Sharp pinching theorems for complete submanifolds in the sphere

Abstract

We prove that every complete, minimally immersed submanifold $f{M}^n\to {\mathbb{S}}^{n+p}$ whose second fundamental form satisfies $|A{|}^2\le np/(2p-1)$, is either totally geodesic, or (a covering of) a Clifford torus or a Veronese surface in ${\mathbb{S}}^4$, thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete $M^n$. We also obtain the corresponding result for complete hypersurfaces with nonvanishing constant mean curvature, due to Alencar & do Carmo in the compact case, under the optimal bound on the umbilicity tensor. In dimension $n\le 6$, a pinching theorem for complete higher-codimensional submanifolds with non-vanishing parallel mean curvature is proved, partly generalizing previous work of Santos. Our approach is inspired by the conformal method of Fischer-Colbrie, Shen & Ye and Catino, Mastrolia & Roncoroni.

The title has been changed; references updated, original result extended to higher codimensions