AN OPTIMAL DIFFERENTIABLE SPHERE THEOREM FOR COMPLETE MANIFOLDS

作者：Xu, Hong-Wei^*; Gu, Juan-Ru

来源：Mathematical Research Letters, 2010, 17(6): 1111-1124.

• Complete manifolds
• differentiable sphere theorem
• Ricci flow
• stable currents
• second fundamental form

A new differentiable sphere theorem is obtained from the view of submanifold geometry. We introduce a new scalar quantity involving both the scalar curvature and the mean curvature of an oriented complete submanifold M-n in a space form Fn+p(c) with c >= 0. Making use of the convergence results of Hamilton and Brendle for Ricci flow and the Lawson-Simons formula for the nonexistence of stable currents, we prove that if the infimum of this scalar quantity is positive, then M is diffeomorphic to S-n. We then introduce an intrinsic invariant I( M) for oriented complete Riemannian n-manifold M via the scalar quantity, and prove that if I(M) > 0, then M is diffeomorphic to Sn. It should be emphasized that our differentiable sphere theorem is optimal for arbitrary n(>= 2). Moreover, we generalize the Brendle-Schoen differentiable sphere theorem for manifolds with strictly 1/4-pinched curvatures in the pointwise sense to the cases of submanifolds in a Riemannian manifold with codimension p(>= 0).

• 出版日期2010-11
• 单位浙江大学