zbMATH — the first resource for mathematics

Quantitative volume space form rigidity under lower Ricci curvature bound. II. (English) Zbl 1394.53045
Let $$M$$ be a compact $$n$$-manifold of Ricci curvature bounded below by $$(n-1)H,\;H=\pm 1$$ or $$0$$. The authors investigate the following space form rigidity conjecture: $$M$$ is diffeomorphic to an $$H$$-space form if for every ball of definite size on $$M$$, the lifting ball on the Riemannian universal covering space of the ball achieves an almost maximal volume, provided the diameter of $$M$$ is bounded for $$H\neq 1$$.
This paper is the second part of an earlier paper [the authors, “Quantitative volume space form rigidity under lower Ricci curvature bound”, Preprint, arXiv:1604.06986]. In the first paper, the authors verified the conjecture for the case that the Riemannian universal covering space $$\tilde M$$ is not collapsed. In the reviewed paper, they verify this conjecture for the case when Ricci curvature is also bounded above, but the non-collapsing condition on $$\tilde M$$ is not required. The authors conclude this paper with three questions related to the approach in this paper.

MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 53C24 Rigidity results
Full Text:
References:
 [1] Anderson, Michael T., Convergence and rigidity of manifolds under Ricci curvature bounds, Invent. Math., 102, 2, 429-445, (1990) · Zbl 0711.53038 [2] Brendle, Simon; Schoen, Richard, Manifolds with $$1/4$$-pinched curvature are space forms, J. Amer. Math. Soc., 22, 1, 287-307, (2009) · Zbl 1251.53021 [3] Cheeger, Jeff; Colding, Tobias H., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. (2), 144, 1, 189-237, (1996) · Zbl 0865.53037 [4] Cheeger, Jeff; Colding, Tobias H., On the structure of spaces with Ricci curvature bounded below. I, J. Differential Geom., 46, 3, 406-480, (1997) · Zbl 0902.53034 [5] Cheeger, Jeff; Fukaya, Kenji; Gromov, Mikhael, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc., 5, 2, 327-372, (1992) · Zbl 0758.53022 [6] Colding, Tobias H., Large manifolds with positive Ricci curvature, Invent. Math., 124, 1-3, 193-214, (1996) · Zbl 0871.53028 [7] Colding, Tobias H., Ricci curvature and volume convergence, Ann. of Math. (2), 145, 3, 477-501, (1997) · Zbl 0879.53030 [8] [CRX]CRX L. Chen, X. Rong, and S. Xu, \emph Quantitive volume rigidity of space form under lower Ricci curvature bound I, to appear in J. Differential Geom. [9] Dai, Xianzhe; Wei, Guofang; Ye, Rugang, Smoothing Riemannian metrics with Ricci curvature bounds, Manuscripta Math., 90, 1, 49-61, (1996) · Zbl 0880.53034 [10] Grove, Karsten; Karcher, Hermann, How to conjugate $$C^{1}$$-close group actions, Math. Z., 132, 11-20, (1973) · Zbl 0245.57016 [11] Hamilton, Richard S., Three-manifolds with positive Ricci curvature, J. Differential Geom., 17, 2, 255-306, (1982) · Zbl 0504.53034 [12] Gromov, M., Almost flat manifolds, J. Differential Geom., 13, 2, 231-241, (1978) · Zbl 0432.53020 [13] Heintze, Ernst, Mannigfaltigkeiten negativer Kr\`‘ummung, Bonner Mathematische Schriften [Bonn Mathematical Publications] 350, ii+36 pp., (2002), Universit\'’at Bonn, Mathematisches Institut, Bonn · Zbl 1032.53024 [14] [HKRX]HKRX H. Huang, L. Kong, X. Rong, and S. Xu, Collapsed manifolds with local Ricci bounded covering geometry. In preparation. [15] Ledrappier, Fran\ccois; Wang, Xiaodong, An integral formula for the volume entropy with applications to rigidity, J. Differential Geom., 85, 3, 461-477, (2010) · Zbl 1222.53040 [16] Perelman, G., Manifolds of positive Ricci curvature with almost maximal volume, J. Amer. Math. Soc., 7, 2, 299-305, (1994) · Zbl 0799.53050 [17] Petersen, Peter, Riemannian geometry, Graduate Texts in Mathematics 171, xvi+401 pp., (2006), Springer, New York · Zbl 1220.53002 [18] Shi, Wan-Xiong, Deforming the metric on complete Riemannian manifolds, J. Differential Geom., 30, 1, 223-301, (1989) · Zbl 0676.53044 [19] Shi, Wan-Xiong, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom., 30, 2, 303-394, (1989) · Zbl 0686.53037 [20] Tian, Gang; Wang, Bing, On the structure of almost Einstein manifolds, J. Amer. Math. Soc., 28, 4, 1169-1209, (2015) · Zbl 1320.53052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.