Mathematical Research Letters

Volume 17 (2010)

Number 6

An optimal differentiable sphere theorem for complete manifolds

Pages: 1111 – 1124



Juan-Ru Gu

Hong-Wei Xu


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

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