Another proof of the local curvature estimate for the Ricci flow

TL;DR

This paper presents a new, simplified proof of a recent result on local curvature estimates in Ricci flow, utilizing the De Giorgi iteration method to establish bounds based on initial conditions and Ricci curvature constraints.

Contribution

It introduces a novel, streamlined proof technique for existing local curvature estimates in Ricci flow using De Giorgi iteration.

Findings

Established local boundedness of Riemannian curvature tensor

Provided a simpler proof of existing curvature estimates

Linked curvature bounds to initial data and Ricci curvature bounds

Abstract

By using the De Giorgi iteration method we will give a new simple proof of the recent result of B.Kotschwar, O.Munteanu, J.Wang [KMW] and N.Sesum [S] on the local boundedness of the Riemmanian curvature tensor of solutions of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci curvature.

Equations

$$\frac{\partial g_{ij}}{\partial t}=-2R_{ij}$$

$$\underset{t\nearrow T}{\lim }|Rm{|}_{g(t)}=\infty .$$

$$\frac{\partial }{\partial t}|Rm{|}^2\le \Delta |Rm{|}^2-2|\nabla Rm{|}^2+C|Rm{|}^3$$

$$\frac{\partial }{\partial t}|Rm|\le \Delta |Rm|+C|Rm{|}^2\text{mbox}in(0,T).$$

$$|Ric|\le K\text{mbox}in{B}_{g(0)}\left(x_0,\frac{2\rho }{\sqrt{K}}\right)\times [0,T)$$

$${\Lambda }_0:=\underset{B_{g(0)}\left(x_0,\frac{2\rho }{\sqrt{K}}\right)}{\sup }|Rm|(x,0)<\infty .$$

$$|Rm|(x,t)$$

$$\le$$

$$E_p(t)=C{e}^{CKt}t\left[{\Lambda }_0^{2p}{V}_{x_0}\left(2\rho /\sqrt{K}\right)+K^{2p}(1+{\rho }^{-4p})V_{x_0}\left(\rho /\sqrt{K}\right)\right]$$

$$|Rm|(x,t)$$

$$\le$$

$$\cdot (1+(4\rho /\sqrt{K})\sqrt{t{V}_{x_0}\left(2\rho /\sqrt{K}\right)}{E}_p(t{)}^{\frac{1}{2}}{)}^{\frac{7}{2p}}$$

$${\int }_{B_{g(0)}\left(x_0,\frac{\rho }{\sqrt{K}}\right)}|Rm|(x,t{)}^{q}d{v}_t\le c{e}^{cKt}\left\{{\int }_{B_{g(0)}\left(x_0,\frac{2\rho }{\sqrt{K}}\right)}|Rm|(x,0{)}^{q}d{v}_0+c{K}^q(1+{\rho }^{-2q})V_{x_0}\left(\rho /\sqrt{K},t\right)\right\}$$

$$\{\begin{array}{rl}& |\nabla Ric{|}^2\le \frac{1}{2}(\Delta -{\partial }_t)|Ric{|}^2+C{K}^2|Rm|\\ & {\partial }_t{R}_{ijk}^l=g^{lq}({\nabla }_i{\nabla }_q{R}_{jk}+{\nabla }_j{\nabla }_i{R}_{kq}+{\nabla }_j{\nabla }_k{R}_{iq})-g^{lq}({\nabla }_i{\nabla }_j{R}_{kq}+{\nabla }_i{\nabla }_k{R}_{jq}+{\nabla }_j{\nabla }_q{R}_{ik}),\end{array}$$

$$\frac{d}{dt}\left({\int }_M|Rm{|}^p{\varphi }^{2p}d{v}_t+\frac{1}{K}{\int }_M|Ric{|}^2|Rm{|}^{p-1}{\varphi }^{2p}d{v}_t+c_{1}K{\int }_M|Rm{|}^{p-1}{\varphi }^{2p}d{v}_t\right)$$

$$\le$$

$$R_{ij}\ge -(n-1)k_1\text{mbox}onB(x_0,\rho )$$

$${\left({\int }_{B(x_0,\rho )}|f{|}^{\frac{2n}{n-2}}\right)}^{\frac{n-2}{n}}dv\le c_1\frac{{\rho }^2{e}^{c_2\rho \sqrt{k_1}}}{V_{x_0}(\rho {)}^{2/n}}{\int }_{B(x_0,\rho )}|\nabla f{|}^{2}dv$$

$$|Rm|(x,t)\le$$

$$\cdot {\left(1+{\iint }_{Q_0}|Rm{|}^{p}d{v}_{0}dt\right)}^{\frac{n+4}{2p}}$$

$$|Rm|(x,t)\le$$

$$\cdot {\left(1+\frac{4\rho }{\sqrt{K}}{\iint }_{Q_0}|Rm{|}^{p}d{v}_{0}dt\right)}^{\frac{7}{2p}}$$

$$e^{-2Kt}{g}_{ij}(x,0)\le g(x,s)\le e^{2Kt}{g}_{ij}(x,0)\forall x\in B_{g(0)}\left(x_0,\frac{2\rho }{\sqrt{K}}\right),0\le s<T$$

$$\Rightarrow$$

$$B_{2\rho /\sqrt{K}}\times (t/4,t)=Q_0\supseteq Q_1\supseteq \cdots \supseteq Q_{m-1}\supseteq Q_m\supseteq \cdots \supseteq Q_{\infty }=B_{\rho /\sqrt{K}}\times (t/2,t).$$

$$\{\begin{array}{rl}& |\nabla {\varphi }_m|\le \frac{C\sqrt{K}{2}^m}{\rho }\text{mbox}in{Q}_m\\ & 0\le {\varphi }_{m,t}\le \frac{C{2}^m}{t}\text{mbox}in{Q}_m.\end{array}$$

$$\frac{1}{p}{\iint }_{Q_m^s}{\varphi }_m^2\frac{\partial }{\partial t}(v-k_{m+1}{)}_{+}^{p}d{v}_{t}dt$$

$$+{\iint }_{Q_m^s}\nabla (v-k_{m+1}{)}_{+}\cdot [(p-1)(v-k_{m+1}{)}_{+}^{p-2}{\varphi }_m^2\nabla (v-k_{m+1}{)}_{+}+2(v-k_{m+1}{)}_{+}^{p-1}{\varphi }_m\nabla {\varphi }_m]d{v}_{t}dt$$

$$\le$$

$${\iint }_{Q_m^s}{\varphi }_m^2\frac{\partial }{\partial t}(v-k_{m+1}{)}_{+}^{p}d{v}_{t}dt$$

$$=$$

$$+{\iint }_{Q_m^s}(v-k_{m+1}{)}_{+}^p{\varphi }_m^{2}Rd{v}_{t}dt$$

$$\ge$$

$$-CK{\iint }_{Q_m^s}(v-k_{m+1}{)}_{+}^p{\varphi }_m^{2}d{v}_{t}dt.$$

$${\int }_{B_{{\rho }_m}}|\nabla ({\varphi }_m(v-k_{m+1}{)}_{+}^{\frac{p}{2}}){|}^{2}d{v}_t\le \frac{11}{10}{\int }_{B_{{\rho }_m}}{\varphi }_m^2|\nabla (v-k_{m+1}{)}_{+}^{\frac{p}{2}}{|}^{2}d{v}_t+11{\int }_{B_{{\rho }_m}}(v-k_{m+1}{)}_{+}^p|\nabla {\varphi }_m{|}^{2}d{v}_t$$

$$2{\int }_{B_{{\rho }_m}}{\varphi }_m(v-k_{m+1}{)}_{+}^{p-1}\nabla {\varphi }_m\cdot \nabla (v-k_{m+1}{)}_{+}d{v}_t$$

Full text

Key words: Ricci flow, local boundedness, Riemmanian curvature, Ricci curvature

AMS 2010 Mathematics Subject Classification: Primary 58J35, 35B45 Secondary 35K10

1 Introduction

There is a lot of interest on Ricci flow ([CK], [CLN], [H2], [MF], [MT]) because it is a very powerful tool in the study of the geometry of manifolds. Recently G. Perelman [P1], [P2], by using the the Ricci flow technique solved the famous Poincare conjecture in geometry. Let $(M,g(t))$, $0<t<T$, be a $n$-dimensional Riemannian manifold. We say that the metric $g(t)=(g_{ij}(t))$ evolves by the Ricci flow if it satisfies

[TABLE]

on $M\times(0,T)$ where $R_{ij}$ is the Ricci curvature of the metric $g(t)=(g_{ij}(t))$. Short time existence of solution of Ricci flow on compact Riemannian manifolds with any initial metric at $t=0$ was proved by R. Hamilton in [H1]. Short time existence of solution of Ricci flow on complete non-compact manifolds with bounded curvature initial metric at time $t=0$ was proved by W.X. Shi in [Sh1], [Sh2]. When $M$ is a compact manifold, R. Hamilton [H1] proved that either the Ricci flow solution exists globally or there exists a maximal existence time $0<T<\infty$ for the solution of Ricci flow and

[TABLE]

Hence in order to know whether the solution of Ricci flow can be extended beyond its interval of existence $(0,T)$, it is important to prove boundedness of the Riemannian curvature for the solution of Ricci flow near the time $T$. Uniform boundedness of the Riemannian curvature of the solution of Ricci flow on a compact manifold when the solution has uniform bounded Ricci curvature on $(0,T)$ was proved by N. Sesum in [S] using a blow-up contradiction argument and G. Perelman’s noncollapsing result [P1]. Local boundedness of the Riemannian curvature for $\kappa$-noncollapsing solutions of Ricci flow in term of its local $L^{\frac{n}{2}}$ norm when its local $L^{\frac{n}{2}}$ norm is sufficiently small was also proved by R. Ye in [Y1], [Y2], using Moser iteration technique and the point picking technique of G. Perelman [P1]. Similar result was also obtained by X Dai, G Wei and R Ye in [DWY].

Local boundedness of the Riemannian curvature of the solution of Ricci flow in terms of its inital value on a given ball and a local uniform bound on the Ricci curvature was proved by B. Kotschwar, O. Munteanu, J. Wang using Moser iteration technique and results of P. Li [L] in [KMW]. A similar local Riemannian curvature result was proved recently by C.W. Chen [C] using the point picking technique of Perelman [P1], M.T. Anderson’s harmonic coordinates [A] and elliptic regularity results [GT]. In this paper we will use the De Giorgi iteration method to give a new simple proof of this result.

We will assume that $(M,g(t))$ is a smooth solution of the Ricci flow (1.1) in $[0,T)$ for the rest of the paper. For any $x_{0}\in M$, $\rho>0$ and $0\leq t<T$, we let $B_{g(t)}(x_{0},\rho)=\{x\in M:\mbox{dist}_{g(t)}\,(x,x_{0})<\rho\}$, $V_{x_{0}}(\rho,t)=\mbox{vol}_{g(t)}\,(B_{g(0)}(x_{0},\rho))$, $V_{x_{0}}(\rho)=V_{x_{0}}(\rho,0)$, $|Ric|(x,t)=|Ric(x,t)|_{g(t)}$ and $|Rm|(x,t)=|Rm(x,t)|_{g(t)}$. We let $dv_{t}$ be the volume element of the metric $g(t)$ and let $C>0$ denote a generic constant that may change from line to line. For any complete Riemannian manifold $(M,g)$, we let $B(x_{0},\rho)=\{x\in M:\mbox{dist}_{g}\,(x,x_{0})<\rho\}$, $V_{x_{0}}(\rho)=\mbox{vol}_{g}\,(B(x_{0},\rho))$ and $dv$ be the volume element of the metric $g$.

Note that by Corollary 13.3 of [H1] or Lemma 7.4 of [CK],

[TABLE]

in $(0,T)$ for some constant $C>0$ depending only on $n$. Since $|\nabla|Rm||\leq|\nabla Rm|$, by (1.2),

[TABLE]

We will prove the following main result in this paper.

Theorem 1.1**.**

(cf. Theorem 1 of [KMW]) Let $g(t)$, $0\leq t<T$, be a smooth solution of Ricci flow on a $n$-dimensional Riemannian manifold $M$. Suppose there exists $x_{0}\in M$ and constants $K>0$, $\rho>0$, such that

[TABLE]

and

[TABLE]

Then for any $n\geq 3$ and $p>\frac{n+2}{2}$ there exist constants $C_{0}>0$ and $C>0$ such that

[TABLE]

holds for any $x\in B_{g(0)}\left(x_{0},\rho/\sqrt{K}\right)$ and $0<t<T$ where

[TABLE]

and for $n=2$ and any $p>\frac{5}{2}$ there exist constants $C_{0}>0$ and $C>0$ such that

[TABLE]

holds for any $x\in B_{g(0)}\left(x_{0},\rho/\sqrt{K}\right)$ and $0<t<T$.

Remark 1.2**.**

Note that the bounds for the Riemannian curvature in (1.1) and (1.1) are slightly different from that of Theorem 1 of [KMW]. When $t\to\infty$, both the right hand side of (1.1), (1.1), and the bound in Theorem 1 of [KMW] are approximately equal to $e^{CKt}$ for some constant $C>0$. However, for $0<t<\rho^{2}/K$ and $t$ close to zero, the right hand side of (1.1) and (1.1) are approximately equal to $Ct^{-\frac{n+2}{2p}}$ and $Ct^{-\frac{5}{2p}}$ respectively for some constant $C>0$, while the bound in Theorem 1 of [KMW] is approximately equal to $Ct^{-\beta}$ for some constant $\beta>0$. Since the constant $\beta$ in Theorem 1 of [KMW] is unknown, Theorem 1.1 is therefore a refinement of the result in Theorem 1 of [KMW].

2 The main result

We first recall a result of [KMW]:

Proposition 2.1**.**

(Proposition 1 of [KMW]) Let $g(t)$, $0\leq t<T$, be a smooth solution of Ricci flow on a $n$-dimensional Riemannian manifold $M$. Suppose there exists $x_{0}\in M$ and constants $K>0$, $\rho>0$, such that (1.4) holds. Then for any $n\geq 2$ and $q\geq 3$ there exists a constant $c=c(n,q)>0$ such that

[TABLE]

holds for any $0\leq t<T$.

Proof: A proof of this result is given in [KMW]. For the sake of completeness we will give a sketch of the proof of this result in this paper. By using (1.2), the inequalities (Chapter 6 of [CK] or Lemma 1 of [KMW]),

[TABLE]

and a direct computation one can show that there exist constants $c_{1}>0$ and $c_{2}>0$ such that

[TABLE]

holds on $M\times(0,T)$ for any Lipschitz function $\phi$ with support in $B\left(x_{0},\frac{2\rho}{\sqrt{K}}\right)$. Proposition 2.1 then follows by choosing an appropriate cut-off function $\phi$ for the set $B\left(x_{0},\frac{\rho}{\sqrt{K}}\right)$ and integrating the above differential inequality over $(0,t)$, $0<t<T$. $\square$

Lemma 2.2**.**

(cf. Theorem 14.3 of [L]) Let $(M,g)$ be a complete Riemannian manifold of dimension $n\geq 3$ with Ricci curvature satisfying

[TABLE]

for some constant $k_{1}\geq 0$. Then there exists constants $c_{1}>0$ and $c_{2}>0$ depending only on $n$ such that for any function $f\in H_{c}^{1,2}(B(x_{0},\rho))$ with compact support in $B(x_{0},\rho)$, $f$ satisfies

[TABLE]

Theorem 2.3**.**

Let $g(t)$, $0\leq t<T$, be a smooth solution of Ricci flow on a $n$-dimensional Riemannian manifold $M$. Suppose there exists $x_{0}\in M$ and constants $K>0$, $\rho>0$, such that (1.4) holds. Then for any $n\geq 3$ and $p>\frac{n+2}{2}$ there exist constants $C_{0}>0$ and $C>0$ such that

[TABLE]

holds for any $x\in B_{g(0)}\left(x_{0},\rho/\sqrt{K}\right)$ and $0<t<T$ where $Q_{0}=B_{g(0)}\left(x_{0},2\rho/\sqrt{K}\right)\times(t/4,t)$ and for $n=2$ and any $p>\frac{5}{2}$ there exist constants $C_{0}>0$ and $C>0$ such that

[TABLE]

holds for any $x\in B_{g(0)}\left(x_{0},\rho/\sqrt{K}\right)$ and $0<t<T$.

Proof: Case 1: $n\geq 3$.

Let $v=|Rm|$, $0<t<T$ and $p>\frac{n+2}{2}$. We will use a modification of the proof of Proposition 2.1 of [DDD] to prove this theorem. By (1.4),

[TABLE]

Let $\rho_{m}=\left(\rho/\sqrt{K}\right)(1+2^{-m})$ and $t_{m}=(1-2^{-m-1})t/2$ for any $m\geq 0$. Then $\rho_{0}=2\rho/\sqrt{K}$ and $t_{0}=t/4$. Moreover $\rho_{m}$ decreases to $\rho/\sqrt{K}$ and $t_{m}$ increases to $t/2$ as $m\to\infty$. Let $B_{\rho_{m}}=B_{g(0)}(x_{0},\rho_{m})$, $Q_{m}=B_{\rho_{m}}\times(t_{m},t)$ and $Q_{m}^{s}=B_{\rho_{m}}\times(t_{m},s)$ for any $t_{m}\leq s\leq t$. Then

[TABLE]

We choose a sequence of Lipschitz continuous functions $\{\phi_{m}\}$ on $M\times(0,t)$ such that $0\leq\phi_{m}\leq 1$ on $M\times(0,t)$, $\phi_{m}(x,s)=1$ for $(x,s)\in Q_{m+1}$, $\phi_{m}(x,s)=0$ for $(x,s)\in M\times(0,t)\setminus Q_{m}$, and satisfying

[TABLE]

Let $k>0$ be a constant to be determined later and $k_{m}=k(1-2^{-m})$ for any $m\geq 0$. Multiplying (1.3) by $(v-k_{m+1})_{+}^{p-1}\phi_{m}^{2}$ and integrating over $Q_{m}^{s}$, $t_{m}\leq s\leq t$,

[TABLE]

Since $\frac{d}{dt}(dv_{t})=-Rdv_{t}$, by (1.4),

[TABLE]

Since

[TABLE]

and

[TABLE]

[TABLE]

By (2), (2) and (2),

[TABLE]

By (2) and (2),

[TABLE]

where

[TABLE]

and

[TABLE]

By Lemma 2.2,

[TABLE]

By the Holder inequality,

[TABLE]

where $E_{m}=\{(x,s)\in Q_{m}:v(x,s)>k_{m+1}\}$. By the Holder inequality and (2) (cf. proof of proposition 3.1 of chapter 1 of [D]),

[TABLE]

By (2), (2) and (2),

[TABLE]

Now (cf. proof on P.645 of [DDD]),

[TABLE]

Hemce by (2.14) and (2),

[TABLE]

for some constant $C_{1}>0$ where $b=4\cdot 2^{\frac{2p}{n+2}}$ and $\alpha=\frac{2}{n+2}-\frac{1}{p}$. Then $0<\alpha<\frac{2}{n+2}$. We now let $\beta>1/\alpha$ and

[TABLE]

We claim that

[TABLE]

In order to prove this claim we observe first that by (2) and (2.17),

[TABLE]

By (2), (2.17) and (2.19),

[TABLE]

Repeating the above argument we get,

[TABLE]

and (2.18) follows. Letting $m\to\infty$ in (2.18),

[TABLE]

Hence $v\leq k$ in $Q_{\infty}$ with $k$ given by (2.17) and $A_{1}$ given by (2.10). Thus (2.3) follows.

Case 2: $n=2$.

Let $\widetilde{M}=M\times{\mathbb{R}}$, $\widetilde{g}(x,t)=g(x,t)+dx^{2}$, and let $\widetilde{Rm}$, $\widetilde{Ric}$, $\widetilde{R}$, be the Riemannian curvature, Ricci curvature and scalar curvature of $(\widetilde{M},\widetilde{g}(t))$. Then

[TABLE]

for all $x\in M$, $y\in{\mathbb{R}}$, $0\leq t<T$ and

[TABLE]

Let $\widetilde{V}_{x_{0}}(r)=\mbox{vol}_{\widetilde{g}(0)}\,(B_{\widetilde{g}(0)}((x_{0},0),r))$ for any $r>0$ and $d\widetilde{v}_{0}=dv_{0}\,dy$ be the volume element of $\widetilde{g}(0)$. By case 1,

[TABLE]

holds for any $(x,y)\in B_{\widetilde{g}(0)}\left((x_{0},0),\rho/\sqrt{K}\right)$ and $0<t<T$ where $\widetilde{Q}_{0}=B_{\widetilde{g}(0)}\left((x_{0},0),2\rho/\sqrt{K}\right)\times(t/4,t)$. Since $B_{\widetilde{g}(0)}\left((x_{0},0),\rho/\sqrt{K}\right)\supset B_{g(0)}\left(x_{0},\rho/\sqrt{2K}\right)\times\left(-\rho/\sqrt{2K},\rho/\sqrt{2K}\right)$,

[TABLE]

Since $B_{\widetilde{g}(0)}\left((x_{0},0),2\rho/\sqrt{K}\right)\subset B_{g(0)}\left(x_{0},2\rho/\sqrt{K}\right)\times\left(-2\rho/\sqrt{K},2\rho/\sqrt{K}\right)$, by (2.21), (2) and (2.23), we get (2.3) and the theorem follows. $\square$

Remark 2.4**.**

By Proposition 2.1, Theorem 2.3, Holder’s inequality and (2), Theorem 1.1 follows.