梯度收缩Ricci孤立子的分类
Classification of Gradient Shrinking Ricci Solitons
DOI: 10.12677/PM.2019.91003, PDF,    科研立项经费支持
作者: 李金楠, 高 翔*：中国海洋大学，数学科学学院，山东 青岛
关键词: 收缩Ricci孤立子；曲率；Weyl张量；分类；Shrinking Ricci Soliton； Curvature； Weyl Tensor； Classification
摘要: Ricci孤立子的研究有两个重要的方向，一个是研究黎曼流形上Ricci孤立子的结构对其拓扑结构的影响，另一个是研究Ricci孤立子的几何性质及几何不变量。本文，我们将系统的阐述满足一定曲率及Weyl张量等条件下梯度收缩孤立子的分类。
Abstract: There are two important aspects of Ricci solitons. One looks at the influence on the topology by the Ricci solitons structure of the Riemannian manifold, and the other looks at its geometric properties and invariant. In this paper, we are interested in summarizing the classification of Ricci soliton and give some results about gradient shrinking Ricci solitons under the assumption of curvature or Weyl tensor.
文章引用：李金楠, 高翔. 梯度收缩Ricci孤立子的分类[J]. 理论数学, 2019, 9(1): 20-28. https://doi.org/10.12677/PM.2019.91003

参考文献

[1] [1] Hamilton, R.S. (1998) The Ricci Flow on Surfaces. Contemporary Mathematics, 71, 237-261.
[Google Scholar] [CrossRef] 
[2] Perelman, G. (2002) The Entropy Formula for the Ricci Flow and It’s Geometric Applications. arXiv: math. DG/0211159
[3] Perelman, G. (2003) Ricci Flow with Surgery on Three-Manifolds. arXiv: math.DG/0303109
[4] Perelman, G. (2003) Finite Extinction Time for the Solutions to the Ricci Flow on Certain Three-Manifolds. arXiv: math. DG/0307245
[5] Hamilton, R.S. (1982) Three Manifolds with Positive Ricci Curvature. Journal of Differential Geometry, 17, 255-306.
[Google Scholar] [CrossRef] 
[6] Hamilton, R.S. (1993) Eternal Solutions to the Ricci Flow. Journal of Differential Geometry, 38, 1-11.
[Google Scholar] [CrossRef] 
[7] Cao, H.D., Chen, B.L. and Zhu, X.P. (2008) Recent Developments on Hamilton’s Ricci Flow. Geometric flows, Surveys in Differential Geometry, 47-112.
[8] Hamilton, R.S. (1993) The Formation of Singularities in the Ricci Flow. Surveys in Differential Geometry, 2, 7-136.
[9] Ivey, T. (1993) Ricci Solitons on Compact Three-Manifolds. Differential Geometry and its Applications, 3, 301-330.
[Google Scholar] [CrossRef] 
[10] Hamilton, R.S. (1986) Four-Manifolds with Positive Cur-vature Operator. Journal of Differential Geometry, 24, 153-179.
[Google Scholar] [CrossRef] 
[11] Böhm, C. and Wilking, B. (2008) Manifolds with Positive Curvature Operator Are Space Forms. Annals of Mathematics, 167, 1079-1097.
[Google Scholar] [CrossRef] 
[12] Löpez, M.F. and Río, E.G. (2008) A Remark on Compact Ricci Solitons. Mathematische Annalen, 340, 893-896.
[Google Scholar] [CrossRef] 
[13] Ni, L. and Wallach, N. (2008) On a Classification of the Gradient Shrinking Solitons. Mathematical Research Letters, 15, 941-955.
[Google Scholar] [CrossRef] 
[14] Naber, A. (2010) Noncompact Shrinking Four Solitons with Nonnegative Curvature. Journal für die Reine und Angewandte Mathematik, 645, 125-153.
[Google Scholar] [CrossRef] 
[15] Petersen, P. and Wylie, W. (2010) On the Classification of Gradient Ricci Solitons. Geometry & Topology, 14, 2277-2300.
[Google Scholar] [CrossRef] 
[16] Zhang, Z.H. (2009) Gradient Shrinking Solitons with Vanishing Weyl Tensor. Pacific Journal of Mathematics, 242, 189-200.
[Google Scholar] [CrossRef] 
[17] Cao, H.D. (2009) Geometry of Complete Gradient Shrinking Ricci Solitons. Mathematics, 17, 1149-1169.
[18] Kotschwar, B. (2008) On Rotationally Invariant Shrinking Ricci Solitons. Pacific Journal of Mathematics, 236, 73-88.
[Google Scholar] [CrossRef] 
[19] Micallef, M. and Moore, J.D. (1998) Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes. Annals of Mathematics, 127, 199-227.
[20] Ni, L. and Wallach, N. (2008) On Four-Dimensional Gradient Shrinking Solitons. International Math-ematics Research Notices, 2008, rnm152.
[21] Gu, H.L. and Zhu, X.P. (2007) The Existence of Type II Singularities for the Ricci Flow on Sn+1. Communications in Analysis and Geometry, 16, 467-494.
[22] Petersen, P. and Wylie, W. (2012) Rigidity of Gradient Ricci Solitons. Pacific Journal of Mathematics, 241, 329-345.
[Google Scholar] [CrossRef] 
[23] Munteanu, O. and Wang, M.T. (2011) The Curvature of Gradient Ricci Solitons. Mathematical Research Letters, 18, 1051-1070.
[Google Scholar] [CrossRef] 
[24] Cai, M.L. (2013) On Shrinking Gradient Ricci Solitons with Nonnegative Sectional Curvature.
[25] Catino, G. and Mantegazza, C. (2013) The Evolution of the Weyl Tensor under the Ricci Flow. Annales de l’Institut Fourier, 61, 1407-1435.
[Google Scholar] [CrossRef] 
[26] Catino, G. (2013) Complete Gradient Shrinking Ricci Solitons with Pinched Curvature. Mathematische Annalen, 355, 629-635.
[Google Scholar] [CrossRef] 
[27] Catino, G., Mastrolia, P., Monticelli, D.D. and Rigoli, M. (2014) Analytic and Geometric Properties of Generic Ricci Solitons. Transactions of the AMS, 368.
[28] Fu, H.P. and Xiao, L.Q. (2015) Rigidity Theorem for Integral Pinched Shrinking Ricci Solitons. Monatshefte Fur Mathematik, 1-8.
[29] Munteanu, O. and Sesum, N. (2013) On Gradient Ricci Solitons. Journal of Geometric Analysis, 23, 539-561.
[Google Scholar] [CrossRef] 
[30] Catino, G., Mastrolia, P. and Monticell, D.D. (2017) Gradient Ricci Solitons with Vanishing Conditions on Weyl. Journal De Mathematiques Pures et Appliquees, 108, 1-13.
[Google Scholar] [CrossRef] 
[31] Chen, X.X. and Wang, Y.Q. (2015) On Four-Dimensional Anti-Self-Dual Gradient Ricci Solitons. Journal of Geometric Analysis, 25, 1335-1343.
[Google Scholar] [CrossRef] 
[32] Wu, J.Y., Wu, P. and Wylie, W. (2014) Gradient Shrinking Ricci Solitons of Half Harmonic Weyl Curvature.
[33] Cao, H.D. and Chen, Q. (2009) On Bach-Flat Gradient Shrinking Ricci Solitons. Journal of Differential Geometry, 85, 175-185.
[Google Scholar] [CrossRef] 

为你推荐