An Optimal Differentiable Sphere Theorem for Complete Manifolds

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 $n(\ge2)$.