# Singular sets on spaces with integral curvature bounds and diffeomorphism
finiteness for manifolds

Xin Qian

###### Abstract

In this paper, we are concerned with noncollapsed Riemannian manifolds
$(M^{n},g)$ with integral curvature bounds, as well as their Gromov-Hausdorff
limits $(M^{n}_{i},g_{i})\xrightarrow{GH}(X,d)$. Our main result generalizes
Cheeger’s Hausdorff dimension estimate for the singular set in [7] and improve
it into Minkowski dimension estimate in the spirit of Cheeger-Naber [12]. We
also prove a difffeomorphism finiteness theorem for manifolds in the critical
case, using Cheeger-Naber’s methods in [13].

##  1 Introduction

Let $(M^{n},g)$ be a Riemannian manifold. For each $x\in M^{n}$, let $h(x)$
denote the smallest eigenvalue for the Ricci tensor $Ric:T_{x}M\to T_{x}M$ and
$Ric_{-}(x)=\max\left\\{0,-h(x)\right\\}$. In this paper, we consider
manifolds $(M^{n},g)$ satisfying

$\displaystyle\int_{M^{n}}|Ric_{-}|^{p}\leqslant\lambda,$ (1.1) $\displaystyle
vol(B(x,r))\geqslant vr^{n},\ \text{for all}\ x\in M^{n}\ \text{and}\
r\leqslant 1,$ (1.2) $\displaystyle\int_{M^{n}}|Rm|^{q}\leqslant\Lambda,$
(1.3)

where $p>\frac{n}{2}$ and $1\leqslant q\leqslant\frac{n}{2}$. We will be
particularly concerned with pointed Gromov-Hausdorff limts

$\displaystyle(M_{i}^{n},g_{i},x_{i})\xrightarrow{pGH}(X,d,x)$

of sequences of such manifolds. Sometimes we suppose in place of (1.1) that

$\displaystyle\int_{M^{n}}|Ric|^{p}\leqslant\lambda.$ (1.4)

The geometry of manifolds with bounded integral Ricci curvature has been
studied extensively. In particular, Petersen-Wei derived a relative volume
comparison theorem in [21]. Afterward, under the non-collapsing condition
(1.2), Petersen-Wei [22] showed some almost rigidity results and Tian-Zhang
[23] developed a regularity theory for manifolds with bounded integral Ricci
curvature. Also, the almost rigidity structures in collapsing case was
discussed by Chen in [14]. Their results were directly derived from similar
arguments as Cheeger-Colding [8, 9, 10] for manifolds with lower Ricci
curvature bound.
The main purpose of this note is to generalize Cheeger’s Hausdorff dimension
estimate for the singular set in [7]. Our main theorem is stated following.

######  Theorem 1.1.

Given $p>\frac{n}{2}$, $1\leqslant q\leqslant\frac{n}{2}$ and positive
constants $v,\lambda$ and $\Lambda$, let
$(M_{i}^{n},g_{i},x_{i})\xrightarrow{pGH}(X,d,x)$ be a Gromov-Hausdorff limit
of manifolds satisfying (1.1)-(1.3). Then the singular set $\mathcal{S}$ of
$X$ satisfies

$\displaystyle\dim_{\mathcal{H}}\mathcal{S}\leqslant n-2q.$ (1.5)

Furthermore, the Hausdorff dimension can be improved to be the Minkowski
dimension if we assume (1.4) instead of (1.1).

Note that if $q>\frac{n}{2}$, then we can get a uniform lower bound on the
harmonic radius (see Theorem 2.14) and hence, the singular set $\mathcal{S}$
must be empty. Indeed, the limit space $X$ is a $C^{\alpha}$-Riemannian
manifold, for any $0<\alpha<2-\frac{n}{q}$. Thus, it is reasonable to consider
$1\leqslant q\leqslant\frac{n}{2}$.
We mention that we use Cheeger-Naber’s methods in [12] to get volume estimates
on the quantitative stratification and then derive the Minkowski dimension
estimate for the singular set.
In the critical case $q=\frac{n}{2}$, we know from Theorem 1.1 that the
singular set $\mathcal{S}$ consists of isolated points. Inspired by the bubble
tree construction in [3] and [13], we prove the following diffeomorphism
finiteness theorem, which generalizes Anderson-Cheeger’s finiteness theorem in
[3].

######  Theorem 1.2.

Given $p>\frac{n}{2}$ and positive constants $v,D,\lambda$ and $\Lambda$,
there exists $C=C(n,p,v,D,\lambda,\Lambda)$ such that the collection of closed
Riemannian n-manifolds $M^{n}$ satisfying (1.2), (1.4),

$\displaystyle diam(M^{n})\leqslant D,$ (1.6)

and

$\displaystyle\int_{M^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda,$ (1.7)

contains at most C diffeomorphism types.

Note that the all scale non-collapsing condition (1.2) is essential and cannot
be replaced by a definite scale non-collapsing condition such as
$vol(B(x,1))\geqslant v$. Indeed, due to an example of Yang [24], for any
$p\geqslant\frac{n}{2}$, there exists a sequence of n-manifolds $M_{k}$ such
that $\left\|Rm\right\|_{p}\leqslant\lambda,\ diam\leqslant D,\ vol\geqslant
v$, but $b_{2}(M_{k})\to\infty$, as $k\to\infty$. In fact,
$vol(B(x_{k},r))=o(r^{n})$ for some $x_{k}\in M_{k}$. In [24, 25], Yang also
proved an orbifold type limit theorem (see Theorem 2.17), which was analogous
to Anderson’s result in [2].
The paper is organized as follows. In Section 2, we supply some known results
about manifolds with integral curvature bounds. In Section 3, we give an
$\epsilon$-regularity theorem and prove Theorem 1.1. In Section 4, we prove
the finite diffeomorphism statement of Theorem 1.2 by using the methods in
[13].

$\mathbf{\bm{\mathbf{Acknowledgement.}\mathbf{}}}$ The author would like to
thank Ruobing Zhang for helpful suggestions on this research project.

##  2 Preliminaries

In this section, we recall some notions and properties we need for manifolds
with integral curvature bounds.

###  2.1 Relative volume comparison

Consider an $n$-manifold $M$. For each $x\in M$, let $h(x)$ denote the
smallest eigenvalue for the Ricci tensor $Ric:T_{x}M\to T_{x}M$ and
$Ric_{-}(x)=\max\left\\{0,-h(x)\right\\}$. Then Petersen-Wei proved the
following relative volume comparison theorem in [21].

######  Theorem 2.1.

Given $p>\frac{n}{2}$, there exists $C(n,p)$ such that for any
$r_{2}>r_{1}>0$,

$\displaystyle\left(\dfrac{vol(B(x,r_{2}))}{r_{2}^{n}}\right)^{\frac{1}{2p}}-\left(\dfrac{vol(B(x,r_{1}))}{r_{1}^{n}}\right)^{\frac{1}{2p}}\leqslant
C(n,p)\left(r_{2}^{2p-n}\int_{B(x,r_{2})}|Ric_{-}|^{p}\right)^{\frac{1}{2p}}.$
(2.1)

Equivalently,

$\displaystyle\left(\dfrac{vol(B(x,r_{1}))}{r_{1}^{n}}\right)^{\frac{1}{2p}}\geqslant\left(\dfrac{vol(B(x,r_{2}))}{r_{2}^{n}}\right)^{\frac{1}{2p}}\left(1-C(n,p)r_{2}\left(\fint_{B(x,r_{2})}|Ric_{-}|^{p}\right)^{\frac{1}{2p}}\right).$
(2.2)

######  Remark 2.2.

By inequality (2.1), if we assume
$\sup\limits_{x}r_{2}\left(\fint_{B(x,r_{2})}|Ric_{-}|^{p}\right)^{\frac{1}{2p}}$
is sufficiently small, then the volume doubling property can be easily derived
(see [22]). Many works on integral Ricci curvature assume this smallness (see
[14, 16, 22] for example) and as pointed out in [16], this smallnes is
necessary in order to get the $L^{p}$ version of Cheeger-Colding’s theory,
which will be reviewed later. Also, under our assumption (1.1) and (1.2), we
only need to consider balls with sufficiently small radius $r$.

Letting $r_{1}\to 0$, we get the following corollary.

######  Corollary 2.3.

Under the asssumption (1.1), we have the volume bound

$\displaystyle vol(B(x,r))\leqslant C(n,p)(r^{n}+\lambda r^{2p}),\ \ \forall
r>0.$ (2.3)

If we further assume (1.2), then we get the volume doubling property

$\displaystyle\frac{vol(B(x,r_{2}))}{vol(B(x,r_{1}))}\leqslant
C(n,p,v,R,\lambda)\dfrac{r_{2}^{n}}{r_{1}^{n}},\ \ \forall
r_{1}<r_{2}\leqslant R,\ r_{1}\leqslant 1.$ (2.4)

###  2.2 Almost rigidity structure

Let $M^{n}$ be a Riemannian manifold satisfying (1.1) and (1.2) with $x\in
M^{n}$. Denote $\omega_{n}$ to be the volume of the unit ball in
$\mathbb{R}^{n}$. Following lines of Cheeger-Colding [8, 9, 10] and Colding
[15], one may obtain the following almost rigidity properties.

######  Theorem 2.4 (Volume cnvergence,[22]).

$\forall\epsilon>0$, there exists $\delta=\delta(n,p,v,\lambda,\epsilon)$ and
$r_{0}=r_{0}(n,p,v,\lambda,\epsilon)$, such that if
$d_{GH}(B(x,r),B(0^{n},r))\leqslant\delta r$, for some $r\leqslant r_{0}$,
then $vol(B(x,r))\geqslant(1-\epsilon)\omega_{n}r^{n}$.

######  Theorem 2.5 (Almost maximal volume implies GH-close,[22, 23]).

$\forall\epsilon>0$, there exists $\delta=\delta(n,p,v,\lambda,\epsilon)$ and
$r_{0}=r_{0}(n,p,v,\lambda,\epsilon)$, such that if
$vol(B(x,r))\geqslant(1-\delta)\omega_{n}r^{n}$, for some $r\leqslant r_{0}$,
then $d_{GH}(B(x,r),B(0^{n},r))\leqslant\epsilon r$.

######  Theorem 2.6 (Almost metric cone,[14, 23]).

$\forall\epsilon>0$, there exists $\delta=\delta(n,p,v,\lambda,\epsilon)$ and
$r_{0}=r_{0}(n,p,v,\lambda,\epsilon)$, such that if

$\dfrac{vol(B(x,r))}{\omega_{n}r^{n}}\geqslant(1-\delta)\dfrac{vol(B(x,r/2))}{\omega_{n}(r/2)^{n}},$

for some $r\leqslant r_{0}$, then there exists a compact length space X with
$diam(X)\leqslant(1+\epsilon)\pi$ such that
$d_{GH}(B(x,r),B(z^{*},r))\leqslant\epsilon r$, where $z^{*}$ is the cone
vertex of $C(X)$.

Finally, recall that an $\epsilon$-splitting map
$b=(b_{1},...,b_{k}):B(x,r)\to\mathbb{R}^{k}$ is a harmonic map such that

$\begin{gathered}\fint_{B(x,r)}\sum_{i,j}|\langle\nabla b_{i},\nabla
b_{j}\rangle-\delta_{ij}|^{2}+r^{2}\sum_{i}|\nabla^{2}b_{i}|^{2}\leqslant\epsilon^{2},\\\
|\nabla b_{i}|\leqslant C(n),\ \forall i.\end{gathered}$ (2.5)

Then we have the following.

######  Theorem 2.7 ($\epsilon$-splitting,[14, 22]).

$\forall\epsilon>0$, there exists positive
$\delta=\delta(n,p,v,\lambda,\epsilon)$ and
$r_{0}=r_{0}(n,p,v,\lambda,\epsilon)$ such that

  1. (1)

If $d_{GH}(B(x,\delta^{-1}r),B((0^{k},y),\delta^{-1}r))\leqslant\delta r$, for
some $r\leqslant r_{0}$ and $(0^{k},y)\in\mathbb{R}^{k}\times Y$, then there
exists an $\epsilon$-splitting map $b:B(x,r)\to\mathbb{R}^{k}$.

  2. (2)

If there exists an $\delta$-splitting map $b:B(x,4r)\to\mathbb{R}^{k}$, for
some $r\leqslant r_{0}$, then $d_{GH}(B(x,r),B((0^{k},y),r))\leqslant\epsilon
r$, for some $(0^{k},y)\in\mathbb{R}^{k}\times Y$.

###  2.3 Harmonic radius and diffeomorphisms

We first recall the notion of the harmonic radius.

######  Definition 2.8.

For $x\in(M^{n},g)$ and $\alpha\in(0,1)$, we define the $C^{\alpha}$ harmonic
radius $r_{h}^{\alpha}(x)$ to be the largest $r>0$ such that there exists a
coordinate map $\mathcal{X}=(x^{1},...,x^{n}):B(x,r)\to\mathbb{R}^{n}$
satisfying

  1. (1)

$\Delta_{g}x^{i}=0$.

  2. (2)

$g_{ij}=g(\nabla x_{i},\nabla x_{j})$ on $B(x,r)$ satisfies

$\displaystyle\left\|g_{ij}-\delta_{ij}\right\|_{C^{0}}+r^{\alpha}[g_{ij}]_{C^{\alpha}}\leqslant
10^{-6}.$ (2.6)

######  Remark 2.9.

(1) We call the above coordinate map $\mathcal{X}:B(x,r)\to\mathbb{R}^{n}$ a
$C^{\alpha}$ harmonic coordinate system. Analogously, one can define the
$W^{2,p}$ harmonic coordinate system and harmonic radius by requiring a
scaling invariant $W^{2,p}$ bound

$\displaystyle\left\|g_{ij}-\delta_{ij}\right\|_{C^{0}}+\sum_{k=1}^{2}r^{k-\frac{n}{p}}\left\|\partial^{k}g_{ij}\right\|_{L^{p}}\leqslant
10^{-6},$ (2.7)

instead of (2.6). We will denote $r_{h}^{2,p}(x)$ to be the $W^{2,p}$ harmonic
radius at $x$. When $p>\frac{n}{2}$ and $0<\alpha<2-\frac{n}{p}$, $W^{2,p}$
harmonic coordinates are also $C^{\alpha}$ harmonic coordinates.
(2) One may replace the bound $10^{-6}$ in (2.6) and (2.7) by any constant
$Q>0$ and denote the $C^{\alpha}$ and $W^{2,p}$ harmonic radius by
$r_{h}^{\alpha}(x,Q)$ and $r_{h}^{2,p}(x,Q)$ respectively. For manifolds
satisfying (1.2) and (1.4), we have $r_{h}^{\alpha}(x,CQ)\geqslant
r_{h}^{2,p}(x,Q)$ for some constant $C=C(n,p,v,\lambda,\alpha)$ when
$p>\frac{n}{2}$ and $0<\alpha<2-\frac{n}{p}$. However, we will omit the
constants $C,Q$ and just write $r_{h}^{\alpha}(x)\geqslant r_{h}^{2,p}(x)$,
since this will not affect our argument.
(3) Harmonic coordinates have many good properties when it comes to regularity
issues. We refer to [20, 19] for a nice introduction. In particular, the
$L^{p}$ bound of the Ricci curvature ($p>\frac{n}{2}$) gives a priori
$C^{\alpha}\cap W^{2,p}$ bounds on the metric tensor $g_{ij}$ in harmonic
coordinates for any $\alpha\in(0,2-\frac{n}{p})$.

We now review the following $\epsilon$-regularity theorem, which can be proved
by modifying arguments in [2]. See also [23, 19].

######  Theorem 2.10.

Given $p>\frac{n}{2}$, there exists $\epsilon=\epsilon(n,p,v,\lambda)>0$ and
$r_{0}=r_{0}(n,p,v,\lambda)>0$ such that if $M^{n}$ satisfies (1.2), (1.4) and
$x\in M^{n}$ satisfies either of the following:

  1. (1)

$vol(B(x,r))\geqslant(1-\epsilon)\omega_{n}r^{n},\ \text{for some}\ r\leqslant
r_{0}$,

  2. (2)

$d_{GH}(B(x,r),B(0^{n},r))\leqslant\epsilon r,\ \text{for some}\ r\leqslant
r_{0}$,

then

$r_{h}^{2,p}(x)\geqslant\frac{1}{2}r.$

As noted in Remark 2.9 (2), we can get the same lower bound for $C^{\alpha}$
harmonic radius at $x$. Then we recall two theorems that will be used to
control diffeomorphism types. See Theorem 8.1 and Theorem 8.4 of [13].

######  Theorem 2.11.

For any $\epsilon>0$, $0<\beta<\alpha<1$, there exists
$\delta=\delta(n,\epsilon,\alpha,\beta)$ such that the following holds. If
$M^{n}_{1}$ and $M^{n}_{2}$ are Riemannian manifolds and $U_{j}\subset M_{j}$
are subsets $r_{h}^{\alpha}(x)>r>0$ for all $x\in U_{j}$ and

$d_{GH}(B(U_{1},r),B(U_{2},r))<\delta r,$

then there exists open sets $B(U_{j},\frac{r}{2})\subseteq
U_{j}^{\prime}\subseteq B(U_{j},r)$ and a $C^{1,\beta}$-diffeomorphism
$\phi:U_{1}^{\prime}\to U_{2}^{\prime}$ such that

$\left\|g_{1}-\phi^{*}g_{2}\right\|_{C^{0}}+r^{\beta}[g_{1}-\phi^{*}g_{2}]_{C^{\beta}}\leqslant\epsilon.$

######  Theorem 2.12.

Let $(M^{n},g)$ be a manifold satisfying (1.6) and (2.4). $U\subseteq M$ is an
open subset such that for some $\alpha\in(0,1)$,
$r_{h}^{\alpha}(x)>rdiam(U)>0$ for all $x\in U$, where $diam(U)$ is the
extrinsic diameter so that $diam(U)\leqslant D$. Then there exists
$C=C(n,p,v,\lambda,D,r,\alpha)$ such that $U$ has at most $C$ diffeomorphism
types.

######  Remark 2.13.

Note that by the volume doubling property (2.4), $U$ may be covered by a
bounded number of harmonic coordinate charts $B(x_{i},\frac{1}{2}rdiam(U))$.
The $C^{\alpha}$ bounds on the metric in harmonic coordinate charts will yield
$C^{1,\alpha}$ control over the transition maps between these charts. Then one
can control the number of diffeomorphism types. See [6, 3] for more details.

If we assume $L^{p}$ bound for Riemannian curvature $Rm$, then we can get the
following (see Theorem 5.4 in [19]).

######  Theorem 2.14.

Given $p>\frac{n}{2},v>0$ and $\Lambda>0$, there exists
$r_{0}=r_{0}(n,p,v,\Lambda)$ such that any n-manifold $(M^{n},g)$ with (1.2)
and

$\displaystyle\int_{M^{n}}|Rm|^{p}\leqslant\Lambda,$ (2.8)

satisfies $r_{h}^{\alpha}(x)\geqslant r_{0}$, for all $x\in M^{n}$ and
$\alpha\in(0,2-\frac{n}{p})$. In particular, the collection of closed
n-manifolds satisfying (1.2), (1.6) and (2.8) contains at most
$C=C(n,p,v,D,\Lambda)$ diffeomorphism types.

######  Remark 2.15.

As noted before, if there is a sequence of manifolds
$(M^{n}_{i},g_{i})\xrightarrow{GH}(X,d)$ with uniform lower bound on harmonic
radius, then $X$ is a Riemannian manifold and $(M^{n}_{i},g_{i})\to(X,d)$ in
the Cheeger-Gromov topology (see [20, 19]). Thus, it is reasonable to consider
$1\leqslant q\leqslant\frac{n}{2}$ in Theorem 1.1.
In addition, when $n=2$ or $3$, the Riemannian curvature is completely
determined by Ricci curvature and $L^{p}$ Ricci curvature bound (1.4) implies
$L^{p}$ bound of Riemannian curvature. Thus, we only need to consider the case
$n\geqslant 4$ when proving Theorem 1.2.

###  2.4 Structure of the limit space

Let $(M_{i}^{n},g_{i})$ be a sequence of Riemannian $n$-manifolds satisfying
(1.1) and (1.2) for some uniform $v>0,\lambda>0$ and $p>\frac{n}{2}$. Then by
Gromov’s precompactness theorem, the volume doubling property (2.4) implies
that up to a subsequence,

$\displaystyle(M_{i}^{n},g_{i},x_{i})\xrightarrow{pGH}(X,d,x),$ (2.9)

for some complete length space $(X,d)$. Following lines of Cheeger-Colding [8,
9, 10], one can easily get similar structure results on the limit space. Let
us define the regular set $\mathcal{R}$ and singular set $\mathcal{S}$ on $X$
by

$\displaystyle\mathcal{R}=\\{x\in X:\exists\text{ tangent cone at {x} is
isometric to }\mathbb{R}^{n}\\},$
$\displaystyle\mathcal{S}=X\setminus\mathcal{R}.$

Moreover, a stratification on the singular set was defined. Namely,

$\displaystyle\emptyset\subset\mathcal{S}^{0}\subset...\subset\mathcal{S}^{n-1}=\mathcal{S}\subset
X,$ (2.10)

where

$\displaystyle\mathcal{S}^{k}=\\{x\in X:\text{ no tangent cone at {x} splits
off }\mathbb{R}^{k+1}\\}.$ (2.11)

We now recall the following structure results on the limit space, which can be
proved by similar arguments as that of Cheeger-Colding [8, 9, 10] (see also
[23] and references therein).

######  Theorem 2.16.

Let $(M_{i}^{n},g_{i},x_{i})\xrightarrow{pGH}(X,d,x_{\infty})$ be a sequence
of manifolds satisfying (1.2).Then the following hold:

  * •

If $(M_{i}^{n},g_{i})$ satisfies (1.1), then

    1. (a1)

For any $r>0$, we have: $vol(B(x_{i},r))\to\mathcal{H}^{n}(B(x_{\infty},r))$.

    2. (a2)

For any $x\in X$, each tangent cone at $x$ is a metric cone and splits off
lines isometrically.

    3. (a3)

$\mathcal{S}=\mathcal{S}^{n-2}$, i.e.,
$\mathcal{S}^{n-1}\setminus\mathcal{S}^{n-2}=\emptyset$.

    4. (a4)

$dim_{\mathcal{H}}\mathcal{S}^{k}\leqslant k$. In particular,
$dim_{\mathcal{H}}\mathcal{S}\leqslant n-2$.

  * •

If $(M_{i}^{n},g_{i})$ satisfies (1.4), then

    1. (b1)

$\mathcal{R}$ is an open subset of $X$.

    2. (b2)

$\mathcal{R}$ is a $C^{1,\alpha}\cap W^{3,p}$ smooth manifold with a
$C^{\alpha}\cap W^{2,p}$ metric $g_{\infty}$ which induces $d$, for any
$\alpha\in(0,2-\frac{n}{p})$, and $g_{i}$ converges to $g_{\infty}$ in the
$C^{\alpha}\cap W^{2,p}$ topology on $\mathcal{R}$.

Now we review Yang’s orbifold type limit theorem in [24, 25].

######  Theorem 2.17.

Let $(M_{i}^{n},g_{i})$ be a sequence of Riemannian manifolds satisfying
(1.2), (1.4), (1.6) and (1.7), for some $v,D,\lambda,\Lambda>0$ and
$p>\frac{n}{2}$. Then there exists a subsequece converging in the Gromov-
Hausdorff topology to an orbifold $(V,g)$ with finite singular points, where
$g$ is a $C^{\alpha}\cap W^{2,p}$ Riemannian metric off the singular points
for all $\alpha\in(0,2-\frac{n}{p})$. Moreover, $g_{i}$ converges to $g$ in
the $C^{\alpha}\cap W^{2,p}$ topology away from the singular points.

######  Remark 2.18.

Theorem 2.17 was analogous to Anderson’s result in [2], where he proved a
similar theorem with the stronger condition $|Ric|\leqslant\lambda$. In this
case, one can obtain $C^{1,\alpha}\cap W^{2,q}$ regularity of the metric for
every $0<\alpha<1$ and $q<\infty$, which was used in Anderson-Cheeger’s proof
for their finiteness theorem in [3].

In Section 4, we will use Cheeger-Naber’s argument in [13], which turns out to
be more effective in nature than the one in [3], to prove our diffeomorphism
finiteness theorem (Theorem 1.2).

###  2.5 Elliptic estimate

In this subsection, we review Cheeger’s elliptic estimate in [7], which is
crucial in proving our $\epsilon$-regularity theorem in Section 3.
Let $[a]$ denote the greatest integer $\leqslant a$ and let
$k\in\mathbb{Z}_{+}$. For $1\leqslant q\leqslant\frac{n}{2}$, define

$\displaystyle\tilde{q}=\begin{cases}[q-\frac{1}{2}]&\text{if
}q\neq\frac{2k+1}{2},\\\ [q-\frac{1}{2}]-1&\text{if
}q=\frac{2k+1}{2}.\end{cases}$ (2.12)

######  Theorem 2.19.

Let $x\in M^{n}$ and $u:B(x,1)\to\mathbb{R}$ satisfy $\Delta u=c$ for some
constant c. Then for $q>\frac{3}{2}$, there exists $\epsilon(q)>0$ such that

$\displaystyle
V^{2\tilde{q}}\fint_{B(x,\frac{3}{4})}|\nabla^{2}u|^{2q-2\tilde{q}}+V^{2q}\fint_{B(x,1)}|Rm|^{q}\geqslant\epsilon(q)\fint_{B(x,\frac{1}{2})}|\nabla^{2}u|^{2q},$
(2.13)

where $V=\sup\limits_{B(x,\frac{7}{8})}|\nabla u|$.

######  Remark 2.20.

Recall that if we assume $M^{n}$ satisfies (1.1) and (1.2), then up to a
rescaling on metric, $V$ can be estimated in terms of
$\sup\limits_{B(x,1)}|u|$ by gradient estimates derived in [16].

######  Remark 2.21.

In (2.13), the constant $\epsilon(q)$ goes to zero as $q\to\frac{3}{2}$. For
$q>\frac{3}{2}$, under the condition (1.2) and (1.3), the estimate (2.13) can
be used to obtain $L^{2q}$ estimates for $\nabla^{2}u$ if we are given $L^{2}$
estimates for $\nabla^{2}u$. See Section 1 in Cheeger’s paper [7] for more
details.

##  3 The $\epsilon$-regularity theorem and dimension estimates for singular
set

In this section, we present an $\epsilon$-regularity theorem which is
analogous to the one in [7] (see also [11] for the critical case $q=1$). And
then we will apply this $\epsilon$-regularity theorem to prove Theorem 1.1.

###  3.1 $\epsilon$-regularity theorems

We first recall some basic facts. As before, we always assume $p>\frac{n}{2}$
and $1\leqslant q\leqslant\frac{n}{2}$.
Consider a manifold $(M^{n},g)$ satisfying (1.1) and (1.2). By rescaling on
metric, we may assume

$\displaystyle\int_{M^{n}}|Ric_{-}|^{p}\leqslant\delta^{2p-n},$ (3.1)
$\displaystyle vol(B(x,r))\geqslant vr^{n}\ \text{for all}\ x\in M^{n}\
\text{and}\ r\leqslant\delta^{-1}.$

We denote a point in $\mathbb{R}^{n-k}\times C(Y)$ by $(z,r,y)$, where
$z=(z_{1},...,z_{n-k})\in\mathbb{R}^{n-k}$ and $(r,y)$ is expressed in polar
coordinates of $C(Y)$. Let $y^{*}$ be the cone vertex of $C(Y)$. Assume

$\displaystyle
d_{GH}(B(x,\delta^{-1}),B((0^{n-k},y^{*}),\delta^{-1}))<\delta.$ (3.2)

Then by Theorem 2.7, there is a $\psi(\delta|n,p,v)$-splitting map

$\displaystyle b=(b_{1},...,b_{n-k}):B(x,3)\to\mathbb{R}^{n-k},$ (3.3)

where we denote $\psi(c_{1},...,c_{s}|n_{1},...,n_{r})$ to be any nonnegative
function such that if $n_{1},...,n_{r}$ are fixed and $c_{1},...,c_{s}$ go to
zero, then $\psi$ tends to zero. In addition, we know from [8] that there is a
Gromov-Hausdorff approximation $F:B(x,\delta^{-1})\to
B((0^{n-k},y^{*}),\delta^{-1})$ realizing (3.2) such that

$\displaystyle b_{j}=z_{j}\circ F.$

Following lines of Cheeger-Colding-Tian [11], we may obtain some additional
facts. There exists $u:B(x,3)\to\mathbb{R}_{+}$ satisfying

$\displaystyle u=r\circ F,$ $\displaystyle\Delta u^{2}=2k,$
$\displaystyle\fint_{B(x,3)}||\nabla u^{2}|-2u|^{2}+\sum_{j}|\langle\nabla
u^{2},\nabla b_{j}\rangle|<\psi(\delta|n,p,v),$ (3.4)
$\displaystyle\fint_{B(x,3)}|\nabla^{2}u^{2}-2(g-\sum_{i,j}db_{i}\otimes
db_{j})|^{2}<\psi(\delta|n,p,v),$ (3.5) $\displaystyle|\nabla u^{2}|\leqslant
C(n).$

Define $\phi:B(x,3)\to\mathbb{R}^{n-k}\times\mathbb{R}_{+}$ by

$\displaystyle\phi=(b,u),$ (3.6)

and let

$\displaystyle U_{r}=\phi^{-1}(B(0^{n-k},1)\times[0,r]).$

Then there exists $C_{\delta}\subset B(0^{n-k},1)$ with
$vol(C_{\delta})\geqslant(1-\psi)vol(B(0^{n-k},1))$, such that for any $z\in
C_{\delta}$,

$\displaystyle\left\lvert vol_{k}(b^{-1}(z)\cap
U_{r})-\frac{r^{k}}{k}vol(Y)\right\rvert\leqslant\psi.$ (3.7)

Similarly, there exists $D_{\delta}\subset B(0^{n-k},1)\times[0,1]$ with
$vol(D_{\delta})\geqslant(1-\psi)vol(B(0^{n-k},1)\times[0,1])$, such that for
any $(z,r)\in D_{\delta}$,

$\displaystyle\left\lvert
vol_{k-1}(\phi^{-1}((z,r)))-r^{k-1}vol(Y)\right\rvert\leqslant\psi.$ (3.8)

Before stating the $\epsilon$-regularity theorem, we first prove a lemma which
is analogous to Lemma 4.14 in [7].

######  Lemma 3.1.

Let $M^{n}$ satisfy (1.1) and (1.2). Let $f:B(m,1)\to\mathbb{R}^{n-k}$ be a
Lipschitz function with $|\textnormal{Lip}f|\leqslant l$ and
$h:B(m,3)\to\mathbb{R}$ be a smooth function. Denote

$S_{\eta}=f\left(\\{x\in B(m,1):|h(x)|\geqslant\eta\\}\right).$

Then for all $\mu,\eta>0$, there exists
$\epsilon=\epsilon(n,p,v,\lambda,l,\mu,\eta,q,k)$ such that if $2q>k$ and

$\displaystyle\fint_{B(m,3)}|\nabla h|^{2q}\leqslant\mu,$ (3.9)
$\displaystyle\fint_{B(m,3)}|h|\leqslant\epsilon,$ (3.10)

then $\mathcal{H}^{n-k}(S_{\eta})\leqslant\eta$.

###### Proof.

Consider the collection

$\mathcal{B}=\\{B(x,r):x\in B(m,2),r\leqslant 1,\fint_{B(x,r)}|\nabla
h|^{2q}\geqslant Ar^{-k}\\},$

where $A$ is a constant to be determined later. By the 5-times covering lemma,
there is a disjoint subcollection $\\{B(m_{t},\frac{r_{t}}{5})\\}$, such that
$\bigcup_{t}B(m_{t},r_{t})\supseteq\bigcup_{B\in\mathcal{B}}B$. Then

$\displaystyle\sum_{t}\frac{vol(B(m_{t},\frac{r_{t}}{5}))}{vol(B(m,3))}\frac{A}{\mu}\left(\frac{r_{t}}{5}\right)^{-k}$
$\displaystyle\leqslant\sum_{t}\frac{vol(B(m_{t},\frac{r_{t}}{5}))}{vol(B(m,3))}\frac{1}{\mu}\fint_{B(m_{t},\frac{r_{t}}{5})}|\nabla
h|^{2q}$ $\displaystyle\leqslant\frac{1}{\mu}\fint_{B(m,3)}|\nabla
h|^{2q}\leqslant 1.$

Thus, by the volume doubling property (2.4),

$\sum_{t}r_{t}^{-k}vol(B(m_{t},r_{t}))\leqslant
C(n,p,v,\lambda)\frac{\mu}{A}vol(B(m,3)).$

By (1.2) and (2.3),

$\sum_{t}r_{t}^{n-k}\leqslant C(n,p,v,\lambda)\frac{\mu}{A}.$

We can now choose $A=A(n,p,v,\lambda,l,\mu,\eta,k)$ such that

$C(n,p,v,\lambda)\frac{\mu}{A}\leqslant\frac{\eta}{\omega_{n-k}l^{n-k}}.$

Since $\mathcal{H}^{n-k}(S_{\eta})\leqslant
l^{n-k}\mathcal{H}^{n-k}\left(\\{x\in B(m,1):|h(x)|\geqslant\eta\\}\right)$,
it suffices to show that there exists
$\epsilon=\epsilon(n,p,v,\lambda,l,\mu,\eta,q,k)$, such that if $x\in B(m,1)$
and $|h(x)|\geqslant\eta$, then $B(x,r)\in\mathcal{B}$ for some $r\leqslant
1$.
Fix $x\in B(m,1)$ and denote $h_{x,r}=\fint_{B(x,r)}h$. If
$B(x,r)\notin\mathcal{B}$ for any $r\leqslant 1$, then for some sufficiently
small $r=r(n,p,v,\lambda,l,\mu,\eta,q,k)$,

$\displaystyle|h_{x,r}-h_{x,\frac{r}{2}}|\leqslant\fint_{B(x,\frac{r}{2})}|h-h_{x,r}|$
$\displaystyle\leqslant C(n,p,v,\lambda)\fint_{B(x,r)}|h-h_{x,r}|$
$\displaystyle\leqslant C(n,p,v,\lambda,q)r\left(\fint_{B(x,r)}|\nabla
h|^{2q}\right)^{\frac{1}{2q}}$ (3.11) $\displaystyle\leqslant
C(n,p,v,\lambda,q)Ar^{1-\frac{k}{2q}}.$

and

$\displaystyle|h(x)|\leqslant|h|_{x,r}+C(n,p,v,\lambda,l,\mu,\eta,q,k)r^{1-\frac{k}{2q}}<|h|_{x,r}+\frac{\eta}{2}.$
(3.12)

In (3.11), we use the $(1,2q)$-type Neumann-Poincaré inequality, which can be
obtained from the work of Dai-Wei-Zhang [16].
Finally, by (3.10) and volume doubling property (2.4), we can choose
$\epsilon$ such that $|h|_{x,r}<\frac{\eta}{2}$ and hence, $|h(x)|<\eta$. This
completes the proof. ∎

Now we can proceed to prove the $\epsilon$-regularity theorem, following the
arguments of Cheeger [7] and Cheeger-Colding-Tian [11].

######  Theorem 3.2.

For any $\eta>0$, there exists $\epsilon=\epsilon(n,p,v,\lambda,\eta,q)$ and
$\delta=\delta(n,p,v,\lambda,\eta,q)$, such that if $(M^{n},g)$ satisfies
(1.1), (1.2),

$\displaystyle\epsilon^{2q}\fint_{B(x,3\epsilon)}|Rm|^{q}\leqslant\delta,$
(3.13)

and for $(0^{n-k},y^{*})\in\mathbb{R}^{n-k}\times C(Y)$,

$\displaystyle
d_{GH}(B(x,\delta^{-1}\epsilon),B((0^{n-k},y^{*}),\delta^{-1}\epsilon))<\delta\epsilon,$
(3.14)

where

$2q>\begin{cases}k,&\ k\ \text{even},\\\ k-1,&\ k\ \text{odd},\end{cases}$

then we have

$d_{GH}(B(x,\epsilon),B(0^{n},\epsilon))\leqslant\eta\epsilon.$

###### Proof.

Without loss of generality, let $\lambda=1$. Rescale the metric by a factor
$\epsilon^{-2}$, we may also assume

$\displaystyle\int_{M^{n}}|Ric_{-}|^{p}\leqslant\epsilon^{2p-n},$ (3.15)
$\displaystyle vol(B(x,r))\geqslant vr^{n}\ \text{for all}\ x\in M^{n}\
\text{and}\ r\leqslant\epsilon^{-1}.$ (3.16)
$\displaystyle\fint_{B(x,3)}|Rm|^{q}\leqslant\delta,$ (3.17) $\displaystyle
d_{GH}(B(x,\delta^{-1}),B((0^{n-k},y^{*}),\delta^{-1}))<\delta.$ (3.18)

We only need to show the theorem for $q>1$. Consider first the case in which
$q>\frac{3}{2}$ and $k$ is even. Let $b,\phi$ be the maps in (3.3) and (3.6).
By (2.5), (2.13) and (3.17), it follows that

$\displaystyle\fint_{B(x,2)}|\nabla^{2}b_{j}|^{2q}+\sum_{i,j}|\langle\nabla
b_{i},\nabla
b_{j}\rangle-\delta_{ij}|^{2}\leqslant\psi(\epsilon,\delta|n,p,v,q).$ (3.19)

For the function $u:B(x,3)\to\mathbb{R}_{+}$, we can get an $L^{2q}$ bound of
$\nabla^{2}u^{2}$ from (2.13), (3.4) and (3.5). Then we choose $q_{1}$
satisfying $2q-1<2q_{1}<2q$. By the elementary inequality

$\displaystyle\int f^{2q_{1}}\leqslant t^{2q_{1}-2}\int
f^{2}+t^{2q_{1}-2q}\int f^{2q},$

we get

$\displaystyle\fint_{B(x,3)}|\nabla^{2}u^{2}-2(g-\sum_{i,j}db_{i}\otimes
db_{j})|^{2q_{1}}<\psi.$

For the map $\phi=(b,u):B(x,3)\to\mathbb{R}^{n-k}\times\mathbb{R}_{+}$, we
denote the level set by

$\displaystyle U_{r}=\phi^{-1}(B(0^{n-k},1)\times[0,r]).$

By (2.5) and the coarea formula, there exists
$(z,r)\in\mathbb{R}^{n-k}\times\mathbb{R}_{+}$ with $1-\psi\leqslant
r\leqslant 1$, $z$ a regular value of $b$ and $(z,r)$ a regular value of
$\phi$, such that (3.7) holds and

$\displaystyle\fint_{b^{-1}(z)\cap U_{r}}|Rm|^{q}\leqslant\psi,$ (3.20)
$\displaystyle\fint_{b^{-1}(z)\cap U_{r}}|\nabla^{2}b_{j}|^{2q}\leqslant\psi.$
(3.21)

From (3.19) and Lemma 3.1 (where we need $2q>k$), with $f=b$ and
$h=\langle\nabla b_{i},\nabla b_{j}\rangle-\delta_{ij}$, we can also assume

$\displaystyle\sup\limits_{b^{-1}(z)\cap U_{r}}|\langle\nabla b_{i},\nabla
b_{j}\rangle-\delta_{ij}|\leqslant\psi.$ (3.22)

Similarly, we may assume (3.8) holds and

$\displaystyle\fint_{\phi^{-1}(z,r)}|Rm|^{q}\leqslant\psi,$ (3.23)
$\displaystyle\fint_{\phi^{-1}(z,r)}|\nabla^{2}b_{j}|^{2q}\leqslant\psi,$
(3.24) $\displaystyle\sup\limits_{\phi^{-1}(z,r)}|\langle\nabla b_{i},\nabla
b_{j}\rangle-\delta_{ij}|\leqslant\psi,$ (3.25)
$\displaystyle\fint_{\phi^{-1}(z,r)}|\nabla^{2}u^{2}-2(g-\sum_{i,j}db_{i}\otimes
db_{j})|^{2q_{1}}\leqslant\psi,$ (3.26)
$\displaystyle\sup\limits_{\phi^{-1}(z,r)}\left(||\nabla
u^{2}|-2u|+|\langle\nabla u^{2},\nabla b_{j}\rangle|\right)\leqslant\psi.$
(3.27)

Applying Gram-Schmidt process to $\nabla b_{1},...,\nabla b_{n-k},\nabla
u^{2}$, we get the normal vector fields $e_{1},e_{2},...,e_{n-k},N$. By
(3.21)-(3.27), we obtain estimates on the second fundamental form
$\mathrm{II}_{b^{-1}(z)\cap U_{r}}$ and $\mathrm{II}_{\phi^{-1}(z,r)}$,

$\displaystyle\fint_{b^{-1}(z)\cap U_{r}}|\mathrm{II}_{b^{-1}(z)\cap
U_{r}}|^{2q}\leqslant\psi,$ (3.28)
$\displaystyle\fint_{\phi^{-1}(z,r)}|\mathrm{II}_{b^{-1}(z)\cap
U_{r}}|^{2q}\leqslant\psi,$ (3.29)
$\displaystyle\fint_{\phi^{-1}(z,r)}|\mathrm{II}_{\phi^{-1}(z,r)}-r^{-1}g_{\phi^{-1}(z,r)}\otimes\nabla
u|^{2q_{1}}\leqslant\psi,$ (3.30)

where $g_{\phi^{-1}(z,r)}$ is the induced metric on $\phi^{-1}(z,r)$.
Note that $b^{-1}(z)\cap U_{r}$ is a $k$-dimensional submanifold with boundary
$\phi^{-1}(z,r)$ and $2q>k$, $2q_{1}>k-1$. The Gauss equation and Chern-Gauss-
Bonnet formula gives

$\displaystyle\chi(b^{-1}(z)\cap U_{r})$ $\displaystyle=\int_{b^{-1}(z)\cap
U_{r}}P_{\chi}+\int_{\phi^{-1}(z,r)}TP_{\chi}$
$\displaystyle=\pm\psi+\frac{vol(Y)}{vol(S^{k-1})}$

Since $\chi(b^{-1}(z)\cap U_{r})$ is an integer and $vol(Y)\leqslant
vol(S^{k-1})+\psi$, it follows that $\chi(b^{-1}(z)\cap U_{r})=1$ and
$vol(Y)=vol(S^{k-1})\pm\psi$. By Theorem 2.4 and Theorm 2.5, this implies
$d_{GH}(B(x,1),B(0^{n},1))\leqslant\psi$, which completes the proof for
$q>\frac{3}{2}$ and $k$ is even.
For $q>\frac{3}{2}$ and $k$ odd, we apply Lemma 3.1 (where we need $2q>k-1$)
with $f=\phi$ and $h=\langle\nabla b_{i},\nabla b_{j}\rangle-\delta_{ij}$. We
still get estimate (3.30) on $\mathrm{II}_{\phi^{-1}(z,r)}$. The Gauss
equation and Chern-Gauss-Bonnet formula gives

$\displaystyle\chi(\phi^{-1}(z,r))$
$\displaystyle=\int_{\phi^{-1}(z,r)}P_{\chi}$
$\displaystyle=2\frac{vol(Y)}{vol(S^{k-1})}\pm\psi.$

Since $\dim\phi^{-1}(z,r)=k-1$ is even and $\phi^{-1}(z,r)$ is a boundary,
$\chi(\phi^{-1}(z,r))$ is even. Thus $\chi(\phi^{-1}(z,r))=2$ and
$vol(Y)=vol(S^{k-1})\pm\psi$. As above, we complete the proof for
$q>\frac{3}{2}$.
When $1<q\leqslant\frac{3}{2}$ and $k=2$ or 3, we do not have the elliptic
estimate (2.13), but we already have the corresponding $L^{2}$-estimates for
$b$ and $u$, which is sufficient for us to follow the above proof steps. Thus,
we complete the proof. ∎

######  Remark 3.3.

The above theorem actually holds in the case $q=1$ and $k=2$ or 3. Note that
for $k$ even, the condition $2q>k$ is only used in applying Lemma 3.1 to
control the second fundamental form. In fact, when $q=1$, Cheeger-Colding-Tian
obtained $L^{2}$-estimates as in (3.28)-(3.30) by direct computation (see
Theorem 3.7 in [11]). We mention that they used Cheeger-Colding’s cutoff
function [8], which can also be constructed on manifolds with integral lower
Ricci bound (see [16]). Once we get the $L^{2}$-estimates for the second
fundamental form, we can apply Gauss-Bonnet formula to $b^{-1}(z)\cap U_{r}$
when $k=2$ and to $\phi^{-1}(z,r)$ when $k=3$ as above (see also Theorem 1.11
in [11]).

###  3.2 The Hausdorff dimension estimate

In this subsection, we will prove the first part of Theorem 1.1, namely the
Hausdorff dimension of the singular set $\leqslant n-2q$.
We first recall some basic facts. The $\epsilon$-regular set
$\mathcal{R}_{\epsilon}$ is defined by

$\displaystyle\mathcal{R}_{\epsilon}=\left\\{x\in X:\text{for all sufficiently
small }r,\ d_{GH}(B(x,r),B(0^{n},r))\leqslant\epsilon r\right\\}.$

Let $\mathring{\mathcal{R}}_{\epsilon}$ denote the interior of
$\mathcal{R}_{\epsilon}$. For any $\epsilon>0$, there exists $\delta>0$, such
that $\mathcal{R}_{\delta}\subset\mathring{\mathcal{R}}_{\epsilon}$. In
particular, $\mathcal{R}=\cap_{\epsilon}\mathring{\mathcal{R}}_{\epsilon}$.
Let $(M^{n}_{i},g_{i},x_{i})\xrightarrow{pGH}(X,d,x)$. If
$\int_{M_{i}^{n}}|Rm_{g_{i}}|^{q}\leqslant\Lambda$, then up to a subsequence,
we may assume
$(M^{n}_{i},g_{i},|Rm_{g_{i}}|^{q}vol_{g_{i}},x_{i})\xrightarrow{pmGH}(X,d,m_{q},x)$
in the pointed measured Gromov-Hausdorff sense, where $m_{q}$ is a Borel
measure on $X$ with $m_{q}(X)\leqslant\Lambda$ (see [9, 17]). Also, from
Theorem 2.16 (a1), we know that

$\displaystyle\fint_{B(x_{i},r)}|Rm|^{q}\to\frac{m_{q}(B(x,r))}{\mathcal{H}^{n}(B(x,r))}.$
(3.31)

######  Theorem 3.4.

Given $p>\frac{n}{2}$, $1\leqslant q\leqslant\frac{n}{2}$ and positive
constants $v,\lambda$ and $\Lambda$, let
$(M_{i}^{n},g_{i},x_{i})\xrightarrow{pGH}(X,d,x)$ be a GH limit of manifolds
satisfying (1.1)-(1.3).

  1. (1)

If $q$ is an integer, then
$\mathcal{H}^{n-2q}(\mathcal{S}\setminus\mathcal{S}_{n-2q})=0$.

  2. (2)

If $q$ is not an integer, then $\mathcal{H}^{n-2q}(\mathcal{S})=0$.

In particular, $\dim_{\mathcal{H}}\mathcal{S}\leqslant n-2q$, for all $q$.

###### Proof.

$\forall a\geqslant 0,\ \epsilon>0$, let $D_{a,\epsilon}(q)$ denote the set of
points such that $y\in X\setminus\mathring{\mathcal{R}}_{\epsilon}$ and

$\displaystyle\limsup_{r\to
0}r^{2q}\frac{m_{q}(B(y,r))}{\mathcal{H}^{n}(B(y,r))}\leqslant a,$ (3.32)
$\displaystyle\lim\limits_{r\to
0}r^{2q}\frac{m_{q}(B(y,r)\cap\mathring{\mathcal{R}}_{\epsilon})}{\mathcal{H}^{n}(B(y,r))}=0.$
(3.33)

Put $D_{a}(q)=\bigcup_{\epsilon>0}D_{a,\epsilon}(q)$ and
$D(q)=\bigcup_{a=1}^{\infty}D_{a}(q)$. Then by a standard covering argument
and $m_{q}(X)\leqslant\Lambda$, it follows that

$\displaystyle\mathcal{H}^{n-2q}(\mathcal{S}\setminus D(q))=0,$ (3.34)
$\displaystyle m_{q}\llcorner D(q)\ll\mathcal{H}^{n-2q}\llcorner D(q).$ (3.35)

Denote $m_{i,s}:=|Rm_{g_{i}}|^{s}vol_{g_{i}}$, for any $1\leqslant s\leqslant
q$. It is clear that $m_{i,s}$ is finite on bounded sets and
$m_{i,s}\ll\mathcal{H}^{n}$ on $M_{i}^{n}$.
Let $(M,\mathcal{H}^{n})$ be a measure space and $f$ be a nonnegative function
with $\int_{M}f^{q}\leqslant\Lambda$. Then for any $q_{1}<q$ and $U\subset M$,

$\displaystyle\int_{U}f^{q_{1}}\leqslant(\mathcal{H}^{n}(U))^{1-\frac{q_{1}}{q}}\left(\int_{U}f^{q}\right)^{\frac{q_{1}}{q}}\leqslant\Lambda^{\frac{q_{1}}{q}}(\mathcal{H}^{n}(U))^{1-\frac{q_{1}}{q}}.$
(3.36)

Therefore, the $L^{q_{1}}$-norm of $f$ cannot concentrate on a set $U$ of
small measure.
By passing to a subsequence in
$(M^{n}_{i},g_{i},m_{i,q},x_{i})\xrightarrow{pmGH}(X,d,m_{q},x)$, we may
assume
$(M^{n}_{i},g_{i},m_{i,q_{1}},x_{i})\xrightarrow{pmGH}(X,d,m_{q_{1}},x)$. This
implies that there exists a metric space $(W,d_{W})$ and isometric embeddings
$\varphi_{i}:M_{i}^{n}\to W$ and $\varphi:X\to W$, such that
$\varphi_{i}(x_{i})\to\varphi(x)$ in $W$ and
$(\varphi_{i})_{\\#}m_{i,q_{1}}\to(\varphi)_{\\#}m_{q_{1}}$ in the weak*
topology on $C_{bs}(W)^{*}$ (see [18]). It is obvious that
$(\varphi_{i})_{\\#}m_{i,q_{1}}\ll\mathcal{H}^{n}$ on $W$.
Then by (3.36), we know that the limit measure
$(\varphi)_{\\#}m_{q_{1}}\ll\mathcal{H}^{n}$ on $W$ and hence,
$m_{q_{1}}\ll\mathcal{H}^{n}$ on $X$. It follows that

$\displaystyle
m_{q_{1}}(B(y,r)\cap\mathring{\mathcal{R}}_{\epsilon})=m_{q_{1}}(B(y,r)),\
\forall y\in X,$ $\displaystyle
D_{0}(q_{1})=D(q_{1})=\\{y\in\mathcal{S}:\lim\limits_{r\to
0}r^{2q_{1}}\frac{m_{q_{1}}(B(y,r))}{\mathcal{H}^{n}(B(y,r))}=0\\}.$

Notice that for any open subset $U_{i}\subset M^{n}_{i}$, we have

$r^{2q_{1}}\fint_{U_{i}}|Rm|^{q_{1}}\leqslant\left(r^{2q}\fint_{U_{i}}|Rm|^{q}\right)^{\frac{q_{1}}{q}}.$

Thus, we get

$\displaystyle D(q)\subset D_{0}(q_{1}),$ (3.37)
$\displaystyle\mathcal{H}^{n-2q}(\mathcal{S}\setminus D_{0}(q_{1}))=0.$ (3.38)

Now we go back and prove the theorem.
If $q$ is an integer, then we can choose $q_{1}$ such that
$2q_{1}\in(2q-1,2q)$ and by Theorem 3.2,
$\mathcal{S}\setminus\mathcal{S}^{n-2q}\subset\mathcal{S}\setminus
D_{0}(q_{1})$. Hence, we complete the proof for (1).
If $q$ is not an integer, then we can choose $q_{1}$ such that
$2q_{1}\in(2[q],2q)$ and choose $k=2[q]+1$ in Theorem 3.2. Thus,
$\mathcal{S}\setminus\mathcal{S}^{n-k-1}\subset\mathcal{S}\setminus
D_{0}(q_{1})$. Note that $k+1>2q$ and hence,
$\mathcal{H}^{n-2q}(\mathcal{S}^{n-k-1})=0$. This completes the proof. ∎

The following corollary can be derived directly from Theorem 3.4 and the
results in [9].

######  Corollary 3.5.

let $(M_{i}^{n},g_{i},x_{i})\xrightarrow{pGH}(X,d,x)$ be a GH limit of
manifolds satisfying (1.1)-(1.3) with $q=\frac{n}{2}$. If $n$ is odd, then the
singular set $\mathcal{S}$ is empty and $(X,d)$ is bi-Hölder equivalent to a
Riemannian manifold.

###  3.3 The Minkowski dimension estimate

In this subsection, we will prove the remaining part of Theorem 1.1. The proof
relies on the Minkowski type estimates on the quantitative stratification of
the singular set, introduced by Cheeger-Naber in [12].

######  Definition 3.6.

A metric space $Y$ is called $k$-symmetric if $Y$ is isometric to
$\mathbb{R}^{k}\times C(Z)$ for some compact metric space $Z$.

######  Definition 3.7.

Given a metric space $Y$ with $y\in Y$, $r>0$ and $\epsilon>0$, we say that
$y$ is $(k,\epsilon,r)$-symmetric if there is a $k$-symmetric metric space $X$
such that $d_{GH}(B(y,r),B(x,r))<\epsilon r$, where $x\in X$ is a vertex.

######  Definition 3.8.

For any $r\in(0,1)$ and $\epsilon>0$, we define the quantitative $k$-stratum
of a metric space $X$ by

$\mathcal{S}^{k}_{\epsilon,r}(X):=\\{x\in X:\text{for no }r\leqslant
s\leqslant 1\ \text{is }x\ \text{a }(k+1,\epsilon,s)\text{-symmetric
point}\\}.$

The first main result in [12] is to show a Minkowski type estimate on
$\mathcal{S}^{k}_{\epsilon,r}$ for noncollapsed manifolds with
$Ric\geqslant-(n-1)$. The proof requires the cone-splitting principle and the
monotonicity of the volume ratio $\frac{vol(B(x,r))}{vol_{-1}(B(r))}$.
However, under our assumption (1.1) and (1.2), the formula (2.2) tells us that
the volume ratio $\frac{vol(B(x,r))}{\omega_{n}r^{n}}$ is almost monotone
whenever the radius $r$ is sufficiently small. Then by a slight modification
of Cheeger-Naber’s arguments in [12], one may generalize their result into the
following form.

######  Theorem 3.9.

Let $(M^{n},g)$ satisfy (1.1) and (1.2) with $x\in M^{n}$. Then there exists
$r_{0}=r_{0}(n,p,v,\lambda)>0$, such that for any $\epsilon,\eta>0$ and
$r\leqslant r_{0}$,

$vol\left(T_{r}(\mathcal{S}^{k}_{\epsilon,r}(M^{n})\cap
B(x,1))\right)\leqslant C(n,p,v,\lambda,\epsilon,\eta)r^{n-k-\eta}.$

The Minkowski estimates require a more rigid statement of
$\epsilon$-regularity theorem. Namely, assuming (1.4) in place of (1.1), we
can combine Theorem 2.10 and Theorem 3.2 together to obtain the following.

######  Theorem 3.10.

There exists $\epsilon=\epsilon(n,p,v,\lambda,q)$ and
$\delta=\delta(n,p,v,\lambda,q)$, such that if $(M^{n},g)$ satisfies (1.2),
(1.4),

$\displaystyle\epsilon^{2q}\fint_{B(x,3\epsilon)}|Rm|^{q}\leqslant\delta,$
(3.39)

and for $(0^{n-k},y^{*})\in\mathbb{R}^{n-k}\times C(Y)$,

$\displaystyle
d_{GH}(B(x,\delta^{-1}\epsilon),B((0^{n-k},y^{*}),\delta^{-1}\epsilon))<\delta\epsilon,$

where

$2q>\begin{cases}k,&\ k\ \text{even},\\\ k-1,&\ k\ \text{odd},\end{cases}$

then for any $\alpha\in(0,2-\frac{n}{p})$, we have

$r^{\alpha}_{h}(x)>\frac{1}{2}\epsilon.$

From the proof of Theorem 3.2, we may fix $\delta=\delta(n,p,v,\lambda,q)<1$
with the property that there exists
$\epsilon_{0}=\epsilon_{0}(n,p,v,\lambda,q)$ such that the above theorem holds
for every $0<\epsilon\leqslant\epsilon_{0}$. Without loss of generality, we
may also assume $\delta^{-1}\epsilon_{0}<r_{0}$, where
$r_{0}=r_{0}(n,p,v,\lambda)$ is the constant in Theorem 3.9.
Now we combine Theorem 3.9 and Theorem 3.10 to prove the following theorem,
which implies the Minkowski dimension estimate of Theorem 1.1.

######  Theorem 3.11.

Let $(M^{n},g)$ satisfy (1.2)-(1.4) with $x\in M^{n}$. Then there exists
$\epsilon_{0}=\epsilon_{0}(n,p,v,\lambda,q)>0$, such that for any $\eta>0$,
$\alpha\in(0,2-\frac{n}{p})$ and $\epsilon\leqslant\epsilon_{0}$,

$\displaystyle vol\left(T_{\epsilon}(\\{y\in
B(x,1):r^{\alpha}_{h}(y)\leqslant\frac{\epsilon}{2}\\})\right)\leqslant
C(n,p,v,\lambda,\Lambda,q,\eta)\epsilon^{2q-\eta}.$ (3.40)

###### Proof.

As noted above, let us fix $\delta=\delta(n,p,v,\lambda,q)<1$ and
$\epsilon_{0}=\epsilon_{0}(n,p,v,\lambda,q)$. Consider first the case in which
$q$ is an integer.
For arbitrary $\epsilon\leqslant\epsilon_{0}$, the above $\epsilon$-regularity
theorem (Theorem 3.10) gives the following inclusion

$\left\\{y\in
B(x,1):r^{\alpha}_{h}(y)\leqslant\frac{1}{2}\epsilon\right\\}\subseteq\left(\mathcal{S}^{n-2q}_{\delta^{2},\delta^{-1}\epsilon}\cup\\{y:\epsilon^{2q}\fint_{B(y,3\epsilon)}|Rm|^{q}>\delta\\}\right)\cap
B(x,1).$

By Theorem 3.9, we know that

$\displaystyle
vol\left(T_{\epsilon}(\mathcal{S}^{n-2q}_{\delta^{2},\delta^{-1}\epsilon}\cap
B(x,1))\right)$ $\displaystyle\leqslant
vol\left(T_{\delta^{-1}\epsilon}(\mathcal{S}^{n-2q}_{\delta^{2},\delta^{-1}\epsilon}\cap
B(x,1))\right)$ $\displaystyle\leqslant
C(n,p,v,\lambda,\delta,\eta)\epsilon^{2q-\eta}$ (3.41) $\displaystyle\leqslant
C(n,p,v,\lambda,q,\eta)\epsilon^{2q-\eta}.$

Then we denote $A=\\{y\in
B(x,1):\epsilon^{2q}\fint_{B(y,3\epsilon)}|Rm|^{q}>\delta\\}$.

Now, our goal is to get volume estimate of the $\epsilon$-tubular neighborhood
$T_{\epsilon}(A)$. Consider the collection

$\displaystyle\mathcal{B}=\left\\{B(y,3\epsilon):y\in A\right\\}.$

By the 5-times covering lemma, there exists a pairwise disjoint subcollection
$\\{B(y_{i},3\epsilon)\\}$ such that

$\displaystyle\bigcup_{B\in\mathcal{B}}B\subset\bigcup_{i}B(y_{i},15\epsilon).$
(3.42)

Thus, we get

$\displaystyle\sum_{i}vol(B(y_{i},3\epsilon))<\frac{\epsilon^{2q}}{\delta}\sum_{i}\int_{B(y_{i},3\epsilon)}|Rm|^{q}\leqslant
C(\delta,\Lambda)\epsilon^{2q},$ (3.43)

and by volume doubling property (2.4),

$\displaystyle
vol(T_{\epsilon}(A))\leqslant\sum_{i}vol(B(y_{i},15\epsilon))\leqslant
C(n,p,v,\lambda,\Lambda,q)\epsilon^{2q}.$ (3.44)

Combining (3.41) and (3.44), we obtain the desired estimate (3.40).
Now, we consider the case in which $q$ is not an integer. Similarly, by
Theorem 3.10, we have the inclusion

$\left\\{y\in
B(x,1):r^{\alpha}_{h}(y)\leqslant\frac{1}{2}\epsilon\right\\}\subseteq\left(\mathcal{S}^{n-2[q]-2}_{\delta^{2},\delta^{-1}\epsilon}\cup\\{y:\epsilon^{2q}\fint_{B(y,3\epsilon)}|Rm|^{q}>\delta\\}\right)\cap
B(x,1).$

Following the same argument as above, we can obtain a stronger estimate:

$\displaystyle vol\left(T_{\epsilon}(\\{y\in
B(x,1):r^{\alpha}_{h}(y)\leqslant\frac{\epsilon}{2}\\})\right)\leqslant
C(n,p,v,\lambda,\Lambda,q)\epsilon^{2q}.$

Thus, we complete the proof. ∎

The Minkowski dimension estimate of Theorem 1.1 can be viewed as an easy
corollary of Theorem 3.11. Thus, we complete the proof of Theorem 1.1.

##  4 Diffeomorphism finiteness

In this section, we will prove our diffeomorphism finiteness theorem (Theorem
1.2). The argument follows Cheeger-Naber’s bubble tree construction in [13].

###  4.1 Neck regions

In this subsection, we prove some technical results to handle the neck regions
in the bubble tree structure.
From formula (2.2), under the assumption (1.2) and (1.4), the volume ratio
$\frac{vol(B(x,r))}{r^{n}}$ will be almost monotone whenever the radius $r$ is
sufficiently small. For convenience in writing, we use the notation

$\displaystyle\mathcal{V}_{r}(x):=-\ln\left(\frac{vol(B(x,r))}{\omega_{n}r^{n}}\right)$
(4.1)

where $\omega_{n}$ is the volume of unit ball in $\mathbb{R}^{n}$.
Let $A_{r_{1},r_{2}}(q)$ denote the geodesic annulus centered at some $q\in
M^{n}$, i.e.,

$A_{r_{1},r_{2}}(q):=B(q,r_{2})\setminus\overline{B(q,r_{1})}.$

We begin with the following annulus estimate. As before, we always assume
$n\geqslant 4,\ p>\frac{n}{2}$ and $\alpha\in(0,2-\frac{n}{p})$. In fact, one
may set $\alpha$ as a fixed value, such as $1-\frac{n}{2p}$.

######  Proposition 4.1.

For any $\epsilon\in(0,\frac{1}{2})$, there exists
$\delta=\delta(n,p,v,\Lambda,\epsilon)$ such that if a manifold $(M^{n},g)$
satisfies

$\displaystyle\int_{M^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda,$ (4.2)
$\displaystyle\int_{M^{n}}|Ric|^{p}\leqslant\delta^{2p-n},$ (4.3)
$\displaystyle vol(B(x,r))\geqslant vr^{n}\ \text{for all}\ x\in M^{n}\
\text{and}\ r\leqslant\delta^{-1},$ (4.4)
$\displaystyle|\mathcal{V}_{4}(q)-\mathcal{V}_{2}(q)|\leqslant\delta,$ (4.5)

then there exists $r_{0}=r_{0}(n,v)$ and a discrete subgroup $\Gamma\subseteq
O(n)$ with $|\Gamma|<C(n,v)$ such that the following hold:

  1. (1)

for any $x\in A_{\epsilon,2}(q)$, $r_{h}^{\alpha}(x)>r_{0}\epsilon$;

  2. (2)

there exists a subset $A_{\epsilon,2}(q)\subseteq U\subseteq
A_{\frac{1}{2}\epsilon,2+\frac{1}{2}\epsilon}(q)$ and a diffeomorphism
$\phi:A_{\epsilon,2}(0)\to U$ where $0\in\mathbb{R}^{n}/\Gamma$, such that

$\left\|\phi^{*}g_{ij}-\delta_{ij}\right\|_{C^{0}}+[\phi^{*}g_{ij}]_{C^{\alpha}}\leqslant\epsilon.$

###### Proof.

Argue by contradiction. We assume that there exists a sequence of manifolds
$(M^{n}_{i},g_{i},q_{i})$ with
$\int_{M_{i}^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda,\
\int_{M_{i}^{n}}|Ric|^{p}\leqslant\delta_{i}^{2p-n}\to 0,\
vol(B(x_{i},r))\geqslant vr^{n}\ \text{for all}\ x_{i}\in
M_{i}^{n},r\leqslant\delta_{i}^{-1}$ and
$|\mathcal{V}_{4}(q_{i})-\mathcal{V}_{2}(q_{i})|\leqslant\delta_{i}\to 0$, but
the conclusions fail. After passing to a subsequence, we get

$(M^{n}_{i},g_{i},q_{i})\xrightarrow{GH}(X,d,q).$

By Theorem 2.6, we know that $B(q,4)=B(z^{*},4)$ where $z^{*}$ is the cone
vertex of a metric cone $C(Z)$ with $diam(Z)\leqslant\pi$. Choose a sequence
$r_{i}\to 0$ and let $\tilde{g}_{i}=r_{i}^{-2}g_{i}$. Then passing to a
subsequence, we may assume

$(M^{n}_{i},\tilde{g}_{i},q_{i})\xrightarrow{GH}(C(Z),d_{C},z^{*}).$

Using Theorem 1.1 and the scaling invariance of metric cone, we know that
$z^{*}$ is the only possible singularity of $C(Z)$ (otherwise, there is a ray
full of singularities) and by Theorem 2.16 (b2), $\tilde{g}_{i}$ converges to
$g_{C}$ in the $C^{\alpha}\cap W^{2,p}$ topology away from the vertex $z^{*}$.
Considering the equation for Ricci curvature in harmonic coordinates and using
$\left\|Ric_{\tilde{g}_{i}}\right\|_{L^{p}}\to 0$, it follows that
$(C(Z),g_{C})$ is smooth and Ricci flat away from the vertex $z^{*}$. Also,
since $\int_{M_{i}^{n}}|Rm_{\tilde{g}_{i}}|^{\frac{n}{2}}\leqslant\Lambda$ and
$\tilde{g}_{i}\xrightarrow{W^{2,p}}g_{C}$ ($p>\frac{n}{2}$) away from $z^{*}$,
we obtain that

$\displaystyle\int_{C(Z)\setminus\\{z^{*}\\}}|Rm_{g_{C}}|^{\frac{n}{2}}\leqslant\Lambda.$
(4.6)

However, by scaling invariance of the metric cone $C(Z)$ and the curvature
energy $\left\|Rm\right\|_{L^{\frac{n}{2}}}$, it follows that for any $r>0$,

$\displaystyle\int_{A_{r,2r}(z^{*})}|Rm_{g_{C}}|^{\frac{n}{2}}=\int_{A_{2r,4r}(z^{*})}|Rm_{g_{C}}|^{\frac{n}{2}}.$
(4.7)

Combining (4.6) and (4.7),
$\int_{C(Z)\setminus\\{z^{*}\\}}|Rm_{g_{C}}|^{\frac{n}{2}}=0$ and $C(Z)$ is a
flat cone. Thus $Z=S^{n-1}/\Gamma$ has constant sectional curvature +1. In
addition, we know from Theorem 2.16 (a1) that $|\Gamma|<C(n,v)$.
Now we come back to the original converging sequence
$(M^{n}_{i},g_{i},q_{i})\xrightarrow{GH}(X,d,q)$. We have already known that
$B(q,4)=B(0,4)$ where $0\in\mathbb{R}^{n}/\Gamma$. It follows that there
exists $r_{0}=r_{0}(n,v)$ such that for $z\in\mathbb{R}^{n}/\Gamma$ with
$|z|=1$, we have $B(z,2r_{0})=B(0^{n},2r_{0})$ where $0^{n}\in\mathbb{R}^{n}$.
In particular, for sufficiently large $i$, we obtain from Theorem 2.10 that
$r^{\alpha}_{h}(x)\geqslant r^{2,p}_{h}(x)>r_{0}\epsilon$, for all $x\in
A_{\epsilon,2}(q_{i})$ and $\alpha\in(0,2-\frac{n}{p})$. Thus, (1) is true and
(2) must fail to hold.
However, by (1) and Theorem 2.11, we have that for sufficiently large $i$,
there exists a diffeomorphism $\phi_{i}:A_{\epsilon,2}(0)\to M_{i}^{n}$ where
$0\in\mathbb{R}^{n}/\Gamma$, such that the pullback metric $\phi_{i}^{*}g_{i}$
is $\epsilon$-close to $\delta_{ij}$ in $C^{\alpha}$ norm. This implies (2); a
contradiction. ∎

######  Remark 4.2.

Note that since we have taken scaling in (4.3) and (4.4), we do not need to
consider small radius $r$ as in Theorem 2.6 and Theorem 2.10.

Motivated by Proposition 4.1, we introduce the following definition and lemma
similar to those in [13] before stating our key neck theorem.

######  Definition 4.3.

Consider the scales $r_{k}=2^{-k}$, $k\in\mathbb{N}$. For any $x\in M^{n}$, we
define

$T_{k}^{\delta}(x):=\begin{cases}1&\text{if
}|\mathcal{V}_{4r_{k}}(x)-\mathcal{V}_{2r_{k}}(x)|>\delta,\\\ 0&\text{if
}|\mathcal{V}_{4r_{k}}(x)-\mathcal{V}_{2r_{k}}(x)|\leqslant\delta.\end{cases}$

We also denote by $|T^{\delta}|(x)=\sum_{k}T_{k}^{\delta}(x)$ the number of
bad scales at $x$.

######  Lemma 4.4.

There exists $\delta=\delta(n,p,v)$ satisfying the following property. Let
$(M^{n},g)$ be a manifold with
$\int_{M^{n}}|Ric_{-}|^{p}\leqslant\delta^{2p-n}$ and $vol(B(x,r))\geqslant
vr^{n}\ \text{for all}\ x\in M^{n}\ \text{and}\ r\leqslant\delta^{-1}$. Then
for any $\delta^{\prime}>0$ and $x\in M^{n}$, there exists at most
$N=N(n,p,v,\delta^{\prime})$ scales $k\in\mathbb{N}$ such that
$T_{k}^{\delta^{\prime}}(x)=1$.

###### Proof.

For any $x\in M^{n}$,
$\mathcal{V}_{1}(x)\leqslant-\ln(\frac{v}{\omega_{n}})=C(n,v)$. And the almost
monotonicity formula (2.2) gives: $\forall r<R\leqslant 1$,

$\mathcal{V}_{R}(x)-\mathcal{V}_{r}(x)\geqslant 2p\ln(1-C(n,p,v)(\delta
R)^{1-\frac{n}{2p}})\geqslant 2p\ln(1-\frac{1}{2}R^{1-\frac{n}{2p}}).$

Here, we have chosen $\delta=(2C(n,p,v))^{\frac{2p}{n-2p}}$. Then

$\displaystyle C(n,v)-1$
$\displaystyle\geqslant\mathcal{V}_{1}(x)-\mathcal{V}_{0}(x)$
$\displaystyle=\sum_{k}(\mathcal{V}_{r_{k}}(x)-\mathcal{V}_{r_{k+1}}(x))$
$\displaystyle\geqslant\sum_{k}|\mathcal{V}_{r_{k}}(x)-\mathcal{V}_{r_{k+1}}(x)|+4p\sum_{k}\ln(1-\frac{1}{2}r_{k}^{1-\frac{n}{2p}})$
$\displaystyle\geqslant\sum_{k}|\mathcal{V}_{r_{k}}(x)-\mathcal{V}_{r_{k+1}}(x)|-8p\sum_{k}2^{-k(1-\frac{n}{2p})}$
$\displaystyle\geqslant\sum_{k}|\mathcal{V}_{r_{k}}(x)-\mathcal{V}_{r_{k+1}}(x)|-C(n,p).$

Then $\sum|\mathcal{V}_{r_{k}}(x)-\mathcal{V}_{r_{k+1}}(x)|\leqslant
C(n,p,v)$. In particular, there are at most $N=C(n,p,v)(\delta^{\prime})^{-1}$
scales $k\in\mathbb{N}$ such that
$|\mathcal{V}_{r_{k}}(x)-\mathcal{V}_{r_{k+1}}(x)|>\delta^{\prime}$. ∎

Note that for any good scale $k\in\mathbb{N}$ at $q$, i.e.,
$T_{k}^{\delta}(q)=0$, we can denote by $\Gamma_{k}\subseteq O(n)$ the
subgroup arising from Proposition 4.1. The following is our key neck theorem,
which is similar to Proposition 4.1 while $\delta$ is independent of the ratio
$r_{1}/r_{2}$ of the annulus $A_{r_{1},r_{2}}(q)$. Generally, we are aimed to
construct a bubble tree decomposition to prove the diffeomorphism finiteness
and in the decomposition, we will encounter two different types of regions:
body regions and neck regions. The neck regions will be diffeomorphic to
cylinders $(0,1)\times S^{n-1}/\Gamma$ and connect different body regions.

######  Theorem 4.5 (Neck Theorem).

For any $0<\epsilon<\epsilon(n,v)$, there exists
$\delta=\delta(n,p,v,\Lambda,\epsilon)$ with the following property. Let
$(M^{n},g)$ satisfy

$\displaystyle\int_{M^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda,$
$\displaystyle\int_{M^{n}}|Ric|^{p}\leqslant\delta^{2p-n},$ $\displaystyle
vol(B(x,r))\geqslant vr^{n}\ \text{for all}\ x\in M^{n}\ \text{and}\
r\leqslant\delta^{-1},$

and assume $k_{1}\in\mathbb{N}$ satisfies $T_{k_{1}}^{\delta}(q)=0$ with
$\Gamma_{k_{1}}$ the corresponding group. If $k_{2}\geqslant k_{1}$ satisfies
$\mathcal{V}_{r_{k_{2}}}(q)\geqslant\ln|\Gamma_{k_{1}}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$,
then there exists a subset $A_{\frac{1}{2}r_{k_{2}},2r_{k_{1}}}(q)\subseteq
U\subseteq A_{\frac{1}{2}(1-\epsilon)r_{k_{2}},2(1+\epsilon)r_{k_{1}}}(q)$ and
a diffeomorphism $\phi:A_{\frac{1}{2}r_{k_{2}},2r_{k_{1}}}(0)\to U$, where
$0\in\mathbb{R}^{n}/\Gamma_{k_{1}}$, such that

$\left\|\phi^{*}g_{ij}-\delta_{ij}\right\|_{C^{0}(A_{\frac{1}{2}r_{k_{2}},2r_{k_{1}}}(0))}+r_{k_{2}}^{\alpha}[\phi^{*}g_{ij}]_{C^{\alpha}(A_{\frac{1}{2}r_{k_{2}},2r_{k_{1}}}(0))}\leqslant\epsilon.$

###### Proof.

Let $\epsilon_{1}>0$ be arbitrary and $\delta(n,p,v,\Lambda,\epsilon_{1})$ be
the corresponding number from Proposition 4.1. This number $\delta$ will be
fixed later to be the number we need. If $T_{k_{1}}^{\delta}(q)=0$, then there
exists a diffeomorphism
$\phi_{k_{1}}:A_{\frac{1}{2}r_{k_{1}},2r_{k_{1}}}(0)\to U_{k_{1}}$, where
$0\in\mathbb{R}^{n}/\Gamma_{k_{1}}$ and
$A_{\frac{1}{2}r_{k_{1}},2r_{k_{1}}}(q)\subseteq U_{k_{1}}\subseteq
A_{\frac{1}{2}(1-\epsilon_{1})r_{k_{1}},2(1+\epsilon_{1})r_{k_{1}}}(q)$, such
that

$\left\|\phi_{k_{1}}^{*}g_{ij}-\delta_{ij}\right\|_{C^{0}}+r_{k_{1}}^{\alpha}[\phi_{k_{1}}^{*}g_{ij}]_{C^{\alpha}}\leqslant\epsilon_{1}.$

Moreover, let $\epsilon>0$ be fixed and $4\delta_{1}(n,p,v,\Lambda,\epsilon)$
be the corresponding number from Proposition 4.1. Then we can choose
$\epsilon_{1}=\epsilon_{1}(n,p,v,\Lambda,\delta_{1})=\epsilon_{1}(n,p,v,\Lambda,\epsilon)$
with
$\delta=\delta(n,p,v,\Lambda,\epsilon_{1})=\delta(n,p,v,\Lambda,\epsilon)$
sufficiently small such that
$\delta^{\frac{1}{2}(1-\frac{n}{2p})}\leqslant\delta_{1}$ and

$\mathcal{V}_{2r_{k_{1}}}(q)<\ln|\Gamma_{k_{1}}|+\delta_{1}.$

Thus, if $k_{2}\geqslant k_{1}$ satisfies
$\mathcal{V}_{r_{k_{2}}}(q)\geqslant\ln|\Gamma_{k_{1}}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}\geqslant\ln|\Gamma_{k_{1}}|-\delta_{1}$,
then by the almost monotonicity formula (2.2), we have
$T_{k}^{4\delta_{1}}(q)=0$ for all scales $k\in[k_{1},k_{2}]\cap\mathbb{N}$.
By Proposition 4.1, for each $k\in[k_{1},k_{2}]$, there exists a
diffeomorphism $\phi_{k}:A_{\frac{1}{2}r_{k},2r_{k}}(0)\to U_{k}$, where
$0\in\mathbb{R}^{n}/\Gamma_{k}$ and $A_{\frac{1}{2}r_{k},2r_{k}}(q)\subseteq
U_{k}\subseteq A_{\frac{1}{2}(1-\epsilon)r_{k},2(1+\epsilon)r_{k}}(q)$, such
that

$\displaystyle\left\|\phi_{k}^{*}g_{ij}-\delta_{ij}\right\|_{C^{0}(A_{\frac{1}{2}r_{k},2r_{k}}(0))}+r_{k}^{\alpha}[\phi_{k}^{*}g_{ij}]_{C^{\alpha}(A_{\frac{1}{2}r_{k},2r_{k}}(0))}\leqslant\epsilon.$
(4.8)

In particular, this implies that $\Gamma_{k}=\Gamma$ is independent of $k$.
Then we consider the inverse maps $\phi_{k}^{-1}:U_{k}\to
A_{\frac{1}{2}r_{k},2r_{k}}(0)$. Observe that after composing $\phi_{k}$ with
a rotation of $\mathbb{R}^{n}/\Gamma$, we can assume that

$\displaystyle|\phi_{k}^{-1}(x)-\phi_{l}^{-1}(x)|\leqslant\epsilon r_{k},\
\forall x\in U_{k}\cap U_{l}.$ (4.9)

Now we can choose a sufficiently small $\epsilon(n,v)$, such that for any
$0<\epsilon<\epsilon(n,v)$, if $y\in\mathbb{R}^{n}/\Gamma$, then
$B(y,4\epsilon|y|)=B(0^{n},4\epsilon|y|)$ where $0^{n}\in\mathbb{R}^{n}$.
For each $k$, let $\eta_{k}:U_{k}\to\mathbb{R}$ be a smooth cutoff function
with

$\eta_{k}(x):=\begin{cases}1&\text{if }x\in
A_{\frac{5}{8}r_{k},\frac{15}{8}r_{k}}(q),\\\ 0&\text{if }x\notin
A_{\frac{1}{2}r_{k},2r_{k}}(q)\end{cases}$

and $|\nabla\eta_{k}|<10r_{k}^{-1}$. If we let $\eta(x)=\sum_{k}\eta_{k}(x)$,
then

$\psi_{k}:=\frac{\eta_{k}}{\eta}:U_{k}\to\mathbb{R}$

satisfies $\sum_{k}\psi_{k}(x)=1$ and $|\nabla\psi_{k}|<200r_{k}^{-1}$. Now we
define the map

$\phi^{-1}:U=\bigcup_{k}U_{k}\to A_{\frac{1}{2}r_{k_{2}},2r_{k_{1}}}(0)$

by $\phi^{-1}(x)=\sum_{k}\psi_{k}(x)\phi_{k}^{-1}(x)$. Notice that all the
$\phi_{k}^{-1}(x)$ live in a Euclidean ball, so the convex combination is well
defined. On each domain $U_{k}$, $\phi^{-1}$ and $\phi_{k}^{-1}$ are
$C^{1}$-close by (4.8) and (4.9). Thus, $\phi^{-1}:U\to
A_{\frac{1}{2}r_{k_{2}},2r_{k_{1}}}(0)$ is a diffeomorphism. Again by (4.8)
and (4.9), we can derive the desired estimate:

$\left\|\phi^{*}g_{ij}-\delta_{ij}\right\|_{C^{0}(A_{\frac{1}{2}r_{k_{2}},2r_{k_{1}}}(0))}+r_{k_{2}}^{\alpha}[\phi^{*}g_{ij}]_{C^{\alpha}(A_{\frac{1}{2}r_{k_{2}},2r_{k_{1}}}(0))}\leqslant
400\epsilon.$

This completes the proof. ∎

To end this section, we prove the following lemma which will be used to
control the number of the neck regions and body regions.

######  Lemma 4.6.

For any sufficiently small $\delta\leqslant\delta(n,p,v)$, there exists a
scale $k=k(n,p,v,\Lambda,\delta)$ with the following property. Let $M^{n}$ be
a manifold with $\int_{M^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda$,
$\int_{M^{n}}|Ric|^{p}\leqslant\delta^{2p-n}$, $vol(B(x,r))\geqslant vr^{n}\
\text{for all}\ x\in M^{n},\ r\leqslant\delta^{-1}$ and
$\mathcal{V}_{1}(q)<\ln N-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$ for some
integer $N\geqslant 2$. Then

$\mathcal{V}_{r_{k}}(q)<\ln(N-1)+\delta^{\frac{1}{2}(1-\frac{n}{2p})}.$

###### Proof.

Let $\delta\leqslant\delta(n,p,v)$ satisfy Lemma 4.4. By Proposition 4.1, we
can choose $\delta^{\prime}(n,p,v,\Lambda,\delta)$ such that if
$\int_{M^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda$,
$\int_{M^{n}}|Ric|^{p}\leqslant(\delta^{\prime})^{2p-n}$,
$vol(B(x,r))\geqslant vr^{n}\ \text{for all}\ x\in
M^{n},r\leqslant(\delta^{\prime})^{-1}$ and $T_{1}^{\delta^{\prime}}(q)=0$,
then

$|\mathcal{V}_{1}(q)-\ln|\Gamma||<\frac{1}{2}\delta^{\frac{1}{2}(1-\frac{n}{2p})}.$

By rescaling, we see that if $\int_{M^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda$,
$\int_{M^{n}}|Ric|^{p}\leqslant\delta^{2p-n}$, $vol(B(x,r))\geqslant vr^{n}\
\text{for all}\ x\in M^{n},r\leqslant\delta^{-1}$ and
$|\mathcal{V}_{2r_{k}}(q)-\mathcal{V}_{r_{k}}(q)|\leqslant\delta^{\prime}$
with $k\geqslant k(n,p,v,\Lambda,\delta)$, then

$|\mathcal{V}_{r_{k}}(q)-\ln|\Gamma||<\frac{1}{2}\delta^{\frac{1}{2}(1-\frac{n}{2p})}.$

Here, the scale $k\geqslant k(n,p,v,\Lambda,\delta)$ satisfies
$\frac{\delta}{\delta^{\prime}}r_{k}\leqslant 1$.
Now, we can apply Lemma 4.4 to see that the scale $k\geqslant
k(n,p,v,\Lambda,\delta)$ such that
$|\mathcal{V}_{2r_{k}}(q)-\mathcal{V}_{r_{k}}(q)|\leqslant\delta^{\prime}$
does exist. Hence, in the context of the lemma, we get

$|\mathcal{V}_{r_{k}}(q)-\ln|\Gamma||<\frac{1}{2}\delta^{\frac{1}{2}(1-\frac{n}{2p})}.$

However, if $\mathcal{V}_{1}(q)<\ln N-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$,
then by the almost monotonicity formula (2.2), $\mathcal{V}_{r_{k}}(q)<\ln
N-\delta^{\frac{1}{2}(1-\frac{n}{2p})}+C(n,p,v)\delta^{1-\frac{n}{2p}}<\ln
N-\frac{1}{2}\delta^{\frac{1}{2}(1-\frac{n}{2p})}$.
Thus, $\ln|\Gamma|<\ln N$ and $|\Gamma|\leqslant N-1$, which completes the
proof. ∎

###  4.2 Bubble tree structure

In this subsection, we will construct the bubble tree structure and prove our
diffeomorphism finiteness Theorem 1.2.
We have built the neck regions of our bubble tree in Theorem 4.5. What is left
is to construct the various body regions. The body regions are designed to
have uniform lower bound on harmonic radius. By the proof of Proposition 4.1,
we see that if the corresponding group $\Gamma$ is a trivial group, then we do
not need to consider the annulus (neck) $A_{\epsilon r,2r}(q)$. Indeed, the
ball $B(q,2r)$ is GH-close to $B(0^{n},2r)$ and serves as a body region. So,
we will first assume $|\Gamma|\geqslant 2$ and try to apply Lemma 4.6. We
begin with the following lemma.

######  Lemma 4.7.

For any sufficiently small $\delta\leqslant\delta(n,p,v)$, there exists
$r_{0}=r_{0}(n,p,v,\Lambda,\delta)$ and $N=N(n,p,v,\Lambda,\delta)$ with the
following property. Let $M^{n}$ be a manifold with
$\int_{M^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda$,
$\int_{M^{n}}|Ric|^{p}\leqslant\delta^{2p-n}$ and $vol(B(x,r))\geqslant
vr^{n}\ \text{for all}\ x\in M^{n}\ \text{and}\ r\leqslant\delta^{-1}$. Then
there exist points $\\{x_{j}\\}_{1}^{m}\subset B(q,1)$ with $m\leqslant N$,
and scales $\\{k_{j}\\}_{1}^{m}\subset\mathbb{N}$ with
$r_{j}:=r_{k_{j}}>r_{0}$ such that

  1. (1)

$T_{k_{j}}^{\delta}(x_{j})=0$;

  2. (2)

if $s_{j}$ denotes the largest integer such that
$\mathcal{V}_{r_{s_{j}}}(x_{j})\geqslant\ln|\Gamma_{j}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$,
then for any $x\in B(x_{j},2r_{s_{j}})$, we have
$\mathcal{V}_{\frac{1}{2}r_{s_{j}}}(x)<\ln|\Gamma_{j}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$;

  3. (3)

if $x\in B(q,1)\setminus\bigcup_{j}B(x_{j},r_{j})$, then
$r_{h}^{\alpha}(x)>r_{0}$.

###### Proof.

Let $\delta\leqslant\delta(n,p,v)$ satisfy Lemma 4.4 with
$\eta=\eta(n,p,v,\Lambda,\delta)$ to be chosen later. For any $x\in M^{n}$,
there exists a scale $k_{x}\in[k(n,p,v,\eta),4k(n,p,v,\eta)]$ so that
$T_{k_{x}}^{\eta}(x)=0$. Pick a covering $\\{B(x,r_{k_{x}})\\}$ of $B(q,1)$
and choose a finite subcovering $\\{B(y_{j},r_{j})\\}_{1}^{m}$, where
$r_{j}=r_{k_{j}}$, $k_{j}=k_{y_{j}}$ and all the balls
$\\{B(y_{j},\frac{1}{10}r_{j})\\}_{1}^{m}$ are disjoint. Thus, $m\leqslant
N(n,p,v,\eta)$ by volume doubling property (2.4).
By Proposition 4.1 and the proof of Lemma 4.6, for any sufficiently small
$\epsilon=\epsilon(n,p,v,\delta)$, we can choose
$\eta(n,p,v,\Lambda,\epsilon,\delta)$ so that for each $x\in A_{\epsilon
r_{j},2r_{j}}(y_{j})$, we have $r_{h}^{\alpha}(x)\geqslant r(n,v)\epsilon
r_{j}>r_{0}(n,p,v,\epsilon,\eta)$ and
$|\mathcal{V}_{\frac{1}{2}\sqrt{\epsilon}r_{j}}(y_{j})-\ln|\Gamma_{j}||<\frac{1}{2}\delta^{\frac{1}{2}(1-\frac{n}{2p})}$.
Also, $\epsilon$ and $\eta$ can be chosen to satisfy $T_{k_{j}}^{\delta}(x)=0$
for all $x\in B(y_{j},4\sqrt{\epsilon}r_{j})$.
For each $x\in B(y_{j},4\sqrt{\epsilon}r_{j})$, let $s_{j}(x)$ denote the
largest integer such that
$\mathcal{V}_{r_{s_{j}(x)}}(x)\geqslant\ln|\Gamma_{j}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$.
Here, we need $\delta^{\frac{1}{2}(1-\frac{n}{2p})}\ll\ln 2$. Consider

$W_{j}:=\\{x\in B(y_{j},4\sqrt{\epsilon}r_{j}):B(x,4r_{s_{j}(x)})\subset
B(y_{j},4\sqrt{\epsilon}r_{j})\\}.$

Note that $r_{s_{j}(y_{j})}\leqslant\sqrt{\epsilon}r_{j}$, so $y_{j}\in
W_{j}$.
Now let $s_{j}:=\max\\{s_{j}(x):x\in W_{j}\\}$ with $x_{j}$ the corresponding
point, i.e., $s_{j}(x_{j})=s_{j}$. Since $B(x_{j},4r_{s_{j}})\subset
B(y_{j},4\sqrt{\epsilon}r_{j})$, we know that for any $x\in
B(x_{j},2r_{s_{j}})$,
$\mathcal{V}_{r_{s_{j}+1}}(x)=\mathcal{V}_{\frac{1}{2}r_{s_{j}}}(x)<\ln|\Gamma_{j}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$.
Consider the collection of balls $\\{B(x_{j},r_{j})\\}_{1}^{m}$. Conditions
(1) and (2) are satisfied by construction. If $x\in
B(q,1)\setminus\bigcup_{j}B(x_{j},r_{j})$, then since
$B(x_{j},\frac{3}{2}r_{j})\supseteq B(y_{j},r_{j})$ and
$\\{B(y_{j},r_{j})\\}_{1}^{m}$ cover $B(q,1)$, we have $x\in
A_{r_{j},\frac{3}{2}r_{j}}(x_{j})\subseteq
A_{\frac{1}{2}r_{j},2r_{j}}(y_{j})$, which implies $r_{h}^{\alpha}(x)\geqslant
r_{0}(n,p,v,\Lambda,\delta)$. This completes the proof. ∎

By Lemma 4.7 and Theorem 2.12, the open set
$B(q,R)\setminus\overline{\bigcup_{j}B(x_{j},r_{j})}$, which will be referred
as a body region, has finite diffeomorphism types.
We are now in the position to prove the following bubble tree decomposition
result, which is stronger than Theorem 1.2. The proof is the same as the one
in Cheeger-Naber’s paper [13], since we have established all the tools they
used in constructing the bubble tree.

######  Theorem 4.8.

Given $p>\frac{n}{2}$ and positive constants $v,D,\lambda$ and $\Lambda$, if
$(M^{n},g)$ is a closed Riemannian manifold satisfying (1.2), (1.4), (1.6) and
(1.7), then there exists a decomposition of $M^{n}$

$\displaystyle
M^{n}=\mathcal{B}_{1}^{0}\cup\bigcup_{i_{1}=1}^{N_{1}}\mathcal{N}_{i_{1}}^{1}\cup\bigcup_{i_{1}=1}^{N_{1}}\mathcal{B}_{i_{1}}^{1}\cup\cdots\cup\bigcup_{i_{m}=1}^{N_{m}}\mathcal{N}_{i_{m}}^{m}\cup\bigcup_{i_{m}=1}^{N_{m}}\mathcal{B}_{i_{m}}^{m}$
(4.10)

into open sets with the following properties:

  1. (1)

if $x\in\mathcal{B}_{i}^{l}$, then $r_{h}^{\alpha}(x)\geqslant
r_{0}(n,p,v,\lambda,\Lambda,D)diam(\mathcal{B}_{i}^{l})$;

  2. (2)

each neck $\mathcal{N}_{i}^{l}$ is diffeomorphic to $(0,1)\times
S^{n-1}/\Gamma_{i}^{l}$, for some $\Gamma_{i}^{l}\subset O(n)$ with
$|\Gamma_{i}^{l}|\leqslant C(n,v)$;

  3. (3)

$\mathcal{N}_{i}^{l}\cap\mathcal{B}_{i}^{l}$ is diffeomorphic to $(0,1)\times
S^{n-1}/\Gamma_{i}^{l}$;

  4. (4)

$\mathcal{B}_{j}^{l-1}\cap\mathcal{N}_{i}^{l}$ are either empty or
diffeomorphic to $(0,1)\times S^{n-1}/\Gamma_{i}^{l}$;

  5. (5)

$N_{l}\leqslant N(n,p,v,\lambda,\Lambda,D)$ and $m\leqslant m(n,v)$.

###### Proof.

Without loss of generality, we can assume $\lambda=1$. Fix
$\epsilon=\epsilon(n,v)$ from Theorem 4.5 with
$\delta=\delta(n,p,v,\Lambda,D,\epsilon)=\delta(n,p,v,\Lambda,D)$ sufficiently
small to satisfy Proposition 4.1, Theorem 4.5, Lemma 4.6 and Lemma 4.7. Since
we want to apply Lemma 4.7 on the whole manifold $M^{n}$ instead of $B(q,1)$,
we need $\delta$ to be dependent on $D$. After rescaling, we may suppose that
$(M^{n},g)$ is a closed manifold with $diam(M^{n})\leqslant D\delta^{-1}$,
$\int_{M^{n}}|Rm|^{\frac{n}{2}}\leqslant\Lambda$,
$\int_{M^{n}}|Ric|^{p}\leqslant\delta^{2p-n}$, $vol(B(x,r))\geqslant vr^{n}\
\text{for all}\ x\in M^{n}$ and $r\leqslant\delta^{-1}$.
To begin with, we apply Lemma 4.7 to produce a collection of balls
$\\{B(x_{i}^{1},r_{i}^{1})\\}_{1}^{N_{1}}$ such that
$r_{i}^{1}=r_{k_{i}^{1}}>r_{0}(n,p,v,\Lambda,D)D$, $N_{1}\leqslant
N(n,p,v,\Lambda,D)$, $T_{k_{i}^{1}}^{\delta}(x_{i}^{1})=0$ and if $x\in
M^{n}\setminus\bigcup_{i}B(x_{i}^{1},r_{i}^{1})$, then
$r_{h}^{\alpha}(x)>r_{0}(n,p,v,\Lambda,D)D$. Moreover, if we denote by
$\Gamma_{i}^{1}$ the group associated to $B(x_{i}^{1},r_{1}^{1})$, then if
$s_{i}^{1}$ denotes the largest integer such that
$\mathcal{V}_{r_{s_{i}^{1}}}(x_{i}^{1})\geqslant\ln|\Gamma_{i}^{1}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$,
then for any $x\in B(x_{i}^{1},2r_{s_{i}^{1}})$, we have

$\mathcal{V}_{\frac{1}{2}r_{s_{i}^{1}}}(x)<\ln|\Gamma_{i}^{1}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}.$

Define
$\mathcal{B}_{1}^{0}=M^{n}\setminus\overline{\bigcup_{i}B(x_{i}^{1},r_{i}^{1})}$
as the first body region. Then we can write

$\displaystyle
M^{n}=\mathcal{B}_{1}^{0}\cup\bigcup_{i}B(x_{i}^{1},2r_{i}^{1}),$ (4.11)

and $B(x_{i}^{1},2r_{i}^{1})\cap\mathcal{B}_{1}^{0}$ is diffeomorphic to
$(0,1)\times S^{n-1}/\Gamma_{i}^{1}$ by Proposition 4.1.
Now, let us inductively construct a decomposition of $M^{n}$,

$\displaystyle
M^{n}=\mathcal{B}_{1}^{0}\cup\bigcup_{i_{1}=1}^{N_{1}}\mathcal{N}_{i_{1}}^{1}\cup\bigcup_{i_{1}=1}^{N_{1}}\mathcal{B}_{i_{1}}^{1}\cup\cdots\cup\bigcup_{i_{t}=1}^{N_{t}}\mathcal{N}_{i_{t}}^{t}\cup\bigcup_{i_{t}=1}^{N_{t}}\mathcal{B}_{i_{t}}^{t}\cup\bigcup_{a=1}^{N_{t+1}}B(x_{a},2r_{a}^{t+1})$
(4.12)

with the following properties:

  1. (1)

if $x\in\mathcal{B}_{i}^{l}$, then $r_{h}^{\alpha}(x)\geqslant
r_{0}(n,p,v,\Lambda,D)diam(\mathcal{B}_{i}^{l})$;

  2. (2)

each neck $\mathcal{N}_{i}^{l}$ is diffeomorphic to $(0,1)\times
S^{n-1}/\Gamma_{i}^{l}$, for some $\Gamma_{i}^{l}\subset O(n)$ with
$|\Gamma_{i}^{l}|\leqslant C(n,v)$;

  3. (3)

$\mathcal{N}_{i}^{l}\cap\mathcal{B}_{i}^{l}$ is diffeomorphic to $(0,1)\times
S^{n-1}/\Gamma_{i}^{l}$;

  4. (4)

$\mathcal{B}_{j}^{l-1}\cap\mathcal{N}_{i}^{l}$ are either empty or
diffeomorphic to $(0,1)\times S^{n-1}/\Gamma_{i}^{l}$;

  5. (5)

$N_{l}\leqslant N(n,p,v,\Lambda,D)$;

  6. (6)

if $\mathcal{B}_{j}^{l-1}\cap\mathcal{N}_{i}^{l}\neq\emptyset$, then
$|\Gamma_{i}^{l}|\leqslant|\Gamma_{j}^{l-1}|-1$;

  7. (7)

$r_{a}^{t+1}=r_{k_{a}^{t+1}}$ with $T_{k_{a}^{t+1}}^{\delta}(x_{a})=0$, and
$\mathcal{B}_{i}^{t}\cap B(x_{a},2r_{a}^{t+1})\subseteq
A_{\frac{1}{2}r_{a}^{t+1},2r_{a}^{t+1}}(x_{a})$;

  8. (8)

if $s_{a}^{t+1}$ is the largest integer such that
$\mathcal{V}_{r_{s_{a}^{t+1}}}(x_{a})\geqslant\ln|\Gamma_{a}^{t+1}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$,
then for any $x\in B(x_{a},2r_{s_{a}^{t+1}})$, we have
$\mathcal{V}_{\frac{1}{2}r_{s_{a}^{t+1}}}(x)<\ln|\Gamma_{a}^{t+1}|-\delta^{\frac{1}{2}(1-\frac{n}{2p})}$.

Note that (4.11) provides the case $t=0$. So let us suppose that the
decomposition (4.12) has been built for $t\geqslant 0$, and we are going to
construct for $t+1$.
We first use Theorem 4.5 to see that there exists an open set

$A_{\frac{1}{2}r_{s_{a}^{t+1}},2r_{k_{a}^{t+1}}}(x_{a})\subset\mathcal{N}_{a}^{t+1}\subset
A_{\frac{1}{2}(1-\epsilon)r_{s_{a}^{t+1}},2(1+\epsilon)r_{k_{a}^{t+1}}}(x_{a})$

and a diffeomorphism $\phi_{a}^{t+1}:\mathcal{N}_{a}^{t+1}\to
A_{\frac{1}{2}r_{s_{a}^{t+1}},2r_{k_{a}^{t+1}}}(0)$ where
$0\in\mathbb{R}^{n}/\Gamma_{a}^{t+1}$. By Lemma 4.6 and condition (8), there
exists a radius
$r_{a}=r(n,p,v,\Lambda,\delta)r_{s_{a}^{t+1}}=r(n,p,v,\Lambda,D)r_{s_{a}^{t+1}}<1$
such that for any $x\in B(x_{a},2r_{s_{a}^{t+1}})$,

$\mathcal{V}_{r_{a}}(x)<\ln(|\Gamma_{a}^{t+1}|-1)+\delta^{\frac{1}{2}(1-\frac{n}{2p})}.$

Now we apply Lemma 4.7 to each ball $B(x_{a},r_{s_{a}^{t+1}})$ to construct a
collection of balls $\\{B(x_{aj},r_{aj}^{t+2})\\}$ with
$r_{aj}^{t+2}=r_{k_{aj}^{t+2}}$ and $T_{k_{aj}^{t+2}}^{\delta}(x_{aj})=0$.
There are at most $N(n,p,v,\Lambda,D)$ such balls in total and for any $x\in
B(x_{a},r_{s_{a}^{t+1}})\setminus\bigcup_{j}B(x_{aj},r_{aj}^{t+2})$,
$r_{h}^{\alpha}(x)>r_{0}(n,p,v,\Lambda,D)r_{s_{a}^{t+1}}$. Also, from the
proof of Lemma 4.7, we can assume that
$r_{aj}^{t+2}<r(n,p,v,\Lambda,D)r_{s_{a}^{t+1}}=r_{a}$ and
$|\mathcal{V}_{r_{aj}^{t+2}}(x_{aj})-\ln|\Gamma_{aj}^{t+2}||<\delta^{\frac{1}{2}(1-\frac{n}{2p})}$
where $\Gamma_{aj}^{t+2}$ is the corresponding group of
$B(x_{aj},r_{aj}^{t+2})$.
On the other hand, by the almost monotonicity formula (2.2), we have

$\mathcal{V}_{r_{aj}^{t+2}}(x_{aj})<\mathcal{V}_{r_{a}}(x_{aj})+C(n,p,v)\delta^{1-\frac{n}{2p}}<\ln(|\Gamma_{a}^{t+1}|-1)+2\delta^{\frac{1}{2}(1-\frac{n}{2p})},$

which implies $|\Gamma_{aj}^{t+2}|\leqslant|\Gamma_{a}^{t+1}|-1$. If we put

$\mathcal{B}_{a}^{t+1}=B(x_{a},r_{s_{a}^{t+1}})\setminus\overline{\bigcup_{j}B(x_{aj},r_{aj}^{t+2})},$

and write

$M^{n}=\mathcal{B}_{1}^{0}\cup\cdots\cup\bigcup_{i_{t}=1}^{N_{t}}\mathcal{N}_{i_{t}}^{t}\cup\bigcup_{i_{t}=1}^{N_{t}}\mathcal{B}_{i_{t}}^{t}\cup\bigcup_{a=1}^{N_{t+1}}\mathcal{N}_{a}^{t+1}\cup\bigcup_{a=1}^{N_{t+1}}\mathcal{B}_{a}^{t+1}\cup\bigcup
B(x_{aj},2r_{aj}^{t+2}),$

then one can check the inductive conditions below (4.12) easily by shrinking
neck regions if necessary.
Once we have the inductive decomposition, we could finish the proof
immediately. In fact, by condition (6) and the upper bound
$|\Gamma_{i}^{l}|\leqslant C(n,v)$, we see that the decomposition will stop
within $m$ steps where $m\leqslant m(n,v)$. This completes the proof. ∎

Once we have the bubble tree decomposition, our diffeomorphism finiteness
statement of Theorem 1.2 can be proved directly. Notice that all body regions
have finite diffeomorphism types by Theorem 2.12 and any neck region is
diffeomorphic to $(0,1)\times S^{n-1}/\Gamma$ for some $\Gamma\subseteq O(n)$
with $|\Gamma|<C(n,v)$. See [13] for more details. Hence, we complete the
proof of Theorem 1.2.

## References

  * [1] M. T. Anderson, Ricci curvature bounds and Einstein metrics on compact manifolds, Journal of the American Mathematical Society, 2 (1989), pp. 455–490.
  * [2] , Convergence and rigidity of manifolds under Ricci curvature bounds, Inventiones mathematicae, 102 (1990), pp. 429–445.
  * [3] M. T. Anderson and J. Cheeger, Diffeomorphism finiteness for manifolds with Ricci curvature and ${L}^{n/2}$-norm of curvature bounded., Geometric and functional analysis, 1 (1991), pp. 231–252.
  * [4] S. Bando, Bubbling out of Einstein manifolds, Tohoku Mathematical Journal, Second Series, 42 (1990), pp. 205–216.
  * [5] S. Bando, A. Kasue, and H. Nakajima, On a construction of coordinates at infinity on manifolds with fast curvature decay and maximal volume growth, Inventiones mathematicae, 97 (1989), pp. 313–349.
  * [6] J. Cheeger, Comparison and finiteness theorems for Riemannian manifolds, Princeton University, 1967.
  * [7] , Integral bounds on curvature, elliptic estimates and rectifiability of singular sets, Geometric and Functional Analysis, 13 (2003), pp. 20–72.
  * [8] J. Cheeger and T. H. Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Annals of mathematics, 144 (1996), pp. 189–237.
  * [9] , On the structure of spaces with Ricci curvature bounded below. I, Journal of Differential Geometry, 46 (1997), pp. 406–480.
  * [10] , On the structure of spaces with Ricci curvature bounded below. II, Journal of Differential Geometry, 54 (2000), pp. 13–35.
  * [11] J. Cheeger, T. H. Colding, and G. Tian, On the singularities of spaces with bounded Ricci curvature, Geometric and Functional Analysis, 12 (2002), pp. 873–914.
  * [12] J. Cheeger and A. Naber, Lower bounds on Ricci curvature and quantitative behavior of singular sets, Inventiones mathematicae, 191 (2013), pp. 321–339.
  * [13] , Regularity of Einstein manifolds and the codimension 4 conjecture, Annals of Mathematics, (2015), pp. 1093–1165.
  * [14] L. Chen, Segment inequality and almost rigidity structures for integral Ricci curvature, International Mathematics Research Notices, 2022 (2022), pp. 11732–11760.
  * [15] T. H. Colding, Ricci curvature and volume convergence, Annals of mathematics, 145 (1997), pp. 477–501.
  * [16] X. Dai, G. Wei, and Z. Zhang, Local sobolev constant estimate for integral Ricci curvature bounds, Advances in Mathematics, 325 (2018), pp. 1–33.
  * [17] K. Fukaya, Collapsing of Riemannian manifolds and eigenvalues of Laplace operator, Inventiones mathematicae, 87 (1987), pp. 517–547.
  * [18] N. Gigli, A. Mondino, and G. Savaré, Convergence of pointed non-compact metric measure spaces and stability of Ricci curvature bounds and heat flows, Proceedings of the London Mathematical Society, 111 (2015), pp. 1071–1129.
  * [19] P. Petersen, Convergence theorems in Riemannian geometry, Comparison geometry (Berkeley, CA, 1993–94), 30 (1997), pp. 167–202.
  * [20] , Riemannian geometry, vol. 171, Springer, 2006.
  * [21] P. Petersen and G. Wei, Relative volume comparison with integral curvature bounds, Geometric & Functional Analysis GAFA, 7 (1997), pp. 1031–1045.
  * [22] , Analysis and geometry on manifolds with integral Ricci curvature bounds. II, Transactions of the American Mathematical Society, 353 (2001), pp. 457–478.
  * [23] G. Tian and Z. Zhang, Regularity of Kähler–Ricci flows on Fano manifolds, Acta Math, 216 (2016), pp. 127–176.
  * [24] D. Yang, Convergence of riemannian manifolds with integral bounds on curvature. I, in Annales scientifiques de l’Ecole normale supérieure, vol. 25, 1992, pp. 77–105.
  * [25] , Convergence of riemannian manifolds with integral bounds on curvature. II, in Annales scientifiques de l’École normale supérieure, vol. 25, 1992, pp. 179–199.

Xin Qian<EMAIL_ADDRESS>
_School of Mathematical Sciences, Fudan University, Shanghai 200433, People’s
Republic of China_