Vol. 254, No. 2, 2011

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ISSN: 0030-8730
A differentiable sphere theorem inspired by rigidity of minimal submanifolds

Hong-Wei Xu and Ling Tian

Vol. 254 (2011), No. 2, 499–510
Abstract

We prove a vanishing theorem for the fundamental group of a compact submanifold in a space form, and then present a refined version of Ejiri’s rigidity theorem for minimal submanifolds in a sphere. Inspired by the refined Ejiri theorem, we verify a new differentiable sphere theorem for compact submanifolds in space forms. We also show that our differentiable sphere theorem is sharp. We emphasize that our method of Ricci flow in the proof of the sphere theorem seems useful in the study of curvature and topology. Also, we obtain a differentiable pinching theorem for compact submanifolds in a Riemannian manifold.

Keywords
Submanifolds, differentiable sphere theorem, Ricci curvature, Ricci flow, stable currents
Mathematical Subject Classification 2010
Primary: 53C20, 53C24, 53C40
Milestones
Received: 21 April 2011
Revised: 29 July 2011
Accepted: 10 August 2011
Published: 27 February 2012
Authors
Hong-Wei Xu
Center of Mathematical Sciences
Zhejiang University
Hangzhou, 310027
China
Ling Tian
Center of Mathematical Sciences
Zhejiang University
Hangzhou, 310027
China