A differentiable sphere theorem inspired by rigidity of minimal submanifolds

Hong-Wei Xu and Ling Tian

Vol. 254 (2011), No. 2, 499–510

DOI: 10.2140/pjm.2011.254.499

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