An Optimal Differentiable Sphere Theorem for Complete Manifolds

TL;DR

This paper establishes a new optimal differentiable sphere theorem for complete manifolds, linking scalar curvature and mean curvature to topological classification, using Ricci flow convergence and stable current nonexistence.

Contribution

It introduces a novel scalar invariant combining curvature measures and proves a sharp sphere theorem for all dimensions using Ricci flow techniques.

Findings

If the scalar invariant is positive, the manifold is diffeomorphic to a sphere.

The intrinsic invariant I(M) provides a new criterion for sphere diffeomorphism.

The theorem is proven to be optimal across all dimensions.

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 $M^n$ in a space form $F^{n+p}(c)$ with $c\ge0$. 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 $S^n$. 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 $S^n$. It should be emphasized that our differentiable sphere theorem is optimal for arbitrary $n(\ge2)$.