Sharp pinching theorems for complete submanifolds in the sphere

Abstract

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