拟爱因斯坦度量的分类
Classification of Quasi-EinsteinMetric
DOI: 10.12677/MP.2019.91002, 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: 拟爱因斯坦度量；Ricci孤立子；稳定；扩张；Quasi-Einstein Metric； Ricci Soliton； Steady； Expanding
摘要: 拟爱因斯坦度量(又称Ricci孤立子)是爱因斯坦度量的自然推广，在规范场论与超弦理论中有重要的应用，其研究有两个重要的方向：一是研究黎曼流形上拟爱因斯坦度量的结构对其几何及拓扑结构的影响；另一个是研究其几何性质与几何不变量。本文，我们将系统的阐述满足一定曲率条件、Weyl张量条件及Bach平坦等条件下梯度稳定及扩张拟爱因斯坦度量的分类。
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 met-ric 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 cur-vature, Weyl tensor and Bach flat.
文章引用：李金楠, 高翔. 拟爱因斯坦度量的分类[J]. 现代物理, 2019, 9(1): 12-18. https://doi.org/10.12677/MP.2019.91002

参考文献

[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 Its 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 Ge-ometry, 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. 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.
[Google Scholar] [CrossRef] 
[9] Ivey, T. (1993) Ricci Solitons on Compact Three-Manifolds. Differential Geometry and Its Applications, 3, 301-330.
[Google Scholar] [CrossRef] 
[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.
[Google Scholar] [CrossRef] 
[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.
[Google Scholar] [CrossRef] 
[12] Brendle, S. (2010) Uniqueness of Gradient Ricci Solitons. arXiv: math. DG/1010.3684v2.
[13] Brendle, S. (2013) Rotational Symmetry of Self-Similar Solutions to the Ricci Flow. Inventiones Mathematicae, 194, 731-764.
[Google Scholar] [CrossRef] 
[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.
[Google Scholar] [CrossRef] 
[15] Deruelle, A. (2012) Steady Gradient Ricci Soliton with Curvature in L1. Commu-nications in Analysis and Geometry, 20, 31-53.
[Google Scholar] [CrossRef] 
[16] Chen, B.L. (2009) Strong Uniqueness of the Ricci Flow. Journal of Differential Geometry, 82, 363-382.
[Google Scholar] [CrossRef] 
[17] Israel, W. (1967) Event Horizons in Static Vacuum Space-Times. Physical Review, 164, 1776-1779.
[Google Scholar] [CrossRef] 
[18] Robnsono, D.C. (1997) A Simple Proof of the Generalization of Isarel’s Theo-rem. General Relativity and Gravitation, 8, 695-698.
[19] Bryant, R.L. (1987) Locally Existence of Gradient Ricci Soli-tons.
[20] Catino, G. and Mantegazza, C. (2013) The Evolution of the Weyl Tensor under the Ricci Flow. Annales-Institut Fourier, 61, 1407-1435.
[Google Scholar] [CrossRef] 
[21] Kim, J. (2016) On a Classification of 4-D Gradient Ricci Solitons with Harmonic Weyl Curvature.
[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.
[Google Scholar] [CrossRef] 
[23] Cao, H.D. and Chen, Q. (2009) On Complete Gradient Shrinking Ricci Solitons. Journal of Differential Geometry, 85, 175-185.
[Google Scholar] [CrossRef] 
[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.
[Google Scholar] [CrossRef] 
[25] Hamilton, R.S. (1993) The Harnack Estimate for the Ricci Flow. Journal of Differential Geometry, 37, 225-243.
[Google Scholar] [CrossRef] 
[26] Chow, B., Lu, P. and Ni, L. (2006) Hamilton’s Ricci Flow. Grad. Stud. Math, Vol. 77, AMS.
[27] Cao, H.D. (1997) Limitations of Solutions to the Kähler-Ricci Flow. Journal of Differential Geometry, 45, 257-272.
[Google Scholar] [CrossRef] 
[28] Chen, B.L. and Zhu, X.P. (2000) Complete Riemannian Manifolds with Pointwise Pinched Curvature. Inventiones Mathematicae, 140, 423-452.
[Google Scholar] [CrossRef] 
[29] Petersen, P. and Wylie, W. (2010) On the Classification of Gradient Ricci Solitons. Geometry & Topology, 14, 2277-2300.
[Google Scholar] [CrossRef] 
[30] Pigola, S., Rimoldi, M. and Setti, A.G. (2011) Remarks on Non-Compact Gradient Ricci Solitons. Mathematische Zeitschrift, 268, 777-790.
[Google Scholar] [CrossRef] 
[31] Brendle, S. (2013) Rotational Symmetry of Ricci Solitons in Higher Dimensions. Journal of Differential Geometry, 97, 191-214.
[Google Scholar] [CrossRef] 
[32] Jablonski, M. (2015) Homogeneous Ricci Solitons. Journal für die Reine und Angewandte Mathematik, 699, 159-182.
[Google Scholar] [CrossRef]