The Stability of Generalized Ricci Solitons

被引:6
作者
Lee, Kuan-Hui [1 ]
机构
[1] Univ Calif Irvine, Dept Math, Irvine, CA 92697 USA
关键词
Generalized Ricci flow; Generalized geometry; Generalized Ricci solitons; DYNAMICAL STABILITY; EINSTEIN-METRICS; INSTABILITY;
D O I
10.1007/s12220-023-01331-9
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
In Garcia-Fernandez and Streets (Generalized Ricci flow, volume 76 of university lecture series, American Mathematical Society, Providence, 2021) and Oliynyk et al. (Nucl Phys B 739(3):441-458, 2006), it was shown that the generalized Ricci flow is the gradient flow of a functional A. generalizing Perelman's A. functional for Ricci flow. In this work, we further computed the second variation formula and proved that a Bismut-flat, Einstein manifold is linearly stable under some curvature assumptions. In the last part of this paper, I proved that dynamical stability and linear stability are equivalent on a steady gradient generalized Ricci soliton (g , H , f). This generalizes the results in Haslhofer and Miller (Math Ann 360(1-2):547-553, 2014), Kroncke (Stability of Einstein Manifolds, 2014, Commun Anal Geom 28(2):351-394, 2020), Raffero and Vezzoni (On the dynamical behaviour of the generalized Ricci flow, 2020) and Sesum (Duke Math J 133(1):1-26, 2006).
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页数:52
相关论文
共 33 条
[1]   A note on flat metric connections with antisymmetric torsion [J].
Agricola, Ilka ;
Friedrich, Thomas .
DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS, 2010, 28 (04) :480-487
[2]  
[Anonymous], 1979, Osaka J. Math.
[3]  
Besse A.L., 1987, Results in Mathematics and Related Areas (3)
[4]   Linear stability of Perelman's ν-entropy on symmetric spaces of compact type [J].
Cao, Huai-Dong ;
He, Chenxu .
JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK, 2015, 709 :229-246
[5]   On second variation of Perelman's Ricci shrinker entropy [J].
Cao, Huai-Dong ;
Zhu, Meng .
MATHEMATISCHE ANNALEN, 2012, 353 (03) :747-763
[6]  
Chow B., 2007, Math. Surv. Monogr., V135
[7]  
Chow B., 2004, RICCI FLOW INTRO, DOI 10.1090/surv/110
[8]  
Chow B., 2006, Graduate Studies in Mathematics, V77
[9]   On uniqueness of tangent cones for Einstein manifolds [J].
Colding, Tobias Holck ;
Minicozzi, William P., II .
INVENTIONES MATHEMATICAE, 2014, 196 (03) :515-588
[10]  
Dai XZ, 2004, Arxiv, DOI arXiv:math/0311253