Abstract
LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound.
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References
Brown, M.: A proof of the generalized Schoenflies theorem, Bull. Amer. Math. Soc.66, 74–76 (1960)
Cheeger, J and Ebin, D.: Comparison theorems in Riemannian geometry, North-Holland: Amsterdam 1975
Cheng, S. Y.: Eigenvalue comparison theorems and its geometric applications, Math. Z.143, 289–297 (1975)
Choi, H. I. and Schoen, R.: The space of minimal embeddings of a surface into a three-dimensional manifold of positive Ricci curvature, Invent. Math.81, 387–394 (1985)
Coghlan, L. and Itokawa, Y.: A sphere theorem for reverse volume pinching on even-dimensional manifolds, Proc. Amer. Math. Soc.111, 815–819 (1991)
Frankel, T.: On the fundamental group of a compact minimal submanifold, Ann. Math.83, 68–73 (1966)
Gromov, G.: Groups of polynomial growth and expanding maps, Inst. Hautes Etudes Sci. Pub. Math.53, 183–215 (1981)
Grove, K. and Petersen, V. P.: Manifolds near the boundary of existence, J. Differential Geometry33, 379–394 (1991)
Hadamard, J.: Sur certaines propretes des trajectories en Dynamique, J. Math. Pures Appl.5, 331–387 (1897)
Hartman, P.: Oscillation criteria for self-adjoint second-order differential systems and “principal sectional curvature”, J. Differential Equations34, 326–338 (1979)
Klingenberg, W.: Riemannian Geometry, Berlin, New York: de Gruyter 1982
Myers, S. B.: Curvature of closed hypersurfaces and non-existence of closed minimal hypersurfaces, Trans. Amer. Math. Soc.71, 211–217 (1951)
Peters, S.: Convergence of Riemannian manifolds, Compositio Math.62, 3–16 (1987)
Reilly, R.: Applications of the Hessian operator in a Riemannian manifold, Indiana Univ. Math. J.26, 459–472 (1977)
Toponogov, V. A.: Computation of the length of a closed geodesic on a convex surface, Dokl. Akad. Nauk SSSR124, 282–284 (1959)
Wu, J. Y.: Hausdorff convergence and sphere theorems, Proc. Sym. Pure Math.54, 685–692 (1993)
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Supported by the JSPS postdoctoral fellowship and NSF of China
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Xia, C. Rigidity and sphere theorem for manifolds with positive Ricci curvature. Manuscripta Math 85, 79–87 (1994). https://doi.org/10.1007/BF02568185
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DOI: https://doi.org/10.1007/BF02568185