\right)^{2}\mathcal{H}(t,\mathbf{x},\mathbf{y})d\mathcal{L}^{N}(\mathbf{y})$
$\displaystyle\geq-\frac{U(t,\mathbf{x})}{2t}|\mathbf{v}|^{2}+U^{-1}(t,\mathbf{x})\left(\int_{\mathbb{R}^{N}}u_{0}(\mathbf{y})\frac{\mathbf{v}\cdot\nabla_{\mathbb{R},\mathbf{x}}\mathcal{H}(t,\mathbf{x},\mathbf{y})}{\mathcal{H}(t,\mathbf{x},\mathbf{y})}\mathcal{H}(t,\mathbf{x},\mathbf{y})d\mathcal{L}^{N}(\mathbf{y})\right)^{2}$
$\displaystyle=-\frac{U(t,\mathbf{x})}{2t}|\mathbf{v}|^{2}+U^{-1}(t,\mathbf{x})(\mathbf{v}\cdot\nabla_{\mathbb{R}}U(t,\mathbf{x}))^{2}.$

That is, for $t>0$, we get the claimed inequality

$\nabla^{2}_{\mathbb{R}}\log U(t,\mathbf{x})\geq-\frac{1}{2t}g_{\mathbb{R}},$

∎

We next establish a useful pointwise bound on elements of
$\mathcal{VT}_{1}^{+}(T)$.

######  Lemma 2.2.

There is a $C_{0}=C_{0}(N)>0$ so that, for $f\in\mathcal{VT}_{1}^{+}(T)$ and
$\mathbf{x}_{0}\in\mathbb{R}^{N}$,

$f(\mathbf{x}_{0})\leq
C_{0}T^{-\frac{N}{2}}\int_{B_{\sqrt{T}}(\mathbf{x}_{0})}fd\mathcal{L}^{N}\leq
C_{0}T^{-\frac{N}{2}}.$

###### Proof.

The fact that $f\in\mathcal{VT}_{1}^{+}(T)$ implies

$-\frac{N}{2T}\leq\Delta_{\mathbb{R}}\log f.$

Hence, for the given $\mathbf{x}_{0}$, the function

$F_{\mathbf{x}_{0}}(\mathbf{x})=\log
f(\mathbf{x})+\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4T}$

is sub harmonic and so, by the mean value inequality, for any $R>0$,

$\log f(\mathbf{x}_{0})=\log
F_{\mathbf{x}_{0}}(\mathbf{x}_{0})\leq\frac{1}{\omega_{N}R^{N}}\int_{B_{R}(\mathbf{x}_{0})}\log
f+\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4T}d\mathcal{L}^{N},$

where $\omega_{N}=|B_{1}|$, the volume of unit $N$-ball. Taking $R=\sqrt{T}$,
yields

$\log
f(\mathbf{x}_{0})\leq\frac{1}{\omega_{N}T^{\frac{N}{2}}}\int_{B_{\sqrt{T}}(\mathbf{x}_{0})}\log
fd\mathcal{L}^{n}+\frac{1}{4(N+2)}.$

It follows from Jensen’s inequality that

$f(\mathbf{x}_{0})\leq
C_{0}T^{-\frac{N}{2}}\int_{B_{\sqrt{T}}(\mathbf{x}_{0})}fd\mathcal{L}^{N},\mbox{
where }C_{0}=C_{0}(N)=\omega_{N}^{-1}\exp(\frac{1}{4(N+2)}).$

To complete the proof, observe that as $f\geq 0$,

$f(\mathbf{x}_{0})\leq\int_{B_{\sqrt{T}}(\mathbf{x}_{0})}fd\mathcal{L}^{N}\leq
C_{0}T^{-\frac{N}{2}}\int_{\mathbb{R}^{N}}fd\mathcal{L}^{N}=C_{0}T^{-\frac{N}{2}}.$

∎

We now establish some useful estimates for elements of
$\mathcal{VT}_{1}^{+}(T)$.

######  Lemma 2.3.

Suppose $u\in\mathcal{VT}_{1}^{+}(T)$, $T>0$. The following hold:

  1. (1)

$\lim_{|\mathbf{x}|\to\infty}u(\mathbf{x})=0$ and there is a point
$\mathbf{z}_{0}=\mathbf{z}_{0}(u)\in\mathbb{R}^{N}$ so that

$0<u(\mathbf{z}_{0})e^{-\frac{|\mathbf{x}-\mathbf{z}_{0}|^{2}}{4T}}\leq
u(\mathbf{x})\leq u(\mathbf{z}_{0})\leq(4\pi T)^{-\frac{N}{2}};$

  2. (2)

$\lim_{|\mathbf{x}|\to\infty}|\nabla_{\mathbb{R}}u(\mathbf{x})|=0$ and

$|\nabla_{\mathbb{R}}u(\mathbf{x})|^{2}\leq\frac{4\pi}{e}(4\pi
T)^{-\frac{N+2}{2}}u(\mathbf{x})\leq\frac{4\pi}{e}(4\pi T)^{-N-1};$

  3. (3)

$|\nabla^{2}_{\mathbb{R}}u|\in L^{1}(\mathbb{R}^{N})$ and there is a constant
$C_{1}=C_{1}(N)>0$ so

$\int_{\mathbb{R}^{N}}|\nabla^{2}_{\mathbb{R}}u|d\mathcal{L}^{N}\leq
C_{1}T^{-1}.$

######  Remark 2.4.

In particular, for $u\in\mathcal{VT}_{1}^{+}(T)$, $T>0$,
$\tau(u)<\frac{1}{4\pi}u(\mathbf{z}_{0})^{-\frac{2}{N}}<\infty$.

###### Proof.

As $u\in L^{1}(\mathbb{R}^{N})$ and $u>0$, for every $\epsilon>0$, there is
$R_{\epsilon}$ so that

$0<\int_{\mathbb{R}^{N}\setminus B_{R_{\epsilon}}}ud\mathcal{L}^{N}<\epsilon.$

It follows from Lemma 2.2 that, for any $\mathbf{x}_{0}\not\in
B_{R_{\epsilon}+\sqrt{T}}$,

$u(\mathbf{x}_{0})\leq C_{0}T^{-\frac{N}{2}}\int_{\mathbb{R}^{N}\setminus
B_{R_{\epsilon}}}ud\mathcal{L}^{N}\leq C_{0}T^{-\frac{N}{2}}\epsilon.$

In other words,

$\lim_{|\mathbf{x}|\to\infty}u(\mathbf{x})=0.$

As $u$ is also continuous, there is a $\mathbf{z}_{0}$ so that

$\max_{\mathbf{x}\in\mathbb{R}^{N}}u(\mathbf{x})=u(\mathbf{z}_{0}).$

Hence, $\nabla_{\mathbb{R}}u(\mathbf{z}_{0})=\mathbf{0}$. Observe that as
$U\in\mathcal{VT}_{1}^{+}(T)$, $u$ satisfies the convexity property given by
(1.3) with $t=T$. This implies that

$-\frac{|\mathbf{x}-\mathbf{z}_{0}|^{2}}{4T}+\log u(\mathbf{z}_{0})\leq\log
u(\mathbf{x}).$

This yields the claimed lower bound $u$. The upper bound on
$u(\mathbf{z}_{0})$, and hence on $u$, follows by integrating over
$\mathbb{R}^{N}$ and using that $u$ has mass one. We have established Item
(1).

For the gradient estimate pick $\mathbf{x}_{0}\in\mathbb{R}^{N}$. Set
$\nabla_{\mathbb{R}}\log u(\mathbf{x}_{0})=\mathbf{v}$ and

$Q(\mathbf{x})=-\frac{1}{4T}|\mathbf{x}-\mathbf{x}_{0}-2T\mathbf{v}|^{2}+\log
u(\mathbf{x}_{0})+T|\mathbf{v}|^{2}.$

Thus, $Q(\mathbf{x}_{0})=\log u(\mathbf{x}_{0}),$
$\nabla_{\mathbb{R}}Q(\mathbf{x}_{0})=\mathbf{v}=\nabla_{\mathbb{R}}\log
u(\mathbf{x}_{0})$ and $\nabla^{2}_{\mathbb{R}}Q=-\frac{g_{\mathbb{R}}}{2T}$.
The convexity of $\log u$, i.e., (1.3), ensures that

$Q(\mathbf{x})\leq\log u(\mathbf{x}).$

Hence, integrating $\exp(Q)$ and using the mass of $u$ is one yields,

$(4\pi T)^{\frac{N}{2}}u(\mathbf{x}_{0})e^{T|\mathbf{v}|^{2}}\leq 1.$

Using the elementary identity

$-x\log x\leq e^{-1}$

and the upper bound on $u$ yields the claimed bounds

$\displaystyle|\nabla u(\mathbf{x}_{0})|^{2}$
$\displaystyle\leq-T^{-1}u^{2}(\mathbf{x}_{0})\log\left((4\pi
T)^{\frac{N}{2}}u(\mathbf{x}_{0})\right)$
$\displaystyle\leq\frac{4\pi}{e}(4\pi
T)^{-\frac{N}{2}-1}u(\mathbf{x}_{0})\leq\frac{4\pi}{e}(4\pi T)^{-N-1}.$

Moreover, as $u(\mathbf{x})$ tends to zero as $|\mathbf{x}|\to\infty$ this
implies same is true of $\nabla u(\mathbf{x})$. This completes proof of Item
(2). While we don’t use it, we note the lower bound on $u$ yields

$|\nabla_{\mathbb{R}}u(\mathbf{x}_{0})|^{2}\leq\frac{|\mathbf{x}_{0}-\mathbf{z}_{0}|^{2}}{4T^{2}}u^{2}(\mathbf{x}_{0}).$

For Item (3), we first observe that, in the sense of symmetric matrices,

$-\frac{1}{2T}g_{\mathbb{R}}\leq\nabla^{2}_{\mathbb{R}}\log
u=\frac{\nabla^{2}_{\mathbb{R}}u}{u}-\frac{\nabla_{\mathbb{R}}u\otimes\nabla_{\mathbb{R}}u}{u^{2}}\leq\frac{\nabla^{2}_{\mathbb{R}}u}{u}.$

By Item (1), $0<u<(4\pi T)^{-\frac{N}{2}}$ so

(2.1) $-2\pi(4\pi
T)^{-\frac{N+2}{2}}g_{\mathbb{R}}\leq-\frac{u}{2T}g_{\mathbb{R}}\leq\nabla^{2}_{\mathbb{R}}u\mbox{
and }-\frac{N}{2T}u\leq\Delta_{\mathbb{R}}u.$

An immediate consequence of this is that

$|\Delta_{\mathbb{R}}u|\leq\Delta_{\mathbb{R}}u+NT^{-1}u.$

For $R>0$, let $\phi_{R}=(1-(4R)^{-2}|\mathbf{x}|^{2})^{2}$ this is a smooth
function which vanishes to first order on $\partial B_{2R}$ and satisfies
$\phi_{R}\leq 1$ and $\Delta_{\mathbb{R}}\phi_{R}\leq C(N)R^{-2}$ on $B_{2R}$
where $C(N)>0$ is a dimensional constant. The above inequalities and an
integration by parts implies

$\displaystyle\frac{1}{4}$
$\displaystyle\int_{B_{R}}|\Delta_{\mathbb{R}}u|d\mathcal{L}^{N}\leq\int_{B_{2R}}\phi_{R}|\Delta_{\mathbb{R}}u|d\mathcal{L}^{N}\leq\int_{B_{2R}}\phi_{R}\left(\Delta_{\mathbb{R}}u+NT^{-1}u\right)d\mathcal{L}^{N}$
$\displaystyle=\int_{B_{2R}}\left(\Delta_{\mathbb{R}}\phi_{R}+NT^{-1}\phi_{R}\right)ud\mathcal{L}^{N}\leq\left(C(N)R^{-2}+NT^{-1}\right)\int_{B_{2R}}ud\mathcal{L}^{N}.$

As $u$ has mass one, sending $R\to\infty$ shows that $\Delta_{\mathbb{R}}u\in
L^{1}(\mathbb{R}^{N})$ and

$\int_{\mathbb{R}^{N}}|\Delta_{\mathbb{R}}u|d\mathcal{L}^{N}\leq 4NT^{-1}.$

Finally, we observe that the lower bounds on the eigenvalues of
$\nabla^{2}_{\mathbb{R}}u$ imply that

$|\nabla^{2}_{\mathbb{R}}u|^{2}\leq
N\left(|\Delta_{\mathbb{R}}u|+\frac{(N-1)u}{2T}\right)^{2}\mbox{ and so
}|\nabla^{2}_{\mathbb{R}}u|\leq\sqrt{N}|\Delta_{\mathbb{R}}u|+\frac{N^{\frac{3}{2}}}{2T}u.$

The final claim follows after making an appropriate choice of
$C_{1}=C_{1}(N)>0$. ∎

Next we establish some properties of the evolution by (1.2) of elements in
$\mathcal{VT}_{1}^{+}(\tau_{0})$.

######  Lemma 2.5.

For $\tau_{0}>0$ and $u_{0}\in\mathcal{VT}_{1}^{+}(\tau_{0})$, let
$U(t,\cdot)=\mathcal{H}_{t}u_{0}$ be the solution of the heat equation on
$[0,\infty)\times\mathbb{R}^{N}$ given by (1.2). This solution has the
following properties:

  1. (1)

$\int_{\mathbb{R}^{N}}U(t,\mathbf{x})=1$, for all $t\geq 0$;

  2. (2)

$U(t,\mathbf{x})\geq
u_{0}(\mathbf{z}_{0})(4\pi(t+\tau_{0}))^{-\frac{N}{2}}e^{-\frac{|\mathbf{x}-\mathbf{z}_{0}|^{2}}{4(t+\tau_{0})}}>0.$

  3. (3)

$U(t,\mathbf{x})+\tau_{0}^{\frac{1}{2}}|\nabla_{\mathbb{R}}U(t,\mathbf{x})|\leq
C_{2}\tau_{0}^{-\frac{N}{2}}$, for all $t\geq 0$;

  4. (4)

$|\nabla^{2}_{\mathbb{R}}U(t,\mathbf{x})|\leq
C_{2}\tau_{0}^{-1}t^{-\frac{N}{2}}$ for all $t>0$;

Here $C_{2}=C_{2}(N)>0$ and $\mathbf{z}_{0}=\mathbf{z}_{0}(u_{0})$ is given by
Lemma 2.3. Moreover,

  1. (5)

$\nabla^{2}_{\mathbb{R}}U(t,\mathbf{x})\geq-\frac{U(t,\mathbf{x})}{2\tau_{0}}g_{\mathbb{R}}$
for all $t\geq 0$;

  2. (6)

$\lim_{t\to 0^{+}}U(t,\cdot)=u_{0}$, in $L^{1}(\mathbb{R}^{N})$,
$C^{1}(\mathbb{R}^{N})$ and $C^{2}_{loc}(\mathbb{R}^{N})$;

  3. (7)

$U,\nabla_{\mathbb{R}}U,\nabla^{2}_{\mathbb{R}}U\in
C^{0}([0,\infty)\times\mathbb{R}^{N}).$

######  Remark 2.6.

In fact, $U$ is the unique continuous and non-negative solution of the heat
equation on $[0,\infty)\times\mathbb{R}^{N}$ with initial data $u_{0}$ – see
[25, 26, 33].

###### Proof.

The first item is a consequence of the fact that $u_{0}$ has mass one and
Fubini’s theorem. The second follows from Item (1) of Lemma 2.3 and an
explicit integration of (1.2).

For the remaining items we observe that as $u_{0}$ is integrable and for $t>0$
and $i\geq 0$,

$|\nabla_{\mathbb{R},\mathbf{x}}^{i}\mathcal{H}(t,\mathbf{x},\mathbf{y})|\leq
C_{N,i}t^{-\frac{N+2i}{2}}$

we may differentiate under the integral sign and obtain

$\nabla^{i}_{\mathbb{R}}U(t,\mathbf{x})=\int_{\mathbb{R}^{N}}u_{0}(\mathbf{y})\nabla^{i}_{\mathbb{R},\mathbf{x}}\mathcal{H}(t,\mathbf{x},\mathbf{y})d\mathcal{L}^{N}.$

As the symmetries of the heat kernel ensure

$\nabla_{\mathbb{R},\mathbf{x}}^{i}\mathcal{H}(t,\mathbf{x},\mathbf{y})=(-1)^{i}\nabla_{\mathbb{R},\mathbf{y}}^{i}\mathcal{H}(t,\mathbf{x},\mathbf{y}),$

The uniform boundedness of $u_{0}$ and $\nabla_{\mathbb{R}}u_{0}$ and fact
that $|\nabla^{2}_{\mathbb{R}}u_{0}|\in L^{1}(\mathbb{R}^{N})$ allows one to
integrate by parts and obtain, for a fixed parallel unit vector field
$\mathbf{v}$ and $t>0$,

$\mathbf{v}\cdot\nabla_{\mathbb{R}}U(t,\mathbf{x})=\int_{\mathbb{R}^{N}}\mathbf{v}\cdot\nabla_{\mathbb{R}}u_{0}(\mathbf{y})\mathcal{H}(t,\mathbf{x},\mathbf{y})d\mathcal{L}^{N}(\mathbf{y});$
$\nabla^{2}_{\mathbb{R}}U(t,\mathbf{x})(\mathbf{v},\mathbf{v})=\int_{\mathbb{R}^{N}}\nabla^{2}_{\mathbb{R}}u_{0}(\mathbf{v},\mathbf{v})\mathcal{H}(t,\mathbf{x},\mathbf{y})d\mathcal{L}^{N}(\mathbf{y}).$

Items (3) is then immediate consequences of the uniform bounds of Item (1) and
(2) of Lemma 2.3. Likewise, Item (4) follows directly from the integral bound
of Item (3) of Lemma 2.3. It is also clear Item (5) follows from (2.1) and
representation formula (1.2).

As both $u_{0}$ and $\nabla_{\mathbb{R}}u_{0}$ are continuous and tend to zero
at infinity, they are both uniformly continuous and bounded. Standard results
about approximations to the identity give

$\lim_{t\to
0^{+}}\|U(t,\cdot)-u_{0}(\cdot)\|_{\infty}+\|\nabla_{\mathbb{R}}U(t,\cdot)-\nabla_{\mathbb{R}}u_{0}(\cdot)\|_{\infty}=0.$

It is also an elementary fact that because $|\nabla^{2}_{\mathbb{R}}u_{0}|\in
L^{1}(\mathbb{R}^{N})$ and $u_{0}$ is $C^{2}$, for any $\mathbf{x}_{0}$

$\lim_{t\to
0^{+},\mathbf{x}\to\mathbf{x}_{0}}\nabla^{2}_{\mathbb{R}}U(t,\mathbf{x})=\nabla^{2}_{\mathbb{R}}u_{0}(\mathbf{x}_{0}).$

Moreover, this convergence can be taken to be uniform on any compact subset of
$\mathbb{R}^{N}$. Together this gives Item (6). Item (7) is an immediate
consequence. ∎

We can now establish the linear increase in virtual time under heat flow.

######  Proposition 2.7.

Suppose $u_{0}\in\mathcal{VT}_{1}^{+}(\tau_{0})$ for $\tau_{0}>0$ and let
$U(t,\cdot)=\mathcal{H}_{t}u_{0}$ be the solution of the heat equation on
$[0,\infty)\times\mathbb{R}^{N}$ given by (1.2). For all $t\geq 0$,

$U(t,\cdot)\in\mathcal{VT}_{1}^{+}(\tau_{0}+t).$

Moreover, if $2(\tau_{0}+t_{0})\nabla^{2}_{\mathbb{R}}\log
U(t_{0},\mathbf{x}_{0})+g_{\mathbb{R}}=0$ for some $t_{0}>0$ and
$\mathbf{x}_{0}\in\mathbb{R}^{N}$, then
$U(t,\cdot)=\mathcal{H}(\tau_{0}+t,\cdot,\mathbf{x}_{0}^{\prime})\in\mathcal{G}_{1}(\tau_{0}+t)$
for some $\mathbf{x}_{0}^{\prime}\in\mathbb{R}^{N}$ and all $t\geq 0$.

###### Proof.

The positivity and unit mass condition are preserved by Lemma 2.5. Hence, it
suffices to establish that, for all $t>0$,

$(\tau_{0}+t)\nabla^{2}_{\mathbb{R}}\log U(t,\mathbf{x})+g_{\mathbb{R}}\geq
0.$

To that end, fix a parallel unit length vector field $\mathbf{v}$ on
$\mathbb{R}^{N}$. By the rotational symmetry of the problem, is enough to take
$\mathbf{v}=\mathbf{e}_{1}$. Set

$\displaystyle H(t,\mathbf{x})$
$\displaystyle=U(t,\mathbf{x})\nabla^{2}_{\mathbb{R}}\log
U(t,\mathbf{x})(\mathbf{v},\mathbf{v})=U(t,\mathbf{x})\partial^{2}_{1}\log
U(t,\mathbf{x})$
$\displaystyle=\partial^{2}_{1}U(t,\mathbf{x})-\frac{(\partial_{1}U(t,\mathbf{x}))^{2}}{U(t,\mathbf{x})}.$

The hypotheses ensure that

$2\tau_{0}H(0,\mathbf{x})+u_{0}(\mathbf{x})\geq 0.$

We compute that

$\displaystyle\partial_{t}H=\partial_{1}^{2}\partial_{t}U-\frac{2\partial_{1}U\partial_{1}\partial_{t}U}{U}+\frac{(\partial_{1}U)^{2}\partial_{t}U}{U^{2}}$
$\displaystyle=\partial_{1}^{2}\Delta_{\mathbb{R}}U-\frac{2\partial_{1}U\partial_{1}\Delta_{\mathbb{R}}U}{U}+\frac{(\partial_{1}U)^{2}\Delta_{\mathbb{R}}U}{U^{2}}$
$\displaystyle=\Delta_{\mathbb{R}}\partial_{1}^{2}U-\frac{\Delta_{\mathbb{R}}(\partial_{1}U)^{2}-2|\nabla_{\mathbb{R}}\partial_{1}U|^{2}}{U}-(\partial_{1}U)^{2}(\Delta_{\mathbb{R}}U^{-1})+2\frac{(\partial_{1}U|\nabla_{\mathbb{R}}U|)^{2}}{U^{3}}$
$\displaystyle=\Delta_{\mathbb{R}}\partial_{1}^{2}U-\Delta_{\mathbb{R}}\left(\frac{(\partial_{1}U)^{2}}{U}\right)+\frac{2|\nabla_{\mathbb{R}}\partial_{1}U|^{2}}{U}-\frac{4\partial_{1}U\nabla_{\mathbb{R}}U\cdot\nabla_{\mathbb{R}}\partial_{1}U}{U^{2}}+2\frac{(\partial_{1}U|\nabla_{\mathbb{R}}U|)^{2}}{U^{3}}$
$\displaystyle=\Delta_{\mathbb{R}}H+2U^{-1}|\nabla_{\mathbb{R}}\partial_{1}U-U^{-1}\partial_{1}U\nabla_{\mathbb{R}}U|^{2}\geq\Delta_{\mathbb{R}}H+2U^{-1}\left|\partial_{1}^{2}U-U^{-1}(\partial_{1}U)^{2}\right|^{2}$
$\displaystyle=\Delta_{\mathbb{R}}H+2U^{-1}H^{2}.$

Now set

$N(t,\mathbf{x})=2(t+\tau_{0})H(t,\mathbf{x})+U(t,\mathbf{x})$

we see that $N(0,\mathbf{x})\geq 0$ and

$(\partial_{t}-\Delta_{\mathbb{R}})N\geq
2H+4(t+\tau_{0})U^{-1}H^{2}=2U^{-1}HN$

As $U(0,\mathbf{x})=u_{0}\in\mathcal{VT}_{1}^{+}(\tau_{0})$, Lemmas 2.3 and
2.5 together imply that

$\displaystyle H(t,\mathbf{x})$
$\displaystyle\geq-\frac{1}{2\tau_{0}}u_{0}(\mathbf{x})-C(t+\tau_{0})^{\frac{N}{2}}e^{\frac{|\mathbf{x}-\mathbf{z}_{0}|^{2}}{4(t+\tau_{0})}}$
$\displaystyle\geq-
C-C(t+\tau_{0})^{\frac{N}{2}}e^{\frac{|\mathbf{x}-\mathbf{z}_{0}|^{2}}{4(t+\tau_{0})}}$

where $C=C(\tau_{0},N,u_{0})>0$ is a constant and
$\mathbf{z}_{0}=\mathbf{z}_{0}(u_{0})\in\mathbb{R}^{N}$ is a fixed point.

Now let

$\Psi_{T}(t,\mathbf{x})=e^{t}(T-t)^{-\frac{N}{2}}e^{\frac{|\mathbf{x}-\mathbf{z}_{0}|^{2}}{4(T-t)}}$

One verifies that $\Psi_{T}$ satisfies

$\left(\partial_{t}-\Delta_{\mathbb{R}}\right)\Psi_{T}=\Psi_{T}$

on $(-\infty,T]$. Moreover, $\Psi_{T}$ has very rapid growth as
$\mathbf{x}\to\infty$.

Set

$N_{\epsilon}(t,\mathbf{x})=N(t,\mathbf{x})+\epsilon\Psi_{\frac{\tau_{0}}{2}}(t,\mathbf{x})$

which is smooth for $t\in[0,\frac{\tau_{0}}{2})$ One readily checks that

$N_{\epsilon}(0,\mathbf{x})\geq\epsilon\Psi_{\frac{\tau_{0}}{2}}(0,\mathbf{x})\geq
T^{-\frac{N}{2}}\epsilon>0$

where $C=C(\tau_{0},N)$. Likewise, for $t\in(0,\frac{\tau_{0}}{2})$ and
outside of a very large ball,

$N_{\epsilon}(t,\mathbf{x})\geq\frac{1}{2}\epsilon\Psi_{\frac{\tau_{0}}{2}}(t,\mathbf{x})>0.$

Now suppose that $N_{\epsilon}(t_{0},{\mathbf{x}}_{0})<0$ for some
$t_{0}\in(0,\frac{\tau_{0}}{2})$. It follows that there is a value
$t_{1}\in(0,t_{0}]$ and $\mathbf{x}_{1}\in\mathbb{R}^{N}$

$0>N_{\epsilon}(t_{0},{\mathbf{x}}_{0})\geq
N_{\epsilon}(t_{1},\mathbf{x}_{1})=\min_{[0,t_{1}]\times\mathbb{R}^{N}}N_{\epsilon}.$

Hence, at $(t_{1},\mathbf{x}_{1})$,

$\displaystyle 0$ $\displaystyle\geq\left(\frac{\partial}{\partial
t}-\Delta_{\mathbb{R}}\right)N_{\epsilon}=2HU^{-1}N=2U^{-1}HN+\epsilon\Psi_{\frac{\tau_{0}}{2}}>2H(1+2(t_{1}+\tau_{0})U^{-1}H),$

which is incompatible with $H(t_{1},\mathbf{x}_{1})\geq 0$ and so at
$(t_{1},\mathbf{x}_{1})$

$0>H\mbox{ and }U^{-1}N=1+2(t_{1}+\tau_{0})U^{-1}H>0.$

That is,

$N_{\epsilon}(t_{1},\mathbf{x}_{1})>N(t_{1},\mathbf{x}_{1})>0$

which is a contradiction. Hence, $N_{\epsilon}>0$ on $[0,\frac{\tau_{0}}{2})$.
By taking $\epsilon\to 0^{+}$ we see that

$N(t,\cdot)\geq 0$

for $t\in(0,\frac{\tau_{0}}{2})$. In particular,

$U\left(\frac{\tau_{0}}{4},\cdot\right)\in\mathcal{VT}_{1}^{+}\left(\tau_{0}+\frac{\tau_{0}}{4}\right).$

As $\tau_{0}+\frac{\tau_{0}}{4}\geq\tau_{0}$, one may translate in time and
apply the same argument to obtain

$N(t,\cdot)\geq 0$

on $[0,\frac{3}{4}\tau_{0})$. Iterating this completes the proof that
$U(t,\cdot)\in\mathcal{VT}_{1}^{+}(\tau_{0}+t)$.

Finally, if for some $t_{0}>0$, $\mathbf{x}_{0}$,
$2(\tau_{0}+t_{0})\nabla^{2}_{\mathbb{R}}\log
U(t_{0},\mathbf{x}_{0})+g_{\mathbb{R}}=0$, then for any choice of $\mathbf{v}$
we have that $N\geq 0$ and $N(t_{0},\mathbf{x}_{0})=0$. By the strict maximum
principle this means $N$ identically vanishes and the claim follows.

∎

While we don’t directly use it in this paper, the fact that the long time
behavior of a solution of the heat equation is towards the Gaussian, i.e.,
$\mathcal{H}(t,\mathbf{x},\mathbf{0})$ explains some of the results and so for
the reader’s benefit we include a precise statement. This may be thought of as
a version of the Central Limit Theorem – see [37] for a proof and further
discussion.

######  Proposition 2.8.

Suppose $u_{0}\in\mathcal{VT}_{1}^{+}(T)$, $T>0$. Let
$U(t,\cdot)=\mathcal{H}_{t}u_{0}$ be the solution of the heat equation on
$[0,\infty)\times\mathbb{R}^{N}$ given by (1.2). For any $T_{0}\in\mathbb{R}$
and $\mathbf{x}_{0}\in\mathbb{R}^{N}$:

  1. (1)

$\lim_{t\to\infty}\|U(t,\cdot)-\mathcal{H}(t-{T}_{0},\cdot,\mathbf{x}_{0})\|_{1}=0$;

  2. (2)

$\lim_{t\to\infty}t^{\frac{N}{2}}\|U(t,\cdot)-\mathcal{H}(t-T_{0},\cdot,\mathbf{x}_{0})\|_{\infty}=0$.

##  3\. Properties of Virtual Entropy

In this section we establish some properties of of the virtual entropy. First,
we recall that finite Colding-Minicozzi entropy is equivalent to uniform
bounds on area ratios.

######  Lemma 3.1.

For $\Sigma\subset\mathbb{R}^{N}$ an $n$-dimensional proper submanifold

$|\Sigma\cap B_{R}(p)|\leq e^{\pi}\lambda_{CM}^{n}[\Sigma]R^{n}.$

Conversely, suppose there is a $\Theta\geq 1$ so, for all $p$ and $R>0$,

$|\Sigma\cap B_{R}(p)|\leq\Theta\omega_{n}R^{n},$

where $\omega_{n}$ is the volume of the unit ball in $\mathbb{R}^{n}$, then

$\lambda_{CM}^{n}[\Sigma]\leq\Theta.$

###### Proof.

Let $T=(4\pi)^{-1}R^{2}$, one has

$R^{-n}e^{-\pi}1_{B_{R}(p)}\leq(4\pi
T)^{-\frac{n}{2}}e^{-\frac{|\mathbf{x}-\mathbf{x}(p)|^{2}}{4T}}$

Hence,

$|\Sigma\cap B_{R}(p)|=\int_{B_{R}(p)\cap\Sigma}d\mathcal{H}^{n}\leq
e^{\pi}\lambda_{CM}^{n}[\Sigma]R^{n}.$

This establishes the first claim.

For the second we observe that arbitrary scalings and translations of $\Sigma$
satisfy the same area ratio bounds. Hence, it is enough to check using the co-
area formula that

$\displaystyle\int_{\Sigma}e^{-\frac{|\mathbf{x}|^{2}}{4}}d\mathcal{H}^{n}$
$\displaystyle=\int_{0}^{\infty}e^{-\frac{R^{2}}{4}}\frac{d}{dR}|\Sigma\cap
B_{R}|dR=\int_{0}^{\infty}\frac{R}{2}e^{-\frac{R^{2}}{4}}|\Sigma\cap B_{R}|dR$
$\displaystyle\leq\frac{\Theta\omega_{n}}{2}\int_{0}^{\infty}R^{n+1}e^{-\frac{R^{2}}{4}}dR=\Theta(4\pi)^{\frac{n}{2}}.$

Where the final equality can be verified by computing on the flat $n$-plane.
It follows that

$\lambda_{CM}^{n}[\Sigma]\leq\Theta.$

∎

We next establish that the virtual and Colding-Minicozzi entropies are
comparable.

######  Proposition 3.2.

Let $\Omega\subset\mathbb{R}^{N}$ be a non-empty open subset and
$\Sigma\subset\mathbb{R}^{N}$ an $n$-dimensional submanifold. There is a
constant $C_{3}=C_{3}(N)\geq 1$ so that if $f\in\mathcal{VT}_{1}^{+}(T)$,
$T>0$, then

$(4\pi T)^{\frac{N-n}{2}}\int_{\Sigma\cap\Omega}fd\mathcal{H}^{n}\leq
C_{3}\lambda_{CM}^{n}[\Sigma]\int_{\mathcal{T}_{2\sqrt{T}}(\Omega)}fd\mathcal{L}^{N}.$

where $\mathcal{T}_{r}(\Omega)=\bigcup_{p\in\Omega}B_{r}(p)$. In particular,

$\lambda_{CM}^{n}[\Sigma]\leq\lambda_{V}^{n}[\Sigma]\leq
C_{3}\lambda_{CM}^{n}[\Sigma].$

###### Proof.

It follows from Lemma 2.2 that for any $q\in\mathbb{R}^{N}$

$\displaystyle\int_{\Sigma\cap B_{\sqrt{T}}(q)}fd\mathcal{H}^{n}$
$\displaystyle\leq C_{0}\frac{|\Sigma\cap
B_{\sqrt{T}}(q)|}{\omega_{N}T^{\frac{N}{2}}}\int_{B_{2\sqrt{T}}(q)}fd\mathcal{L}^{N}$
$\displaystyle\leq\frac{C_{0}e^{\pi}}{\omega_{N}T^{\frac{N-n}{2}}}\lambda_{CM}^{n}[\Sigma]\int_{B_{2\sqrt{T}}(q)}fd\mathcal{L}^{N}$

where the second inequality used Lemma 3.1 and $f>0$.

Pick a maximal set of points $\left\\{q_{i}\right\\}$ so
$q_{i}\in\Sigma\cap\Omega$ are extrinsic distance $\geq\sqrt{T}$ and so every
point in $\Sigma\cap\Omega$ is distance $\leq\sqrt{T}$ from some $q_{i}$. The
doubling property of $\mathbb{R}^{N}$ and choice of points implies there is a
constant $D(N)>0$ so for any $q_{i}$ the set of points $\left\\{q_{j}:q_{j}\in
B_{4\sqrt{T}}(q_{i})\right\\}$ has at most $D(N)$ points.

It follows that

$\int_{\Sigma\cap\Omega}fd\mathcal{H}^{n}\leq\sum_{i=1}^{\infty}\int_{\Sigma\cap\Omega\cap
B_{\sqrt{T}}(q_{i})}fd\mathcal{H}^{n}\leq\sum_{i=1}^{\infty}\frac{C_{0}e^{\pi}}{\omega_{N}T^{\frac{N-n}{2}}}\lambda_{CM}^{n}[\Sigma]\int_{B_{2\sqrt{T}}(q_{i})}fd\mathcal{L}^{N}.$

Hence, as $f>0$,

$\int_{\Sigma\cap\Omega}fd\mathcal{H}^{n}\leq\frac{C_{0}D(N)e^{\pi}}{\omega_{N}T^{\frac{N-n}{2}}}\lambda_{CM}^{n}[\Sigma]\int_{\mathcal{T}_{2\sqrt{T}}(\Omega)}fd\mathcal{L}^{N}.$

The first claim follows with

$C_{3}=C_{3}(N)=(4\pi)^{\frac{N}{2}}\omega_{N}^{-1}e^{\pi}C_{0}D(N)\geq(4\pi)^{\frac{N-n}{2}}\omega_{N}^{-1}e^{\pi}C_{0}D(N).$

By taking $\Omega=\mathbb{R}^{N}$ we see that for all
$f\in\mathcal{VT}_{1}^{+}(T)$,

$(4\pi T)^{\frac{N-n}{2}}\int_{\Sigma}fd\mathcal{H}^{n}\leq
C_{3}\lambda_{CM}^{n}[\Sigma].$

As $\lambda_{CM}^{n}[\Sigma]\leq\lambda_{V}^{n}[\Sigma]$ follows from
$\mathcal{G}_{1}(T)\subset\mathcal{VT}_{1}^{+}(T)$, it follows that

$\lambda_{CM}^{n}[\Sigma]\leq\lambda_{V}^{n}[\Sigma]\leq
C_{3}\lambda_{CM}^{n}[\Sigma]$

which verifies the claimed bounds and forces $C_{3}\geq 1$. ∎

Virtual entropy is invariant under the natural symmetries of $\mathbb{R}^{N}$.

######  Lemma 3.3.

Let $\Sigma\subset\mathbb{R}^{N}$ be an $n$-dimensional submanifold. If
$\Sigma^{\prime}$ is obtained from $\Sigma$ by a rigid motion and scaling,
then

$\lambda^{n}_{V}[\Sigma]=\lambda^{n}_{V}[\Sigma^{\prime}].$

###### Proof.

By Proposition 3.2, $\lambda^{n}_{V}[\Sigma]=\infty$ if and only if
$\lambda_{CM}^{n}[\Sigma]=\infty$. In this case,
$\lambda_{CM}^{n}[\Sigma^{\prime}]=\infty$ and the same is true of
$\lambda^{n}_{V}[\Sigma^{\prime}]$.

Hence, we may suppose
$\lambda_{CM}^{n}[\Sigma],\lambda_{CM}^{n}[\Sigma^{\prime}]<\infty$. If
$R:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is a rigid motion, then
$f\in\mathcal{VT}_{1}^{+}(T)$ if and only if $f\circ
R\in\mathcal{VT}_{1}^{+}(T)$ and

$\int_{\Sigma}f\circ Rd\mathcal{H}^{n}=\int_{R(\Sigma)}fd\mathcal{H}^{n}.$

Likewise, if $S_{\rho}(\mathbf{x})=\rho\mathbf{x}$ is dilation by $\rho>0$,
then $f\in\mathcal{VT}_{1}^{+}(T)$ if and only if $\rho^{N}f\circ
S_{\rho}\in\mathcal{VT}_{1}^{+}(\rho^{-2}T)$ and

$\int_{\Sigma}f\circ
S_{\rho}d\mathcal{H}^{n}=\rho^{-n}\int_{S_{\rho}(\Sigma)}fd\mathcal{H}^{n}.$

These observations and the definition of virtual entropy conclude the proof. ∎

Following [27], we reinterpret the monotonicity property of [30] and take care
to establish it for non-closed submanifolds.

######  Proposition 3.4.

Let $t\mapsto\Sigma_{t}\subset\mathbb{R}^{N},t\in[0,T_{0})$ be a flow of
$n$-dimensional proper submanifolds with
$\lambda_{CM}^{n}[\Sigma_{t}]\leq\Lambda<\infty$ for $t\in[0,T_{0})$. Fix
$[T_{1},T_{2}]\subset[0,T_{0})$ and $\tau>0$. If
$W:[T_{1},T_{2}]\times\mathbb{R}^{N}\to\mathbb{R}$ satisfies

  1. (1)

$\left(\frac{\partial}{\partial t}+\Delta_{\mathbb{R}^{N}}\right)W=0$;

  2. (2)

$W(t,\cdot)\in\mathcal{VT}_{1}^{+}(T_{2}+\tau-t)$, $t\in[T_{1},T_{2}]$,

then

$(4\pi\tau)^{\frac{N-n}{2}}\int_{\Sigma_{T_{2}}}W(T_{2},\cdot)d\mathcal{H}^{n}\leq(4\pi(T_{2}-T_{1}+\tau))^{\frac{N-n}{2}}\int_{\Sigma_{T_{1}}}W(T_{1},\cdot)d\mathcal{H}^{n}.$

###### Proof.

Fix $R\geq 1$ and let $\phi_{R}\geq 0$ be a smooth cutoff function that is
identically $1$ in $B_{R}$ and supported in $B_{2R}$ and so

$\phi_{R}+R|\nabla_{\mathbb{R}}\phi_{R}|+R^{2}|\nabla^{2}_{\mathbb{R}}\phi_{R}|\leq\kappa$

where $\kappa=\kappa(N)>1$. Set

$V(t,\mathbf{x})=(4\pi(T_{2}+\tau-t))^{\frac{N-n}{2}}W(t,\mathbf{x})>0.$

Using the notation of [27, Section 3.14], for $t\in(T_{1},T_{2})$ if we set

$F_{R}(t)=\int_{\Sigma_{t}}\phi_{R}(\cdot)V(t,\cdot)d\mathcal{H}^{n},$

then, following [27, Section 3.14],

$F_{R}^{\prime}(t)=\int_{\Sigma_{t}}\left(\left(\frac{d}{dt}-\Delta_{\Sigma_{t}}\right)\phi_{R}-\left|\mathbf{H}_{\Sigma_{t}}+\frac{\nabla^{\perp}_{\Sigma_{t}}V}{V}\right|^{2}\phi\right)V+\phi_{R}Q_{V}d\mathcal{H}^{n}$

where

$Q_{V}=Q_{V}(t,\mathbf{x},N_{p}\Sigma_{t})=\left(\frac{\partial}{\partial
t}+\Delta_{\mathbb{R}}\right)V-\sum_{i=1}^{N-n}\left(\nabla^{2}_{\mathbb{R}}V(E_{i},E_{i})-\frac{(E_{i}\cdot\nabla_{\mathbb{R}}V)^{2}}{V}\right)$

and $E_{1},\ldots,E_{N-n}$ is a choice of orthonormal frame of the normal
bundle. One computes

$\displaystyle Q_{V}(t,\mathbf{x},N_{p}\Sigma_{t})$
$\displaystyle=-V\sum_{i=1}^{N-n}\left(\nabla_{\mathbb{R}}^{2}\log
V(E_{i},E_{i})+\frac{1}{2(T_{2}+\tau-t)}\right)$
$\displaystyle=-V\sum_{i=1}^{N-n}\left(\nabla_{\mathbb{R}}^{2}\log
W(E_{i},E_{i})+\frac{g_{\mathbb{R}}(E_{i},E_{i})}{2(T_{2}+\tau-t)}\right).$

In particular, as $W(t,\cdot)\in\mathcal{VT}_{1}^{+}(T_{2}+\tau-t)$ and $V>0$,

$Q_{V}\leq 0.$

Using Proposition 3.2, it follows that

$F_{R}(T_{2})-F_{R}(T_{1})\leq\kappa
R^{-1}\int_{T_{1}}^{T_{2}}\int_{\Sigma_{t}}Vd\mathcal{H}^{n}dt\leq
C_{3}\kappa\Lambda R^{-1}(T_{2}-T_{1}).$

Hence, sending $R\to\infty$ yields

$\int_{\Sigma_{T_{2}}}Vd\mathcal{H}^{n}\leq\int_{\Sigma_{T_{1}}}Vd\mathcal{H}^{n}.$

Expressing $V$ in terms of $W$ completes the proof. ∎

We can now prove the claimed monotonicity of virtual entropy, i.e., Theorem
1.6

###### Proof of Theorem 1.6.

If $\lambda_{V}^{n}[\Sigma_{0}]<\infty$, then the same is true of
$\lambda_{CM}^{n}[\Sigma_{0}]$ and so the monotonicity of Colding-Minicozzi
entropy ensures this for each $\Sigma_{t}$. In particular,
$\lambda_{V}^{n}[\Sigma_{t}]$ is also finite. Hence, for any $T_{2}\in(0,T)$
and $\epsilon>0$, there is a $\tau_{\epsilon}>0$ and
$u_{T_{2},\epsilon}\in\mathcal{VT}_{1}^{+}(\tau_{\epsilon})$ so

$\infty>\lambda_{V}^{n}[\Sigma_{T_{2}}]\geq(4\pi\tau_{\epsilon})^{\frac{N-n}{2}}\int_{\Sigma_{T_{2}}}u_{T_{2},\epsilon}d\mathcal{H}^{n}\geq\lambda_{V}^{n}[\Sigma_{T_{2}}]-\epsilon.$

Let $U_{T_{2},\epsilon}(t,\cdot)=\mathcal{H}_{t}u_{T_{2},\epsilon}$ so

$U_{T_{2},\epsilon}(t,\cdot)\in\mathcal{VT}_{1}^{+}(\tau_{\epsilon}+t).$

Fix a $T_{1}\in[0,T_{2})$ and, for $t\in[T_{1},T_{2}]$, let

$W(t,\cdot)=U(T_{2}-t,\cdot).$

One readily checks $W$ satisfies the hypotheses of Proposition 3.4. Hence,

$\displaystyle(4\pi\tau_{\epsilon})^{\frac{N-n}{2}}\int_{\Sigma_{T_{2}}}W(T_{2},\cdot)d\mathcal{H}^{n}\leq(4\pi(T_{2}-T_{1}+\tau_{\epsilon}))^{\frac{N-n}{2}}\int_{\Sigma_{T_{1}}}W(T_{1},\cdot)d\mathcal{H}^{n}.$

It follows that

$\displaystyle\lambda_{V}^{n}[\Sigma_{T_{2}}]-\epsilon$
$\displaystyle\leq(4\pi\tau_{\epsilon})^{\frac{N-n}{2}}\int_{\Sigma_{T_{2}}}u_{T_{2},\epsilon}d\mathcal{H}^{n}=(4\pi\tau_{\epsilon})^{\frac{N-n}{2}}\int_{\Sigma_{T_{2}}}W(T_{2},\cdot)d\mathcal{H}^{n}$
$\displaystyle\leq(4\pi(T_{2}-T_{1}+\tau_{\epsilon}))^{\frac{N-n}{2}}\int_{\Sigma_{T_{1}}}W(T_{1},\cdot)d\mathcal{H}^{n}\leq\lambda_{V}^{n}[\Sigma_{T_{1}}].$

Where we used that
$W(T_{1},\cdot)\in\mathcal{VT}_{1}^{+}(T_{2}-T_{1}+\tau_{\epsilon})$ for the
final inequality. As $\epsilon>0$ and $0\leq T_{1}<T_{2}<T$ are arbitrary the
result follows. ∎

Using this monotonicity and an observation from [34] we obtain precise
estimates on the integrals in the definition of $\lambda_{V}^{n}$ in terms of
the Colding-Minicozzi entropy.

######  Lemma 3.5.

For $T>0$, let $t\in[-T,0]\mapsto\Sigma_{t}\subset\mathbb{R}^{N}$ be the mean
curvature flow of $n$-dimensional proper submanifolds with
$\lambda_{V}^{n}[\Sigma_{t}]<\infty$. If $\tau_{0}>0$ and
$u_{0}\in\mathcal{VT}_{1}^{+}(\tau_{0})$, then

$(4\pi\tau_{0})^{\frac{N-n}{2}}\int_{\Sigma_{0}}u_{0}d\mathcal{H}^{n}\leq(1+T^{-1}\tau_{0})^{\frac{N-n}{2}}\lambda_{CM}^{n}[\Sigma_{-T}].$

###### Proof.

Set $U(t,\cdot)=\mathcal{H}_{t}u_{0}$. By Proposition 3.4,

$\displaystyle(4\pi\tau_{0})^{\frac{N-n}{2}}\int_{\Sigma_{0}}u_{0}d\mathcal{H}^{n}$
$\displaystyle\leq(4\pi(\tau_{0}+T))^{\frac{N-n}{2}}\int_{\Sigma_{-T}}U(T,\cdot)d\mathcal{H}^{n}.$

By (1.2), Fubini’s theorem, the definition of Colding-Minicozzi entropy and
fact that $u_{0}$ has mass one yields

$\displaystyle\int_{\Sigma_{-T}}U(T,\cdot)d\mathcal{H}^{n}$
$\displaystyle=\int_{\Sigma_{-T}}\int_{\mathbb{R}^{N}}u_{0}(\mathbf{y})\mathcal{H}(T,\mathbf{x},\mathbf{y})d\mathcal{L}^{N}(\mathbf{y})d\mathcal{H}^{n}(\mathbf{x})$
$\displaystyle=\int_{\mathbb{R}^{N}}u_{0}(\mathbf{y})\int_{\Sigma_{-T}}\mathcal{H}(T,\mathbf{x},\mathbf{y})d\mathcal{H}^{n}(\mathbf{x})d\mathcal{L}^{N}(\mathbf{y})$
$\displaystyle\leq(4\pi
T)^{\frac{n-N}{2}}\lambda_{CM}^{n}[\Sigma_{-T}]\int_{\mathbb{R}^{N}}u_{0}(\mathbf{y})d\mathcal{L}^{N}(\mathbf{y})$
$\displaystyle=(4\pi T)^{\frac{n-N}{2}}\lambda_{CM}^{n}[\Sigma_{-T}].$

The claimed inequality follows immediately. ∎

We now establish (1.6).

######  Theorem 3.6.

For a $n$-dimensional proper self-shrinker, $\Sigma$,
$\lambda_{CM}^{n}[\Sigma]=\lambda_{V}^{n}[\Sigma]$.

###### Proof.

By Proposition 3.2, $\lambda_{V}^{n}[\Sigma]=\infty$ if and only if
$\lambda_{CM}^{n}[\Sigma]=\infty$ and so we may assume both quantities are
finite. The inequality $\lambda_{CM}^{n}[\Sigma]\leq\lambda_{V}^{n}[\Sigma]$
is shown in Proposition 3.2.

Fix $\tau>0$ and chose any $u_{0}\in\mathcal{VT}_{1}^{+}(\tau)$. Let
$t\in(-\infty,0]\mapsto\Sigma_{t}=\sqrt{1-t}\Sigma$ be the self-similar flow
associated to $\Sigma$ shifted in time. For fixed $t<0$, it follows from Lemma
3.5, the definition of $\Sigma_{t}$ and the scaling invariance of Colding-
Minicozzi entropy that

$(4\pi\tau_{0})^{\frac{N-n}{2}}\int_{\Sigma}u_{0}d\mathcal{H}^{n}\leq(1+(-t)^{-1}\tau_{0})^{\frac{N-n}{2}}\lambda_{CM}^{n}[\Sigma].$

Taking $t\to-\infty$ implies

$(4\pi\tau_{0})^{\frac{N-n}{2}}\int_{\Sigma}u_{0}d\mathcal{H}^{n}\leq\lambda_{CM}^{n}[\Sigma].$

As $\tau_{0}>0$ and $u_{0}\in\mathcal{VT}_{1}^{+}(\tau_{0})$ were arbitrary,
$\lambda_{V}^{n}[\Sigma]\leq\lambda_{CM}^{n}[\Sigma]$ completing the proof. ∎

######  Remark 3.7.

This result can also be proved using Proposition 2.8. This provides a
heuristic explanation for its validity for self-shrinkers.

##  4\. Li-Yau Conformal volume and Stable Conformal Volume

For $\rho>0$, recall we defined certain weights on $\mathbb{R}^{N}$ by

$W^{n}_{\rho}(\mathbf{x})=\frac{\rho^{n}}{(1+\frac{1}{4}\rho^{2}|\mathbf{x}|^{2})^{n}}\in
C^{\infty}(\mathbb{R}^{N}).$

Notice that $W^{n}_{\rho}\in L^{1}(\mathbb{R}^{N})$ if and only if $2n>N$.
When $n=N\geq 1$, this is the density on $\mathbb{R}^{N}$ from the volume form
of the unit $N$-sphere $\mathbb{S}^{N}\subset\mathbb{R}^{N+1}$ induced by the
stereographic projection map $\mathcal{S}$ of (1.1) precomposed with the
scaling $\mathbf{x}\mapsto\rho\mathbf{x}$; conjugating with $\mathcal{S}$
gives a Möbius transformation on $\mathbb{S}^{N}$. Likewise, Euclidean
translations correspond to certain Möbius transformations that fix $\infty$,
and the corresponding volume forms are given by the translated weights
$W^{N}_{\rho}(\cdot-\mathbf{x}_{0})$.

When $n\leq N$, these are the densities induced on $n$-dimensional
submanifolds by $\mathcal{S}^{*}g_{\mathbb{S}}$ by same transformations. Thus
for the $n$-dimensional affine plane $\mathbb{R}^{n}\subset\mathbb{R}^{N}$ and
$\mathbf{x}_{0}\in\mathbb{R}^{n}$,

$|\mathbb{S}^{n}|=\int_{\mathbb{R}^{n}}W^{n}_{\rho}(\cdot-\mathbf{x}_{0})d\mathcal{H}^{n}\mbox{
and
}|\mathbb{S}^{N}|=\int_{\mathbb{R}^{N}}W^{N}_{\rho}(\cdot-\mathbf{x}_{0})d\mathcal{L}^{N}.$

Likewise, when $\Sigma\subset\mathbb{R}^{N}$ a general $n$-dimensional
submanifold,

$\lambda_{LY}^{n}[\Sigma]=\sup_{\mathbf{x}_{0}\in\mathbb{R}^{N},\rho>0}\int_{\Sigma}W^{n}_{\rho}(\mathbf{x}-\mathbf{x_{0}}),$

is equal to the Li-Yau $N$-conformal volume of
$\mathcal{S}(\Sigma)\subset\mathbb{S}^{N}$. In what follows it will be
convenient to introduce the following modified family of weights for
$n,M,\rho>0$

$\hat{W}_{M,\rho}^{n}(\mathbf{x})=(4\pi\rho)^{-\frac{n}{2}}(M\rho)^{\frac{M}{2}}W^{M}_{(M\rho)^{-\frac{1}{2}}}(\mathbf{x})=(4\pi\rho)^{-\frac{n}{2}}\frac{1}{(1+\frac{1}{4M\rho}|\mathbf{x}|^{2})^{M}}$

We observe the following elementary integration result.

######  Lemma 4.1.

Suppose that $\Sigma\subset\mathbb{R}^{N}$ is a $n$-dimensional submanifold.
For $m,M\in\mathbb{N}$, and
$\mathbf{x}_{0}=(\mathbf{y}_{0},\mathbf{z}_{0})\in\mathbb{R}^{N}\times\mathbb{R}^{2m}$,

$\int_{\Sigma\times\mathbb{R}^{2m}}W^{M+2m}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2m}=\frac{C_{M,m}}{\rho^{m}}\int_{\Sigma}W^{M+m}_{\rho}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n}$

where
$\Sigma\times\mathbb{R}^{2m}\subset\mathbb{R}^{N}\times\mathbb{R}^{2m}=\mathbb{R}^{N+2m}$
and

$C_{M,m}=\frac{(4\pi)^{m}(M+m-1)!}{(M+2m-1)!}=\frac{2^{m}|\mathbb{S}^{M+2m}||\mathbb{S}^{M+2m-1}|}{|\mathbb{S}^{M+m}||\mathbb{S}^{M+m-1}|}.$

As a consequence,

$\bar{\lambda}_{LY}^{n+2m}[\Sigma]=\hat{C}_{n,m}\sup_{\rho>0,\mathbf{y}_{0}\in\mathbb{R}^{N}}\int_{\Sigma}\hat{W}_{n+m,\rho}^{n}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n},$

where

$\hat{C}_{n,m}=\left(\frac{4\pi}{n+m}\right)^{\frac{n}{2}}|\mathbb{S}^{n+2m}|^{-1}C_{n,m}.$

###### Proof.

Let us denote the coordinates on
$\mathbb{R}^{N+2}=\mathbb{R}^{N}\times\mathbb{R}^{2}$ by
$\mathbf{x}=(\mathbf{y},\mathbf{z})$. Write
$\mathbf{x}_{0}=(\mathbf{y}_{0},\mathbf{z}_{0})$. By replacing $\Sigma$ with
$\Sigma-(\mathbf{y}_{0},\mathbf{0})$ we may, without loss of generality,
assume that $\mathbf{y}_{0}=\mathbf{0}$. By Fubini’s theorem,

$\displaystyle\int_{\Sigma\times\mathbb{R}^{2}}$ $\displaystyle
W^{M+2}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2}=\int_{\Sigma}\int_{\mathbb{R}^{2}}\frac{\rho^{M+2}}{(1+\frac{\rho^{2}}{4}|\mathbf{y}|^{2}+\frac{\rho^{2}}{4}|\mathbf{z}-\mathbf{z}_{0}|^{2})^{M+2}}d\mathcal{L}^{2}(\mathbf{z})d\mathcal{L}^{n}(\mathbf{y})$
$\displaystyle=\int_{\Sigma}\frac{\rho^{M+2}}{(1+\frac{\rho^{2}}{4}|\mathbf{y}|^{2})^{M+2}}\int_{\mathbb{R}^{2}}\left(1+\frac{\rho^{2}}{4+{\rho^{2}}|\mathbf{y}|^{2}}|\mathbf{z}-\mathbf{z}_{0}|^{2}\right)^{-M-2}d\mathcal{L}^{2}(\mathbf{z})d\mathcal{H}^{n}(\mathbf{y})$
$\displaystyle=\frac{4\pi}{M+1}\frac{1}{\rho}\int_{\Sigma}W^{M+1}_{\rho}(\mathbf{y})d\mathcal{H}^{n}(\mathbf{y}).$

Where we used polar coordinates and translation invariance to evaluate

$\int_{\mathbb{R}^{2}}\left(1+\frac{\rho^{2}}{4+{\rho^{2}}|\mathbf{y}|^{2}}|\mathbf{z}-\mathbf{z}_{0}|^{2}\right)^{-M-2}d\mathcal{L}^{2}(\mathbf{z})=\frac{4\pi}{M+1}\frac{1+\frac{\rho^{2}}{4}|\mathbf{y}|^{2}}{\rho^{2}}.$

Iterating this $m$ times verifies the first equality. The expression for the
$C_{M,m}$ follows from the recursion identity
$|\mathbb{S}^{M+2}|=\frac{2\pi}{M+1}|\mathbb{S}^{M}|$.

To conclude the proof, write $\mathbf{x}_{0}\in\mathbb{R}^{N+2m}$ as
$\mathbf{x}_{0}=(\mathbf{y}_{0},\mathbf{z}_{0})\in\mathbb{R}^{N}\times\mathbb{R}^{2m}$.
By translation invariance, a simple change of variables and fact that $\rho>0$
if and only if $((n+m)\rho)^{-\frac{1}{2}}>0$, one has

$\displaystyle\lambda_{LY}^{n+2m}$
$\displaystyle[\Sigma\times\mathbb{R}^{2m}]=\sup_{\mathbf{x}_{0}\in\mathbb{R}^{N+2m},\rho>0}\int_{\Sigma\times\mathbb{R}^{2m}}W^{n+2m}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2m}$
$\displaystyle=\sup_{\mathbf{y}_{0}\in\mathbb{R}^{N},\rho>0}\int_{\Sigma\times\mathbb{R}^{2m}}W^{n+2m}_{((n+m)\rho)^{-\frac{1}{2}}}(\mathbf{x}-(\mathbf{y}_{0},\mathbf{0}))d\mathcal{H}^{n+2m}$
$\displaystyle=\sup_{\rho>0}(4\pi\rho)^{\frac{n}{2}}((n+m)\rho)^{-\frac{n}{2}}C_{n,m}\sup_{\mathbf{y}_{0}\in\mathbb{R}^{N}}\int_{\Sigma}\hat{W}_{n+m,\rho}^{n}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n}.$

Where the last equality follow from the first part of the proof and the
definition of $\hat{W}^{n}_{n+m,\rho}$. The claim follows immediately.

∎

######  Proposition 4.2.

Let $\Sigma\subset\mathbb{R}^{N}$ be an $n$-dimensional submanifold, possibly
with boundary. If $\Sigma$ has a finite conformal volume that is realized,
i.e., there are $\mathbf{x}_{0}\in\mathbb{R}^{N}$ and $\rho_{0}>0$ so

$\lambda_{LY}^{n}[\Sigma]=\int_{\Sigma}W^{n}_{\rho_{0}}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}<\infty,$

then

$\bar{\lambda}_{LY}^{n}[\Sigma]\leq\bar{\lambda}_{LY}^{n+2}[\Sigma\times\mathbb{R}^{2}]\leq
2\bar{\lambda}_{LY}^{n}[\Sigma]<\infty.$

As a consequence, if $\Sigma\subset\mathbb{R}^{N}$ is an $n$-dimensional
compact submanifold, possibly with boundary, then, for all $m\in\mathbb{N}$,
$m\geq 0$,

$\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}]\leq\bar{\lambda}_{LY}^{n+2m+2}[\Sigma\times\mathbb{R}^{2m+2}]\leq\bar{\lambda}^{\infty}_{LY}[\Sigma].$

Moreover, either $\Sigma$ is contained in an affine $n$-dimensional plane or

$1<\lambda_{LY}^{n+2}[\Sigma\times\mathbb{R}^{2}].$

######  Remark 4.3.

The affine $n$-dimensional plane $\mathbb{R}^{n}\subset\mathbb{R}^{N}$
satisfies the hypotheses of the proposition and one computes that

$1=\bar{\lambda}_{LY}^{n}[\mathbb{R}^{n}]=\bar{\lambda}_{LY}^{n+2}[\mathbb{R}^{n}\times\mathbb{R}^{2}]=\bar{\lambda}_{LY}^{n+2}[\mathbb{R}^{n+2}].$

In contrast, round spheres, $\mathbb{S}^{n}\subset\mathbb{R}^{N}$, give

$1=\bar{\lambda}_{LY}^{n}[\mathbb{S}^{n}]<\bar{\lambda}_{LY}^{n+2}[\mathbb{S}^{n}\times\mathbb{R}^{2}].$

Here the second inequality is strict by [14]. See Appendix A where examples of
equality are obtained for non-flat curves.

###### Proof.

One computes, for fixed $\mathbf{x}_{0}\in\mathbb{R}^{N}$ and $\rho>0$,

$\displaystyle\frac{\partial}{\partial\rho}W^{M}_{\rho}(\mathbf{x}-\mathbf{x}_{0})$
$\displaystyle=\frac{M\rho^{M-1}}{(1+\frac{\rho^{2}}{4}|\mathbf{x}-\mathbf{x}_{0}|^{2})^{M}}-\frac{\frac{1}{2}M\rho^{M+1}|\mathbf{x}-\mathbf{x}_{0}|^{2}}{(1+\frac{\rho^{2}}{4}|\mathbf{x}-\mathbf{x}_{0}|^{2})^{M+1}}$
$\displaystyle=-\frac{M}{\rho}W^{M}_{\rho}(\mathbf{x}-\mathbf{x}_{0})+\frac{2M}{\rho^{2}}W^{M+1}_{\rho}(\mathbf{x}-\mathbf{x}_{0}).$

Using the pointwise estimate

(4.1)
$W^{M+1}_{\rho}(\mathbf{x}-\mathbf{x}_{0})\leq\rho(1+\frac{\rho^{2}}{4}|\mathbf{x}-\mathbf{x}_{0}|^{2})^{-1}W^{M}_{\rho}(\mathbf{x}-\mathbf{x}_{0})\leq\rho
W^{M}_{\rho}(\mathbf{x}-\mathbf{x}_{0}),$

one sees that if
$\int_{\Sigma}W^{M}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}<\infty,$
then differentiation by $\rho$ under the integral sign is justified. In this
case,

$\displaystyle\frac{\partial}{\partial\rho}\int_{\Sigma}W^{M}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}=\int_{\Sigma}\frac{2M}{\rho^{2}}W^{M+1}_{\rho}(\mathbf{x}-\mathbf{x}_{0})-\frac{M}{\rho}W^{M}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}.$

Specializing to the case $M=n$, if

$\lambda_{LY}^{n}[\Sigma]=\int_{\Sigma}W^{n}_{\rho_{0}}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}<\infty,$

then there is finite maximum at $\rho=\rho_{0}$ and so

$0=\frac{2n}{\rho^{2}_{0}}\int_{\Sigma}W^{n+1}_{\rho_{0}}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}-\frac{n}{\rho_{0}}\int_{\Sigma}W^{n}_{\rho_{0}}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}.$

It follows from Lemma 4.1 that, with
$\mathbf{x}_{0}=(\mathbf{y}_{0},\mathbf{0})\in\mathbb{R}^{N+2}$,

$\displaystyle\lambda_{LY}^{n}[\Sigma]$
$\displaystyle=\int_{\Sigma}W^{n}_{\rho_{0}}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n}=\frac{2}{\rho_{0}}\int_{\Sigma}W^{n+1}_{\rho_{0}}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n}$
$\displaystyle=\frac{n+1}{2\pi}\int_{\Sigma\times\mathbb{R}^{2}}W^{n+2}_{\rho_{0}}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2}\leq\frac{n+1}{2\pi}\lambda_{LY}^{n+2}[\Sigma\times\mathbb{R}^{2}].$

As $|\mathbb{S}^{n+2}|=\frac{2\pi}{n+1}|\mathbb{S}^{n}|$, it follows that

$\bar{\lambda}_{LY}^{n}[\Sigma]\leq\bar{\lambda}_{LY}^{n+2}[\Sigma\times\mathbb{R}^{2}].$

Likewise, (4.1) and Lemma 4.1 imply that when
$\mathbf{x}_{0}=(\mathbf{y}_{0},\mathbf{z}_{0})\in\mathbb{R}^{N}\times\mathbb{R}^{2}$,

$\displaystyle\int_{\Sigma\times\mathbb{R}^{2}}W^{n+2}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2}$
$\displaystyle=\frac{4\pi}{n+1}\rho^{-1}\int_{\Sigma}W^{n+1}_{\rho}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n}$
$\displaystyle\leq\frac{4\pi}{n+1}\int_{\Sigma}W^{n}_{\rho}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n}\leq\frac{4\pi}{n+1}\lambda_{LY}^{n}[\Sigma]$

Taking the sup over $\rho>0$ and
$\mathbf{x}_{0}\in\mathbb{R}^{N}\times\mathbb{R}^{2}$ yields

$\lambda_{LY}^{n+2}[\Sigma\times\mathbb{R}^{2}]\leq\frac{4\pi}{n+1}\lambda_{LY}^{n}[\Sigma]$

Which gives claimed bound between the normalized conformal volumes.

To complete the proof let
$\Sigma_{m}=\Sigma\times\mathbb{R}^{2m}\subset\mathbb{R}^{N+2m}$, $n+2m\geq
1$. We claim exactly one of the following situations holds:

  1. (1)

$\Sigma$ contained in an affine $n$-plane and, for all $m\geq 0$,
$\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}]=\bar{\lambda}_{LY}[\Sigma]=1$;

  2. (2)

$n=1$ and, for all $m\geq 1$,
$\bar{\lambda}_{LY}^{1+2m}[\Sigma_{m}]>\bar{\lambda}_{LY}^{1}[\Sigma_{0}]=\bar{\lambda}_{LY}^{1}[\Sigma]=1$;

  3. (3)

For all $m\geq 0$, $\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}]>1$.

First of all, by [14], one has for all $m\geq 0$,

$\lambda_{LY}^{n+2m}[\Sigma_{m}]\geq|\mathbb{S}^{n+2m}|\Rightarrow\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}]\geq
1.$

Moreover, when $n+2m>1$, the inequality is strict at a fixed $m$ unless
$\Sigma_{m}$ is contained in a $(n+2m)$-dimensional affine plane. In the
latter situation, $\Sigma$ is contained in an affine $n$-dimensional plane
which then implies the same for all $\Sigma_{m}$, $m\geq 0$. That is, case (1)
holds. When $n=1$ and $m=0$, it may happen that
$\bar{\lambda}_{LY}^{1}[\Sigma]=1$ and $\Sigma$ is not contained in an affine
line. In this case, the same reasoning implies that, for $m\geq 1$,
$\lambda_{LY}^{1+2m}[\Sigma_{m}]>1=\bar{\lambda}_{LY}^{1}[\Sigma]$ and we are
in case (2). Any other situation is covered by case (3), establishing the
claim.

It is immediate that the remaining conclusions of the proposition hold when
case (1) holds for all $n$ and $m$ and they hold in case (2) when $m=0$.
Hence, it suffices to suppose $\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}]>1$. Using
what has already been established, we need only show that this condition and
fact that $\Sigma$ is compact ensures $\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}]$
is realized.

For any fixed
$\mathbf{x}_{0}\in\mathbb{R}^{N}\times\mathbb{R}^{2m}=\mathbb{R}^{N+2m}$ write
$\mathbf{x}_{0}=(\mathbf{y}_{0},\mathbf{z}_{0})$. By Lemma 4.1,

$\displaystyle\int_{\Sigma_{m}}W^{n+2m}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2m}$
$\displaystyle=\frac{C_{n,m}}{\rho^{m}}\int_{\Sigma}W^{n+m}_{\rho}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n}\leq
C_{n,m}|\Sigma|\rho^{n}<\infty.$

Where we also used that $\Sigma$ is compact and of finite area and (4.1) to
estimate

$\rho^{-m}W^{n+m}_{\rho}(\mathbf{y}-\mathbf{y}_{0})\leq
W^{n}_{\rho}(\mathbf{y}-\mathbf{y}_{0})\leq\rho^{n}.$

Thus, there is a uniform $\rho_{m}^{-}>0$ so, for $0<\rho<\rho_{m}^{-}$,

$|\mathbb{S}^{n+2m}|^{-1}\int_{\Sigma_{m}}W^{n+2m}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2m}\leq\frac{3}{4}<\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}].$

When

(4.2) $\inf_{\mathbf{y}\in\Sigma}|\mathbf{y}-\mathbf{y}_{0}|>4,$

iterating (4.1) and combining with Lemma 4.1 yields

$\int_{\Sigma_{m}}W^{n+2m}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2m}\leq\frac{C_{n,m}}{\rho^{m}}\int_{\Sigma}W^{n+m}_{\rho}(\mathbf{y}-\mathbf{y}_{0})d\mathcal{H}^{n}\leq\frac{C_{n,m}}{(1+\rho^{2})^{m}}\lambda_{LY}^{n}[\Sigma]$

and so there is a $\rho_{m}^{+}$ so for $\rho>\rho_{m}^{+}$ and
$\mathbf{y}_{0}$ not close to $\Sigma$ in sense of (4.2)

$|\mathbb{S}^{n+2m}|^{-1}\int_{\Sigma_{m}}W^{n+2m}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2m}\leq\frac{3}{4}<\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}].$

Finally, as $\Sigma$ is compact, the set of $\mathbf{y}_{0}$ not satisfying
(4.2), i.e., so $\inf_{\mathbf{y}\in\Sigma}|\mathbf{y}-\mathbf{y}_{0}|\leq 4$,
is compact and so it follows from the asymptotic expansion of [14] and
$\bar{\lambda}_{LY}^{n+2m}[\Sigma^{\prime}]>1$ that, after possibly increasing
$\rho_{m}^{+}$, when $\rho>\rho_{m}^{+}$ and $\mathbf{y}_{0}$ does not satisfy
(4.2)

$|\mathbb{S}^{2n+m}|^{-1}\int_{\Sigma_{m}}W^{n+2m}_{\rho}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n+2m}\leq\frac{1}{2}(1+\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}])<\bar{\lambda}_{LY}^{n+2m}[\Sigma_{m}].$

Thus, by the translation invariance in the $\mathbb{R}^{2m}$ factor, there is
a compact $K_{m}\subset\mathbb{R}^{N}$ so

$\lambda_{LY}^{n+2m}[\Sigma_{m}]=\sup_{\rho_{m}^{-}\leq\rho\leq\rho_{m}^{+},\mathbf{y}_{0}\in
K_{m}}\int_{\Sigma_{m}}W^{n+2m}_{\rho}(\mathbf{x}-(\mathbf{y}_{0},\mathbf{0}))d\mathcal{H}^{n+2m}.$

As this is the supremum of a continuous function over a compact set, it is a
maximum and so the conformal volume is achieved. This completes the proof. ∎

We can now prove Proposition 1.7

###### Proof of Proposition 1.7.

When $\bar{\lambda}_{LY}^{n}[\Sigma]<\infty$, for any $\epsilon>0$ there are
$\mathbf{x}_{0},\rho_{0}$ so

$\infty>\lambda_{LY}^{n}[\Sigma]\geq\int_{\Sigma}W^{n}_{\rho_{0}}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}\geq\lambda_{LY}[\Sigma]-\epsilon.$

Pick $R_{\epsilon}>0$ large enough so that
$\Sigma_{\epsilon}=\bar{B}_{R_{\epsilon}}\cap\Sigma$ satisfies

$\lambda_{LY}^{n}[\Sigma_{\epsilon}]\geq\int_{\Sigma_{\epsilon}}W^{n}_{\rho_{0}}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}\geq\lambda_{LY}^{n}[\Sigma]-2\epsilon.$

By slightly increasing $R_{\epsilon}$ we may also assume $\Sigma_{\epsilon}$
is a compact manifold with boundary. As $\Sigma_{\epsilon}$ is compact, we can
apply Proposition 4.2 to obtain that

$\bar{\lambda}_{LY}^{n+2m}[\Sigma_{\epsilon}\times\mathbb{R}^{2m}]\geq\bar{\lambda}_{LY}^{n}[\Sigma_{\epsilon}]\geq\bar{\lambda}_{LY}^{n}[\Sigma]-2|\mathbb{S}^{n}|^{-1}\epsilon.$

As $\Sigma_{\epsilon}\times\mathbb{R}^{2m}\subset\Sigma\times\mathbb{R}^{2m}$,

$\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}]\geq\bar{\lambda}_{LY}^{n+2m}[\Sigma_{\epsilon}\times\mathbb{R}^{2m}]\geq\bar{\lambda}_{LY}^{n}[\Sigma]-2|\mathbb{S}^{n}|^{-1}\epsilon.$

The choice of $\epsilon>0$ was arbitrary so we conclude that

$\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}]\geq\bar{\lambda}_{LY}^{n}[\Sigma].$

When $\bar{\lambda}_{LY}^{n}[\Sigma]=\infty$, for any $\Lambda>0$ large there
are $\mathbf{x}_{0}$ and $\rho_{0}$ so

$\int_{\Sigma}W^{n}_{\rho}(\mathbf{x}-\mathbf{x}_{0})\geq 2\Lambda.$

Now pick $R_{\Lambda}>0$ large enough so that
$\Sigma_{\Lambda}=B_{R_{\Lambda}}\cap\Sigma$ satisfies

$\lambda_{LY}^{n}[\Sigma_{\Lambda}]\geq\int_{\Sigma_{\Lambda}}W^{n}_{\rho_{0}}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}\geq\Lambda.$

We may again increase $R_{\Lambda}$ slightly to ensure $\Sigma_{\Lambda}$ is a
compact manifold with boundary. Proposition 4.2 implies

$\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}]\geq\bar{\lambda}_{LY}^{n+2m}[\Sigma_{\Lambda}\times\mathbb{R}^{2m}]\geq\bar{\lambda}_{LY}^{n}[\Sigma_{\Lambda}]\geq|\mathbb{S}^{n}|^{-1}\Lambda.$

As $\Lambda$ is arbitrary, we conclude

$\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}]=\infty.$

∎

We will show, that in general,

$\lambda_{CM}^{n}[\Sigma]\leq\bar{\lambda}^{\infty}_{LY}[\Sigma]\leq\lambda_{V}^{n}[\Sigma].$

######  Proposition 4.4.

One has

$(4\pi\rho)^{-\frac{n}{2}}e^{-\frac{|\mathbf{x}|^{2}}{4\rho}}=\lim_{m\to\infty}\hat{W}_{n+m,\rho}^{n}(\mathbf{x})\leq\hat{W}_{n+m+1,\rho}^{n}(\mathbf{x})\leq\hat{W}_{n+m,\rho}^{n}(\mathbf{x})\leq\hat{W}_{n,\rho}^{n}(\mathbf{x}).$

Likewise,

$\left(\frac{4\pi}{n}\right)^{\frac{n}{2}}|\mathbb{S}^{n}|^{-1}=\hat{C}_{n,0}<\hat{C}_{n,m}<\hat{C}_{n,m+1}<\lim_{m\to\infty}\hat{C}_{n,m}=1.$

As a consequence, for an $n$-dimensional submanifold,
$\Sigma\subset\mathbb{R}^{N}$,

$\hat{C}_{n,m}\lambda_{CM}^{n}[\Sigma]\leq\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}];$
$\lambda_{CM}^{n}[\Sigma]\leq\bar{\lambda}_{LY}^{\infty}[\Sigma].$

###### Proof.

Observe that $\hat{W}^{n}_{n+m+1,\rho}\leq\hat{W}^{n}_{n+m,\rho}$ if and only
if for $t\geq 0$,

$-(n+m+1)\log\left(1+\frac{t}{n+m+1}\right)\leq-(n+m)\log(1+\frac{t}{n+m}).$

Fix $t\geq 0$ and consider the function

$I_{t}(s)=-s\log(1+\frac{t}{s}),s>0.$

It will suffice to show this is a non-increasing function for $s\geq 1$. One
has

$I_{t}^{\prime}(s)=-\log(1+\frac{t}{s})+\frac{t}{s}(1+\frac{t}{s})^{-1}=1-\log(1+\frac{t}{s})-(1+\frac{t}{s})^{-1},$
$I_{t}^{\prime\prime}(s)=\frac{t}{s^{2}}(1+\frac{t}{s})^{-1}-\frac{t}{s^{2}}(1+\frac{t}{s})^{-2}=\frac{t^{2}}{s^{3}}(1+\frac{t}{s})^{-2}\geq
0.$

As $\lim_{s\to\infty}I_{t}^{\prime}(s)=0$, it follows that
$I_{t}^{\prime}(s)\leq 0$ for $s\geq 1$ and $t\geq 0$. That is, $I_{t}(s)$ is
non-increasing as claimed.

For fixed $\mathbf{x}\in\mathbb{R}^{N}$, one has the elementary limit

$\lim_{m\to\infty}(1+\frac{1}{4(n+m)\rho}|\mathbf{x}|^{2})^{-n-m}=e^{-\frac{|\mathbf{x}|^{2}}{4\rho}}.$

Hence, the $\hat{W}^{n}_{n+m,\rho}$ decrease pointwise in $m$ to

$(4\pi\rho)^{-\frac{n}{2}}e^{-\frac{|\mathbf{x}|^{2}}{4\rho}}=\lim_{m\to\infty}\hat{W}_{n+m,\rho}^{n}(\mathbf{x}).$

With $\Sigma=\mathbb{R}^{n}$ the $n$-dimensional affine plane through the
origin, Lemma 4.1 yields:

$1=\bar{\lambda}_{LY}^{n+2m}[\mathbb{R}^{n}\times\mathbb{R}^{2m}]=\hat{C}_{n,m}\int_{\mathbb{R}^{n}}\hat{W}_{n+m,1}^{n}d\mathcal{H}^{n}=\hat{C}_{n,m}\int_{\mathbb{R}^{n}}\hat{W}_{n+m,1}^{n}d\mathcal{L}^{n}.$

As the $\hat{W}_{n+m,1}^{n}$ are decreasing pointwise in $m$, the
$\hat{C}_{n,m}$ are increasing in $m$. Moreover,

$1=\int_{\mathbb{R}^{n}}(4\pi)^{-\frac{n}{2}}e^{-\frac{|\mathbf{x}|^{2}}{4}}d\mathcal{L}^{n}=\lim_{m\to\infty}\int_{\mathbb{R}^{n}}\hat{W}^{n}_{n+m,1}d\mathcal{L}^{n}$

by the dominated convergence theorem and so
$\lim_{m\to\infty}\hat{C}_{n,m}=1$.

Finally, by Lemma 4.1 we see that

$\displaystyle\hat{C}_{n,m}F_{\mathbf{x}_{0},t_{0}}[\Sigma]$
$\displaystyle=\hat{C}_{n,m}\int_{\Sigma}(4\pi
t_{0})^{-\frac{n}{2}}e^{-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4t_{0}}}d\mathcal{H}^{n}$
$\displaystyle\leq\hat{C}_{n,m}\int_{\Sigma}\hat{W}_{n+m,t_{0}}^{n}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}=\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}].$

By taking the supremum over $\mathbf{x}_{0}$ and $t_{0}$ we obtain the first
estimate. The second follows by taking $m\to\infty$.

∎

We record an elementary computation:

######  Lemma 4.5.

For $M\geq 0$,

$\displaystyle\nabla^{2}_{\mathbb{R}}\log W^{M}_{\rho}$
$\displaystyle=-\frac{1}{2}M\rho^{2}g_{\mathbb{R}}+\frac{M}{8}\frac{\rho^{4}|\mathbf{x}|^{2}}{1+\frac{\rho^{2}}{4}|\mathbf{x}|^{2}}g_{\mathbb{R}}+\frac{M}{16}\frac{\rho^{4}d|\mathbf{x}|^{2}\otimes
d|\mathbf{x}|^{2}}{(1+\frac{\rho^{2}}{4}|\mathbf{x}|^{2})^{2}}\geq-\frac{1}{2}M\rho^{2}g_{\mathbb{R}}$

with equality at $\mathbf{x}=\mathbf{0}$. In particular,

$\tau(W^{M}_{\rho})=\frac{1}{M\rho^{2}}\mbox{ and
}\tau(\hat{W}_{n+m,\rho}^{n})=\rho.$

###### Proof.

One directly computes

$\nabla_{\mathbb{R}}\log
W^{M}_{\rho}(\mathbf{x})=-\frac{M\rho^{2}}{2}\frac{\mathbf{x}}{1+\frac{\rho^{2}}{4}|\mathbf{x}|^{2}};$
$\nabla_{\mathbb{R}}^{2}\log
W^{M}_{\rho}(\mathbf{x})=-\frac{M\rho^{2}}{2}\frac{g_{\mathbb{R}}}{1+\frac{\rho^{2}}{4}|\mathbf{x}|^{2}}+\frac{M}{16}\frac{\rho^{4}d|\mathbf{x}|^{2}\otimes
d|\mathbf{x}|^{2}}{(1+\frac{\rho^{2}}{4}|\mathbf{x}|^{2})^{2}}.$

It is straightforward to verify the claims from this. ∎

######  Proposition 4.6.

For $n,m,N\in\mathbb{N}$ with $m\geq N-n$,

$\hat{C}_{N,n-N+m}(4\pi\rho)^{\frac{n-N}{2}}\hat{W}_{n+m,\rho}^{n}\in\mathcal{VT}_{1}^{+}(\rho).$

In particular, if $\Sigma\subset\mathbb{R}^{N}$ is a $n$-dimensional proper
submanifold and $m\geq N=n$, then

$\mathcal{\lambda}_{V}^{n}[\Sigma]\geq\hat{C}_{N,n-N+m}\hat{C}_{n,m}^{-1}\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}]$
$\mathcal{\lambda}_{V}^{n}[\Sigma]\geq\bar{\lambda}^{\infty}_{LY}[\Sigma].$

###### Proof.

First observe that by Lemma 4.5, $\tau(\hat{W}_{n+m,\rho}^{n})=\rho$. Hence,
it suffices to calculate the mass of $\hat{W}_{n+m,\rho}^{n}$ and scale by
it’s inverse. To that end, observe that $m+n\geq N\geq 1$ ensures
$\hat{W}_{n+m,\rho}^{n}\in L^{1}(\mathbb{R}^{N})$. By considering the change
of variables $\mathbf{x}\mapsto\sqrt{\rho}\mathbf{x}$ one has

$\int_{\mathbb{R}^{N}}\hat{W}_{n+m,\rho}^{n}d\mathcal{L}^{N}=\rho^{\frac{N-n}{2}}\int_{\mathbb{R}^{N}}\hat{W}_{n+m,1}^{n}d\mathcal{L}^{N}.$

Define $\alpha_{n,m,N}\in(0,\infty)$ by

$\alpha_{n,m,N}^{-1}=\int_{\mathbb{R}^{N}}\hat{W}_{n+m,1}^{n}d\mathcal{L}^{N}$

For, $m^{\prime}=n-N+m\geq 0$, it is clear that

$\hat{W}_{n+m,1}^{n}=(4\pi)^{\frac{N-n}{2}}\hat{W}_{N+m^{\prime},1}^{N}.$

Hence, for $m\geq N-n$, as
$\bar{\lambda}_{LY}^{N+2m^{\prime}}[\mathbb{R}^{N}\times\mathbb{R}^{2m^{\prime}}]=1$,
Lemma 4.1 gives

$\int_{\mathbb{R}^{N}}\hat{W}_{n+m,1}^{n}=(4\pi)^{\frac{N-n}{2}}\int_{\mathbb{R}^{N}}\hat{W}_{N+m^{\prime},1}^{N}=(4\pi)^{\frac{N-n}{2}}\hat{C}_{N,m^{\prime}}^{-1}$

That is, for $m\geq N-n$,

$\alpha_{n,m,N}=(4\pi)^{\frac{n-N}{2}}\hat{C}_{N,n-N+m}\mbox{ and
}\alpha_{n,m,N}\hat{W}_{n+m,1}^{n}\in\mathcal{VT}_{1}^{+}(\rho).$

To show the first inequality for $\Sigma$ set

$\tilde{W}_{n+m,\rho}^{n}=\alpha_{n,m,N}\rho^{\frac{n-N}{2}}\hat{W}_{n+m,\rho}^{n}=(4\pi)^{\frac{n-N}{2}}\hat{C}_{N,n-N+m}\hat{W}_{n+m,\rho}^{n}.$

Clearly, along with any of its translates one has

$\tilde{W}_{n+m,\rho}^{n}\in\mathcal{VT}_{1}^{+}(\rho).$

Hence, the definition of virtual entropy gives

$\displaystyle\lambda_{V}^{n}[\Sigma]$
$\displaystyle\geq(4\pi\rho)^{\frac{N-n}{2}}\int_{\Sigma}\tilde{W}_{n+m,\rho}^{n}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}$
$\displaystyle=(4\pi)^{\frac{N-n}{2}}\alpha_{n,m,N}\hat{C}_{n,m}^{-1}\hat{C}_{n,m}\int_{\Sigma}\hat{W}_{n+m,\rho}^{n}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{n}.$

By Lemma 4.1, taking the supremum over $\rho>0$ and
$\mathbf{x}_{0}\in\mathbb{R}^{N}$ yields

(4.3)
$\lambda_{V}^{n}[\Sigma]\geq\hat{C}_{N,n-N+m}\hat{C}_{n,m}^{-1}\bar{\lambda}_{LY}^{n+2m}[\Sigma\times\mathbb{R}^{2m}].$

This gives the first inequality for $\Sigma$. Taking $m\to\infty$ and
appealing to Proposition 4.4 gives the second one. Note that plugging
$\Sigma=\mathbb{R}^{n}$ into the (4.3) implies

$\hat{C}_{n,m}\geq\hat{C}_{N,n-N+m}.$

This gives an alternative way to conclude the second inequality. ∎

Theorem 1.8 is an immediate consequence of Propositions 4.4 and 4.6.

##  Appendix A Loxodromes

For $\alpha\in[0,\infty)$ consider the rectifiable set:

$S_{\alpha}^{0}=\left\\{(e^{t}\cos\alpha t,e^{t}\sin\alpha
t):t\in\mathbb{R}\right\\}\subset\mathbb{R}^{2}.$

This is a smooth curve away from $\mathbf{0}$. When $\alpha>0$ this is a
logarithmic spiral (i.e., a special case of a loxodrome or rhumb line) and for
$\alpha=0$ it is the non-negative part of the $x$-axis. If
$R_{\theta}:\mathbb{R}^{2}\to\mathbb{R}^{2}$ is rotation counter clockwise by
$\theta$, then we see that for $\alpha>0$,
$S^{\theta}_{\alpha}=R_{\theta}(S_{\alpha}^{0})=e^{-\alpha^{-1}\theta}\cdot
S_{\alpha}^{0}$, i.e., scaling $S_{\alpha}^{0}$ about the origin is the same
as rotating it. Consider the doubled curve

$S_{\alpha}=S_{\alpha}^{0}\cup S_{\alpha}^{\pi}$

which is symmetric with respect to reflection through the origin.

######  Proposition A.1.

For any $\alpha\geq 0$ one has,

$\lambda_{CM}^{1}[S_{\alpha}]=\bar{\lambda}_{LY}^{\infty}[S_{\alpha}]=\bar{\lambda}_{LY}^{1}[S_{\alpha}]=\sqrt{1+\alpha^{2}}.$

######  Remark A.2.

This shows the existence of multiplicity one curves that have entropy any
value larger than one. In particular, it suggests some of the results of [17]
are optimal.

###### Proof.

When $\alpha=0$ this is elementary as $S_{0}$ is a line. In what follows we
suppose $\alpha>0$. In general, the parameterization of $S_{\alpha}^{0}$ and a
change of variables gives

$\int_{S_{\alpha}}f(|\mathbf{x}|)d\mathcal{H}^{1}=2\int_{-\infty}^{\infty}f(e^{r})\sqrt{1+\alpha^{2}}e^{r}dr=2\sqrt{1+\alpha^{2}}\int_{0}^{\infty}f(u)du.$

As scaling $S_{\alpha}$ is the same as rotating it around the origin, it
follows from the change of variables formula and rotational symmetry of the
weights that

$\lambda_{CM}^{1}[S_{\alpha}]=\sup_{\mathbf{x}_{0}\in\mathbb{R}^{2}}(4\pi)^{-\frac{1}{2}}\int_{S_{\alpha}}e^{-\frac{|\mathbf{x}-\mathbf{x}_{0}|^{2}}{4}}d\mathcal{H}^{1}.$

Hence,

$\lambda_{CM}^{1}[S_{\alpha}]\geq(4\pi)^{-\frac{1}{2}}\int_{S_{\alpha}}e^{-\frac{|\mathbf{x}|^{2}}{4}}d\mathcal{H}^{1}=2(4\pi)^{-\frac{1}{2}}\sqrt{1+\alpha^{2}}\int_{0}^{\infty}e^{-\frac{u^{2}}{4}}du=\sqrt{1+\alpha^{2}}.$

For the same reason,

$\bar{\lambda}_{LY}^{\infty}[S_{\alpha}]\geq\bar{\lambda}_{LY}^{1}[S_{\alpha}]=(2\pi)^{-1}\sup_{\mathbf{x}_{0}\in\mathbb{R}^{2}}\int_{S_{\alpha}}W^{1}_{1}(\mathbf{x}-\mathbf{x}_{0})d\mathcal{H}^{1}\geq\sqrt{1+\alpha^{2}}.$

In order to establish the claim, let us suppose
$\mathbf{x}_{0}=(x_{0},y_{0})\in\mathbb{S}^{1}\subset\mathbb{R}^{2}$ is a
fixed unit vector. For $t\in\mathbb{R}$ and $m\geq 0$ set

$L_{m}^{\alpha,\mathbf{x}_{0}}(t)=\int_{S_{\alpha}}W^{m+1}_{1}(\mathbf{x}-t\mathbf{x}_{0})d\mathcal{H}^{1}.$

Up to a constant factor, these are the integrals that compute
$\bar{\lambda}_{LY}^{1+2m}[S_{\alpha}\times\mathbb{R}^{2m}]$.

Using the parameterization of $S_{\alpha}$ we have

$\displaystyle L_{m-1}^{\alpha,\mathbf{x}_{0}}(t)$
$\displaystyle=\int_{S_{\alpha}}\frac{1}{(1+\frac{1}{4}|\mathbf{x}-t\mathbf{x}_{0}|^{2})^{m}}d\mathcal{H}^{1}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{\sqrt{1+\alpha^{2}}e^{r}}{(1+\frac{1}{4}((e^{r}\cos(\alpha
r)-tx_{0})^{2}+(e^{r}\sin(\alpha r)-ty_{0})^{2}))^{m}}dr$
$\displaystyle+\int_{-\infty}^{\infty}\frac{\sqrt{1+\alpha^{2}}e^{r}}{(1+\frac{1}{4}((e^{r}\cos(\alpha
r)+tx_{0})^{2}+(e^{r}\sin(\alpha r)+ty_{0})^{2}))^{m}}dr$
$\displaystyle=\sqrt{1+\alpha^{2}}\int_{-\infty}^{\infty}\frac{e^{r}}{(I_{1}(r)+I_{2}(r))^{m}}+\frac{e^{r}}{(I_{1}(r)-I_{2}(r))^{m}}dr$

where here

$I_{1}(r)=1+\frac{1}{4}(e^{2r}+t^{2})\geq 1\mbox{ and
}I_{2}(r)=-\frac{1}{2}tx_{0}e^{r}\cos(\alpha
r)-\frac{1}{2}ty_{0}e^{r}\sin(\alpha r).$

By the Cauchy-Schwarz and absorbing inequality,

$I_{2}(r)^{2}\leq
J_{2}(r)^{2}=\frac{1}{4}t^{2}e^{2r}\leq\frac{1}{16}(t^{2}+e^{2r})^{2}\leq
I_{1}(r)^{2}.$

The following inequality holds: for $a^{2}\leq b^{2}$, $x\geq 0$ and an
integer $m\geq 1$,

$(x-a)^{m}+(x+a)^{m}\leq(x-b)^{m}+(x+b)^{m}.$

Using this we obtain the estimate

$\displaystyle\frac{L_{m-1}^{\alpha,\mathbf{x}_{0}}(t)}{\sqrt{1+\alpha^{2}}}$
$\displaystyle=\int_{-\infty}^{\infty}\frac{(I_{1}(r)-I_{2}(r))^{m}+(I_{1}(r)+I_{2}(r))^{m}}{(I_{1}(r)^{2}-I_{2}(r)^{2})^{m}}e^{r}dr$
$\displaystyle\leq\int_{-\infty}^{\infty}\frac{(I_{1}(r)-J_{2}(r))^{m}+(I_{1}(r)+J_{2}(r))^{m}}{(I_{1}(r)^{2}-J_{2}(r)^{2})^{m}}e^{r}dr$
$\displaystyle=\int_{-\infty}^{\infty}\frac{e^{r}}{(I_{1}(r)+J_{2}(r))^{m}}+\frac{e^{r}}{(I_{1}(r)-J_{2}(r))^{m}}dr$
$\displaystyle=\frac{L_{m-1}^{0,\mathbf{e}_{1}}(t)}{\sqrt{1+\alpha^{2}}}=\frac{L_{m-1}^{0,\mathbf{e}_{1}}(0)}{\sqrt{1+\alpha^{2}}}$

Where here the final equality follows from the observation that last line is
exactly what would be obtained from evaluating on $S_{0}$, i.e., the $x$-axis,
and with $\mathbf{x}_{0}=\mathbf{e}_{1}=(1,0)$. In particular, the value is
independent of $t$. We conclude that for all $m\geq 0$,

$\bar{\lambda}_{LY}^{1+2m}[S_{\alpha}\times\mathbb{R}^{2m}]\leq\sqrt{1+\alpha^{2}}\bar{\lambda}_{LY}^{1+2m}[S_{0}\times\mathbb{R}^{2m}]=\sqrt{1+\alpha^{2}}.$

Combined with the lower bound already established this verifies that

$\sqrt{1+\alpha^{2}}=\bar{\lambda}_{LY}^{\infty}[S_{\alpha}]=\bar{\lambda}_{LY}^{1}[S_{\alpha}].$

Finally, by Proposition 4.4 and the previously established lower bound

$\sqrt{1+\alpha^{2}}\leq\lambda_{CM}^{1}[S_{\alpha}]\leq\bar{\lambda}_{LY}^{\infty}[S_{\alpha}]=\sqrt{1+\alpha^{2}},$

which completes the proof. ∎

##  Appendix B Further Questions and Observations

First, we ask whether the inequalities between Colding-Minicozzi entropy,
stable conformal volume and virtual entropy are ever strict when $\Sigma$ is
not a self-shrinker.

######  Question B.1.

Is there a $n$-dimensional submanifold, possibly singular,
$\Sigma\subset\mathbb{R}^{N}$, with
$\lambda_{CM}^{n}[\Sigma]<\lambda_{V}^{n}[\Sigma]$? More generally, can one
ever have
$\lambda_{CM}^{n}[\Sigma]<\bar{\lambda}_{LY}^{\infty}[\Sigma]<\lambda_{V}^{n}[\Sigma]$?
If $\lambda_{CM}^{n}[\Sigma]<\bar{\lambda}_{LY}^{\infty}[\Sigma]$ can occur,
may one also have $\lambda_{CM}^{n}[\Sigma]<\bar{\lambda}_{LY}^{n}[\Sigma]$?

It would be particularly interesting to compute $\lambda_{V}^{1}[S_{\alpha}]$
for the $S_{\alpha}$ from Appendix A.

######  Question B.2.

What is the lowest entropy self-shrinking $\mathbb{RP}^{2}$ in
$\mathbb{R}^{N}$?

When $N=3$, any immersed $\mathbb{RP}^{2}$ must have a triple point [3], and
so entropy at least three. There are no candidate self-shrinkers in this
setting – see however [38] where the existence of a shrinking Möbius band
bounded by a straight line in $\mathbb{R}^{3}$ is shown. For $N\geq 5$, it
follows from [1, Theorem 1.1] that $\Sigma_{\mathbb{RP}^{2}}$ is (Colding-
Minicozzi) entropy unstable, suggesting there should be a lower entropy
unoriented shrinker. Likewise, as $\Sigma_{\mathbb{RP}^{2}}$ is unstable as a
minimal surface in $\mathbb{S}^{4}$, there is a non-constant flow out of it
that remains in the sphere. As observed in [39] such a flow has decreasing
Colding-Minicozzi entropy when viewed in the ambient $\mathbb{R}^{5}$. As the
flow is constrained to lie in $\mathbb{S}^{4}$ and must become singular, this
suggests the existence of an unoriented self-shrinker of lower entropy in
$\mathbb{R}^{4}$. However, the fact that
$\lambda^{2}_{CM}[\Sigma_{\mathbb{RP}^{2}}]>2$ introduces significant
technical issues.

In their proof of the Willmore conjecture [35], Marques-Neves showed that any
positive genus surface in $\mathbb{S}^{3}$ sits inside a five parameter family
and at least one member of the family has area at least that of the Clifford
torus. Together with Theorem 1.1, this suggests:

######  Conjecture B.3.

Let $\Sigma\subset\mathbb{R}^{3}$ be a closed self-shrinker of positive genus,
then

$1.57\approx\frac{\pi}{2}=\bar{\lambda}_{LY}^{2}[\Sigma_{C}]\leq\lambda_{CM}^{2}[\Sigma]$

where here $\Sigma_{C}\subset\mathbb{S}^{3}$ is the Clifford torus.

It was shown in [10] that such a $\Sigma$ satisfies the bound

$1.52\approx\lambda_{CM}^{2}[\mathbb{S}^{1}\times\mathbb{R}]\leq\lambda_{CM}^{2}[\Sigma].$

In [4], numerical methods established that there a rotationally symmetric
genus one self-shrinker, i.e., an Angenent torus [2],
$\Sigma_{A}\subset\mathbb{R}^{3}$, whose entropy satisfies

$\lambda_{CM}^{2}[\Sigma_{A}]\approx 1.852.$

Recently, Chu-Sun [20] have shown that no Angenent torus can be the lowest
entropy positive genus shrinker. Their work suggests that the lowest entropy
is achieved by a non-compact shrinker of genus one.

Finally, by using the heat kernel on hyperbolic space, a notion of Colding-
Minicozzi entropy has been extended to some spaces of non-positive curvature
[5, 7, 8, 39].

######  Question B.4.

Is there a useful notion of virtual entropy in other ambient Riemannian
manifolds – in particular those of non-positive curvature?

## References

  * [1] B. Andrews, H. Li, and Y. Wei, _$\mathcal{F}$ -Stability for Self-Shrinking Solutions to Mean Curvature Flow_, Asian Journal of Mathematics 18 (2014), no. 5, 757–778.
  * [2] S. B. Angenent, _Shrinking doughnuts_ , Nonlinear Diffusion Equations and Their Equilibrium States, 3 (Gregynog, 1989), Progr. Nonlinear Differential Equations Appl., vol. 7, Birkhäuser Boston, Boston, MA, 1992, pp. 21–38.
  * [3] T. Banchoff, _Triple Points and Singularities of Projections of Smoothly Immersed Surfaces_ , Transactions of the American Mathematical Society 46 (1974), no. 3, 402–406.
  * [4] Y. I. Berchenko-Kogan, _The entropy of the Angenent torus is approximately 1.85122_ , Journal of Experimental Mathematics 30 (2021), no. 4, 587–594.
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