﻿ Ricci孤立子的曲率及势函数

# Ricci孤立子的曲率及势函数Curvature and Potential Function of Ricci Solitons

Abstract: In the last decades, there has been an increasing interest in the study of Riemannian manifolds endowed with metrics satisfying special structural equation. One of the most important examples is represented by Ricci flow and Ricci solitons that has become the subject of rapidly increasing investigation since the appearance of the seminal works. It plays a key role in Hamilton and Pe-relman’s proof of the Poincaré conjecture, and has been widely used to study the topology, geometry and complex structure of manifolds. The Ricci flow equation is of own interest as a geometric partial differential equation. It gives a canonical way of a critical metric. There are two important aspects of Ricci solitons. One looks at the influence on the topology by the Ricci soliton structure of the Riemannian manifold, and the other looks at its influence in its geometry. In this paper, we are interested in summarizing some new results about the curvature and potential function estimates of Ricci solitons.

1. 引言

20世纪80年代，Hamilton [1] 提出了Ricci流的概念，实际上Ricci流最初的引进是为了解决3维流形著名的Poincaré猜想。Ricci孤立子是Ricci流的自相似解 [2] 且经常出现在Ricci流方程的奇异点经伸缩变换后的极限中 [3] [4] [5] [6] 。一方面，Ricci孤立子的研究有助于更好的理解Ricci流的奇异结构，从而结合几何手术的方法可以得到一些重要的几何和拓扑结构。另一方面，Ricci孤立子是爱因斯坦度量的自然推广，也被称为quasi-Einetein度量，在规范场论与超弦理论中有重要的应用，因此Ricci孤立子的几何性质及几何不变量对于数学及物理发展均具有重要的研究意义。

1.1. Ricci孤立子介绍

${R}_{ij}+\frac{1}{2}\left({\nabla }_{i}{V}_{j}+{\nabla }_{j}{V}_{i}\right)=\rho {g}_{ij},$

$\rho$ 为常数。此外，若 $V$ 为一个梯度向量场，对光滑函数 $f$ 满足

${R}_{ij}+{\nabla }_{i}{\nabla }_{j}f=\rho {g}_{ij},$

1.2. 典型的Ricci孤立子

1) $\left({R}^{n},{g}_{0},\frac{{|x|}^{2}}{4}\right)$ 为带有势函数 $f=\frac{{|x|}^{2}}{4}$ 的梯度收缩Ricci孤立子， $Ric+{\nabla }^{2}f=\frac{1}{2}{g}_{0}$

2) $\left({R}^{n},{g}_{0},-\frac{{|x|}^{2}}{4}\right)$ 为势函数是 $f=-\frac{{|x|}^{2}}{4}$ 的梯度扩张Ricci孤立子， $Ric+{\nabla }^{2}f=-\frac{1}{2}{g}_{0}$

2. 数量曲率，Ricci曲率及曲率算子的估计

$R\ge -\frac{n}{2\left(t-\alpha \right)}.$

1) $\left({M}^{n},{g}_{ij}\right)$ 有常数量曲率当且仅当 $2{|Ric|}^{2}\le R+c\frac{{|\nabla R|}^{2}}{R+1}$，常数 $c\ge 0$

2) $\left({M}^{n},{g}_{ij}\right)$ 等距于 ${R}^{n}$ 当且仅当 $2{|Ric|}^{2}\le \left(1-\epsilon \right)R+c\frac{{|\nabla R|}^{2}}{R+1}$，常数 $c\ge 0$$\epsilon >0$

1) 若梯度孤立子为稳定或收缩的，则 $R\ge 0$

2) 若该梯度孤立子为扩张的，则存在仅取决于维数的正常数 $C\left(n\right)$ 使得 $R\ge -C\left(n\right)\rho$

$R\ge \frac{1}{\sqrt{\frac{n}{2}}+2}{e}^{f}.$

${|\nabla \mathrm{ln}R|}^{2}\le C\mathrm{ln}\left(f+2\right).$

1) 若 $\left({M}^{n},{g}_{ij},f\right)$ 为稳定或收缩孤立子，则 $R\ge 0$

2) 若 $\left({M}^{n},{g}_{ij},f\right)$ 为扩张Ricci孤立子，则 $R\ge -\frac{n\rho }{2}$。此外，若其数量曲率在某点处达到最小值 $-\frac{n\rho }{2}$，则 $\left({M}^{n},{g}_{ij}\right)$ 为爱因斯坦流形。

$R\left(x\right)\le C\mathrm{exp}\left(-a\left(r\left(x\right)+1\right)\right),$

$R\left(x\right)\le \frac{1}{4}{\left(r\left(x\right)+2\sqrt{f\left({x}_{0}\right)}\right)}^{2}.$

$R\left(x\right)\ge k\mathrm{sec}{\text{h}}^{2}\frac{r\left(x\right)}{2}$

$R\left(x\right)\ge k\mathrm{sec}{\text{h}}^{2}r\left(x\right)$

Ricci曲率条件是研究Ricci孤立子分类的最有力工具之一，通过对Ricci曲率条件的控制可以得到诸多关于孤立子的分类，因此对研究流形的几何性质及拓扑性质都至关重要。另一方面Ricci曲率与黎曼曲率及数量曲率紧密相关，研究Ricci曲率对于研究整体微分流形也有极其重要的意义。

$0\le Rm\le C.$

Munteanu-Wang [20] 证明了任意具有有界Ricci曲率的梯度收缩Ricci孤立子其黎曼曲率张量增长至多为距离函数的多项式形式即对常数 $a>0$ 满足：

$|Rm|\left(x\right)\le C{\left(r\left(x\right)+1\right)}^{a},$

$|Rm|\le cS.$

$|Rm|\le c\left(\frac{|\nabla Ric|}{\sqrt{f}}+\frac{{|Ric|}^{2}+1}{f}+|Ric|\right).$

$|Rm|\le c\left(\frac{|\nabla Ric|}{|\nabla f|}+\frac{{|Ric|}^{2}}{{|\nabla f|}^{2}}+|Ric|\right).$

$|Rm|\le C\left(|\nabla Ric|+{|Ric|}^{2}+|Ric|\right).$

Munteeanu-Wang [15] 又证明了具有有界数量曲率的4维梯度收缩Ricci孤立子的曲率算子也有上界。

$\underset{M}{\mathrm{sup}}\left(|Rm|+|\nabla Rm|\right)\le C.$

$\underset{M}{\mathrm{sup}}\frac{{|\nabla Rm|}^{2}}{S}\le C.$

$\underset{x\in M}{\mathrm{sup}}|Rm|\le C.$

${|Ric|}^{2}\le C{R}^{a},$

$\underset{x\in M}{\mathrm{sup}}|Rm|\le C.$

3. 势函数估计

Perelman [22] 证明了典型的梯度收缩Ricci孤立子的势函数的上下界估计结果：

${R}_{ij}+{\nabla }_{i}{\nabla }_{j}f=\frac{1}{2}{g}_{ij}$$R+{|\nabla f|}^{2}-f=0$。令 $r\left(x\right)=d\left({x}_{0},x\right)$ 表示到定点 ${x}_{0}\in M$ 的距离函数，则存在正常数 ${C}_{1},{C}_{2},{c}_{1}$${c}_{2}$ 使得当 $r\left(x\right)$ 充分大时势函数满足：

$|\nabla f|\left(x\right)\le {C}_{2}\left(r\left(x\right)+1\right),$

$\frac{1}{4}{\left(r\left(x\right)-{c}_{1}\right)}^{2}\le f\left(x\right)\le {C}_{1}{\left(r\left(x\right)+{c}_{2}\right)}^{2}.$

$|\nabla f|\left(x\right)\le Cr\left(x\right),$

$f\left(x\right)\le C{r}^{2}\left(x\right).$

$|\nabla f|\left(x\right)\le \frac{1}{2}r\left(x\right)+\sqrt{f\left({x}_{0}\right)},$

$R\left(x\right)\le \frac{1}{4}{\left(r\left(x\right)+2\sqrt{f\left({x}_{0}\right)}\right)}^{2},$

$f\left(x\right)\le \frac{1}{4}{\left(r\left(x\right)+2\sqrt{f\left({x}_{0}\right)}\right)}^{2}.$

$\frac{1}{4}{\left(r\left(x\right)-{c}_{1}\right)}^{2}\le f\left(x\right)\le \frac{1}{4}{\left(r\left(x\right)+{c}_{2}\right)}^{2}.$

$|\nabla f|\le c\left(1+r\left(x\right)\right),$

$f\left(x\right)\le c\left(1+r{\left(x\right)}^{2}\right).$

$\frac{1}{4}{\left(r\left(x\right)-{c}_{1}\right)}^{2}-{c}_{2}\le -f\left(x\right)\le \frac{1}{4}{\left(r\left(x\right)+2\sqrt{-f\left({x}_{0}\right)}\right)}^{2}.$

$|Ric|\le {c}_{1}\cdot dist{\left(p,x\right)}^{-\epsilon },$

$-r\left(1+\frac{{c}_{2}}{{r}^{\epsilon }}\right)\le {f}^{\prime }\left(x\right)\le -r\left(1-\frac{{c}_{2}}{{r}^{\epsilon }}\right),$

$-\frac{1}{2}{r}^{2}\left(1+\frac{{c}_{3}}{{r}^{\epsilon }}\right)+f\left(p\right)\le f\left(x\right)\le -\frac{1}{2}{r}^{2}\left(1-\frac{{c}_{3}}{{r}^{\epsilon }}\right)+f\left(p\right).$

$\left({M}^{n},{g}_{ij},f\right)$ 的Ricci曲率非负，对梯度稳定孤立子有：

${c}_{1}r\left(x\right)-{c}_{2}\le f\left(x\right)\le \sqrt{{c}_{0}}r\left(x\right)+|f\left({x}_{0}\right)|.$

$\sqrt{\lambda }-\frac{c}{\sqrt{r}}\le \frac{1}{r}\underset{\partial {B}_{p}\left(r\right)}{\mathrm{sup}}f\left(x\right)\le \sqrt{\lambda }+\frac{c}{r}.$

NOTES

*通讯作者。

[1] Hamilton, R.S. (1998) The Ricci Flow on Surfaces. Contemporary Mathematics, 71, 237-261.
https://doi.org/10.1090/conm/071/954419

[2] Hamilton, R.S. (1982) Three Manifolds with Positive Ricci Curvature. Journal of Differential Geometry, 17, 255-306.
https://doi.org/10.4310/jdg/1214436922

[3] Hamilton, R.S. (1993) Eternal Solutions to the Ricci Flow. Journal of Differential Geometry, 38, 1-11.
https://doi.org/10.4310/jdg/1214454093

[4] Chen, B.L. and Zhu, X.P. (2000) Complete Riemannian Manifolds with Point-Wise Pinched Curvature. Inventiones Mathematicae, 140, 423-452.
https://doi.org/10.1007/s002220000061

[5] Cao, H.D. (1996) Existence of Gradient Kähler Ricci Solitons. Elliptic and Parabolic Methods in Geometry (Minneapolis, MM, 1994), A K Peters, Wellesley, MA, 1-16.

[6] Sesum, N. (2004) Limiting Behaviour of the Ricci Flow. arXiv:0402194

[7] Cao, H.D. (2009) Recent Progress on Ricci Solitons. Mathematics, 1-38.

[8] Chu, S.C. (2003) Geometry of 3-Dimensional Gradient Ricci Solitons with Positive Curvature. Communications in Analysis and Geometry, 13, 129-150.
https://doi.org/10.4310/CAG.2005.v13.n1.a4

[9] Chen, B.L. (2009) Strong Uniqueness of the Ricci Flow. Journal of Differ-ential Geometry, 82, 363-382.
https://doi.org/10.4310/jdg/1246888488

[10] Zhang, S.J. (2011) On a Sharp Volume Estimate for Gradient Ricci Solitons with Scalar Curvature Bounded Below. Mathematics, 27, 871-882.

[11] Naber, A. (2006) Some Geometry and Analysis on Ricci Solitons. Mathematics, arXiv:0612532

[12] Petersen, P. and Wylie, W. (2012) Rigidity of Gradient Ricci Solitons. Pacific Journal of Mathe-matics, 241, 329-345.
https://doi.org/10.2140/pjm.2009.241.329

[13] Löpez, M.F. and Río, E.G. (2011) Maximum Principles and Gradient Ricci Solitons. Journal of Differential Geometry, 251, 73-81.
https://doi.org/10.1016/j.jde.2011.03.020

[14] Chow, B., Lu, P. and Yang, B. (2011) Lower Bounds for the Scalar Curvature of Noncompact Steady Gradient Ricci Solitons. Mathematics, 349, 1265-1267.

[15] Munteanu, O. and Wang, J. (2015) Geometry of Shrinking Ricci Solitons. Computational Mathematics, 151, 2273-2300.
https://doi.org/10.1112/S0010437X15007496

[16] Ni, L. (2005) Ancient Solutions to Kähler-Ricci Flow. Mathe-matical Research Letters, 12, 633-654.
https://doi.org/10.4310/MRL.2005.v12.n5.a3

[17] Löpez, M.F. and Río, E.G. (2011) A Sharp Lower Bound for the Scalar Curvature of Certain Steady Gradient Ricci Solitons. arXiv:1104. 1889v1

[18] Zhang, Z.H. (2008) Gradient Shrinking Solitons with Vanishing Weyl Tensor. Pacific Journal of Mathematics, 242, 189-200.
https://doi.org/10.2140/pjm.2009.242.189

[19] Cai, M.L. (2015) On Shrinking Gradient Ricci Solitons with Nonnegative Sectional Curvature. Pacific Journal of Mathematics, 277, 61-76.

[20] Munteanu, O. and Wang, M.T. (2011) The Curvature of Gradient Ricci Solitons. Mathematical Research Letters, 18, 1051-1070.
https://doi.org/10.4310/MRL.2011.v18.n6.a2

[21] Cao, H.D. and Cui, X. (2014) Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons. arXiv:1411.3631v1

[22] Perelman, G. (2003) Ricci Flow with Surgery on Three-Manifolds. arXiv:0303109

[23] Cao, X.D., Wang, B.A. and Zhang, Z. (2011) On Locally Conformally Flat Gradient Shrinking Ricci Solitons. Communications in Contemporary Mathematics, 13, 269-282.
https://doi.org/10.1142/S0219199711004191

[24] Cao, H.D., Chen, B.L. and Zhu, X.P. (2008) Recent Developments on Ham-ilton’s Ricci Flow. Surveys in Differential Geometry, 12, 47-112.
https://doi.org/10.4310/SDG.2007.v12.n1.a3

[25] Cao, H.D. and Zhou, D. (2009) On Complete Gradient Shrinking Ricci Solitons. Journal of Differential Geometry, 85, 175-186.
https://doi.org/10.4310/jdg/1287580963

[26] Zhang, Z.H. (2009) On the Completeness of Gradient Ricci Solitons. Proceedings of the American Mathematical Society, 137, 2755-2759.
https://doi.org/10.1090/S0002-9939-09-09866-9

[27] Chen, C.W. (2011) Volume Estimates and the Asymptotic Behavior of Expanding Gradient Ricci Solitons. Annals of Global Analysis and Geometry, 42, 267-277.
https://doi.org/10.1007/s10455-012-9311-7

[28] Cao, H.D. and Chen, Q. (2012) On Locally Conformally Flat Gradient Steady Ricci Solitons. Transactions of the American Mathematical Society, 364, 2377-2391.
https://doi.org/10.1090/S0002-9947-2011-05446-2

[29] Munteanu, O. and Sesum, N. (2013) On Gradient Ricci Solitons. Journal of Geometric Analysis, 23, 539-561.
https://doi.org/10.1007/s12220-011-9252-6

Top