An Optimal Differentiable Sphere Theorem for Complete Manifolds

2010/05/14 by Hong-Wei Xu And Juan-Ru Gu, Xu, Hong-Wei, Gu, Juan-Ru
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1005.2557

openalex publication_date 2010/05/14 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

A new differentiable sphere theorem is obtained from the view of submanifold geometry. An important scalar is defined by the scalar curvature and the mean curvature of an oriented complete submanifold Mⁿ in a space form F^n+p(c) with c≥0. Making use of the Hamilton-Brendle-Schoen convergence result for Ricci flow and the Lawson-Simons-Xin formula for the nonexistence of stable currents, we prove that if the infimum of this scalar is positive, then M is diffeomorphic to Sⁿ. We then introduce an intrinsic invariant I(M) for oriented complete Riemannian n-manifold M via the scalar, 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).

Related

• The sphere theorems for manifolds with positive scalar curvature
• Some sharp differential sphere theorems for nonnegative scalar curvature manifolds
• Curvature, sphere theorems, and the Ricci flow
• An Optimal Convergence Theorem for Mean Curvature Flow of Arbitrary Codimension in Hyperbolic Spaces
• Sphere Theorems in Geometry