Sharp pinching theorems for complete submanifolds in the sphere

Abstract: We prove that every complete, minimally immersed submanifold $f: Mⁿ \to \mathbb{S}<sup>{n+p}$ 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 $\mathbb{S}<sup>4$, thereby extending the well-known results by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete $M<sup>n$. 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.