﻿ 拟爱因斯坦度量的分类 Classification of Quasi-EinsteinMetric

Modern Physics
Vol. 09  No. 01 ( 2019 ), Article ID: 28328 , 7 pages
10.12677/MP.2019.91002

Classification of Quasi-Einstein Metric

Jinnan Li, Xiang Gao*

School of Mathematical Sciences, Ocean University of China, Qingdao Shandong

Received: Dec. 12th, 2018; accepted: Dec. 27th, 2018; published: Jan. 2nd, 2019

ABSTRACT

The natural extension of Einstein's metrics for quasi-Einstein metrics (also known as Ricci soliton) has important applications in gauge field theory and super-string theory. There are two important aspects of it: One looking at the influence on the geometry and topology by the quasi-Einstein metric structure of the Riemannian manifold; and the other looking at its geometric properties and invariant. In this paper, we are interested in summarizing the classification of it and giving some results about gradient steady and expanding quasi-Einstein metric under the assumption of curvature, Weyl tensor and Bach flat.

Keywords:Quasi-Einstein Metric, Ricci Soliton, Steady, Expanding

Copyright © 2019 by authors and Hans Publishers Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

1. 引言

20世纪80年代，Hamilton [1] 提出了Ricci流的概念，实际上Ricci流最初引进是为了解决3维流形著名的Poincaré猜想(任意单连通的3维闭流形必同胚于3维闭球面)。拟爱因斯坦度量(又称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，拟爱因斯坦度量的研究有助于更好地理解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果，是研究黎曼流形最有力的工具之一，因此它在几何分析和Ricci流的研究中具有重要的地位。另一方面，从微分方程的角度看拟爱因斯坦度量是爱因斯坦度量的自然推广 [7] ，在相对论、规范场论与超弦理论中有重要的应用，因此其分类对于数学及物理发展均具有重要的研究意义。

1.1. 拟爱因斯坦度量介绍

1.2. 典型的拟爱因斯坦度量

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. 梯度稳定拟爱因斯坦度量的分类

2002年，Perelman [2] 提出著名的猜想：非平坦的k-非坍塌梯度稳定拟爱因斯坦度量必为Bryant孤立子。Brendle [13] 最终给出证明：

$n=3$ 时，减弱假设条件下可得一个类似的分类结果：3维完备梯度稳定拟爱因斯坦度量，若满足 $\underset{r\to +\infty }{\mathrm{lim}}\mathrm{inf}\frac{1}{r}{\int }_{{B}_{r}\left(o\right)}R=0$，则 $\left({M}^{3},{g}_{ij}\right)$ 等距于 ${R}^{3}$$R×{\sum }^{2}$ 的一个有限商空间。

Bryant [19] 曾证明存在唯一非平坦的梯度稳定 拟爱因斯坦度量为 ${S}^{n-1}$ 与R的乘积，在此基础上，Catino-Mantegazza [20] 给出以下命题：

${W}_{ijkl}={R}_{ijkl}+\frac{R}{\left(n-1\right)\left(n-2\right)}\left({g}_{ik}{g}_{jl}-{g}_{il}{g}_{jk}\right)-\frac{1}{n-2}\left({R}_{ik}{g}_{jl}-{R}_{il}{g}_{jk}+{R}_{jl}{g}_{ik}-{R}_{jk}{g}_{il}\right)$

Bach张量定义为：

${B}_{ij}=\frac{1}{n-3}{\nabla }^{k}{\nabla }^{l}{W}_{ikjl}+\frac{1}{n-2}{R}_{kl}{W}_{ij}^{kl}$

$n\ge 4$ 时，根据Bach平坦也能得到很多关于稳定拟爱因斯坦度量的分类。特别地，Bach平坦表明局部共形平坦，能否在Weyl张量一阶消失的条件下获得类似的结果依然是一个有待解决的问题。

3维张量 ${D}_{ijk}$ 定义为：

${D}_{ijk}=\frac{1}{n-2}\left({A}_{jk}{\nabla }_{i}f-{A}_{ik}{\nabla }_{j}f\right)+\frac{1}{\left(n-1\right)\left(n-2\right)}\left({g}_{jk}{E}_{il}-{g}_{jk}{E}_{jl}\right){\nabla }_{l}f$

3. 梯度扩张拟爱因斯坦度量的分类

Brendle [13] [31] 证明了具有正截面曲率的梯度稳定拟爱因斯坦度量附加额外条件时必为旋转对称的。在此基础上，扩张情况下Chodosh [29] 证明了：

1) 势函数f为常数函数， ${g}_{ij}$ 为爱因斯坦度量；

2) $\left({M}^{4},{g}_{ij}\right)$ 等距于的 ${R}^{2}×{N}_{\lambda }$ 一个有限商空间，其中 ${N}_{\lambda }$ 为2维黎曼流形且具有常曲率 $\lambda <0$ 、且其势函数为 $f=\frac{\lambda }{2}{|x|}^{2}$

3) $\left({M}^{4},{g}_{ij}\right)$ 为局部共形平坦的。

Ricci流的齐次解相对简单且出现在Ricci流的极限中，长期以来也得到诸多研究及重要进展。此外，在没有爱因斯度量时，齐次Ricci孤立子度量自然是一个非常不错的选择，因此齐次Ricci孤立子空间的分类也一直是微分几何的一个核心问题，其中Jablonski [32] 给出了有关齐次拟爱因斯坦度量的分类定理：任意紧致齐次的拟爱因斯坦度量必为爱因斯坦度量。希望得到的黎曼流形上拟爱因斯坦度量的研究与更为精确的分类能够更好地应用于解决广义相对论及爱因斯坦场方程等问题。

Classification of Quasi-EinsteinMetric[J]. 现代物理, 2019, 09(01): 12-18. https://doi.org/10.12677/MP.2019.91002

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

2. 2. Perelman, G. (2002) The Entropy Formula for the Ricci Flow and Its Geometric Applications. arXiv: math. DG/0211159.

3. 3. Perelman, G. (2003) Ricci Flow with Surgery on Three-Manifolds. arXiv: math.DG/0303109.

4. 4. Perelman, G. (2003) Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds. arXiv: math. DG/0307245.

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

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

7. 7. Cao, H.D., Chen, B.L. and Zhu, X.P. (2008) Recent Developments on Hamilton’s Ricci Flow. Surveys in Differential Geometry, 47-112.

8. 8. Hamilton, R.S. (1993) The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry, 2, 7-136.
https://doi.org/10.4310/SDG.1993.v2.n1.a2

9. 9. Ivey, T. (1993) Ricci Solitons on Compact Three-Manifolds. Differential Geometry and Its Applications, 3, 301-330.
https://doi.org/10.1016/0926-2245(93)90008-O

10. 10. Cao, H.D. and Chen, B.L. (2012) On Locally Conformally Flat Gradient Steady Ricci Solitons. Transactions of the American Mathematical Society, 634, 2377-2391.
https://doi.org/10.1090/S0002-9947-2011-05446-2

11. 11. Chen, X.X. and Wang, Y.Q. (2015) On Four-Dimensional An-ti-Self-Dual Gradient Ricci Solitons. The Journal of Geometric Analysis, 25, 1335-1343.
https://doi.org/10.1007/s12220-014-9471-8

12. 12. Brendle, S. (2010) Uniqueness of Gradient Ricci Solitons. arXiv: math. DG/1010.3684v2.

13. 13. Brendle, S. (2013) Rotational Symmetry of Self-Similar Solutions to the Ricci Flow. Inventiones Mathematicae, 194, 731-764.
https://doi.org/10.1007/s00222-013-0457-0

14. 14. Catino, G., Mastrolia, P. and Monticelli, D.D. (2016) Classification of Expanding and Steady Ricci Solitons with Integral Curvature Decay. Geometry & Topology, 20, 2665-2685.
https://doi.org/10.2140/gt.2016.20.2665

15. 15. Deruelle, A. (2012) Steady Gradient Ricci Soliton with Curvature in L1. Commu-nications in Analysis and Geometry, 20, 31-53.
https://doi.org/10.4310/CAG.2012.v20.n1.a2

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

17. 17. Israel, W. (1967) Event Horizons in Static Vacuum Space-Times. Physical Review, 164, 1776-1779.
https://doi.org/10.1103/PhysRev.164.1776

18. 18. Robnsono, D.C. (1997) A Simple Proof of the Generalization of Isarel’s Theo-rem. General Relativity and Gravitation, 8, 695-698.

19. 19. Bryant, R.L. (1987) Locally Existence of Gradient Ricci Soli-tons.

20. 20. Catino, G. and Mantegazza, C. (2013) The Evolution of the Weyl Tensor under the Ricci Flow. Annales-Institut Fourier, 61, 1407-1435.
https://doi.org/10.5802/aif.2644

21. 21. Kim, J. (2016) On a Classification of 4-D Gradient Ricci Solitons with Harmonic Weyl Curvature.

22. 22. Cao, H.D., Catino, G., Chen, Q., Mantegazza, C. and Mazzieri, L. (2014) Bach-Flat Gradient Steady Ricci Solitons. Calculus of Variations & Partial Differential, 49, 125-138.
https://doi.org/10.1007/s00526-012-0575-3

23. 23. Cao, H.D. and Chen, Q. (2009) On Complete Gradient Shrinking Ricci Solitons. Journal of Differential Geometry, 85, 175-185.
https://doi.org/10.4310/jdg/1287580963

24. 24. Catino, G., Mastrolia, P. and Monticelli, D.D. (2017) Gradient Ricci Solitons with Vanishing Conditions on Weyl. Journal De Mathématiques Pures et Appliquées, 108, 1-13.
https://doi.org/10.1016/j.matpur.2016.10.007

25. 25. Hamilton, R.S. (1993) The Harnack Estimate for the Ricci Flow. Journal of Differential Geometry, 37, 225-243.
https://doi.org/10.4310/jdg/1214453430

26. 26. Chow, B., Lu, P. and Ni, L. (2006) Hamilton’s Ricci Flow. Grad. Stud. Math, Vol. 77, AMS.

27. 27. Cao, H.D. (1997) Limitations of Solutions to the Kähler-Ricci Flow. Journal of Differential Geometry, 45, 257-272.
https://doi.org/10.4310/jdg/1214459797

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

29. 29. Petersen, P. and Wylie, W. (2010) On the Classification of Gradient Ricci Solitons. Geometry & Topology, 14, 2277-2300.
https://doi.org/10.2140/gt.2010.14.2277

30. 30. Pigola, S., Rimoldi, M. and Setti, A.G. (2011) Remarks on Non-Compact Gradient Ricci Solitons. Mathematische Zeitschrift, 268, 777-790.
https://doi.org/10.1007/s00209-010-0695-4

31. 31. Brendle, S. (2013) Rotational Symmetry of Ricci Solitons in Higher Dimensions. Journal of Differential Geometry, 97, 191-214.
https://doi.org/10.4310/jdg/1405447804

32. 32. Jablonski, M. (2015) Homogeneous Ricci Solitons. Journal für die Reine und Angewandte Mathematik, 699, 159-182.
https://doi.org/10.1515/crelle-2013-0044

33. NOTES

*通讯作者。