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Sharp pinching theorems for complete CMC hypersurfaces in the sphere

Luciano MariFernanda RoingAndreas Savas-Halilaj
Feb 2024
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摘要原文
In this paper, we prove that every complete, minimally immersed hypersurface $f:M^n \to \mathbb{S}^{n+1}$ whose second fundamental form satisfies $|A|^2 \le n$, is either totally geodesic or (a covering of) a minimal Clifford torus, thereby extending the well-known result by Simons, Lawson and Chern, do Carmo & Kobayashi from compact to complete $M^n$. We also extend the corresponding result for hypersurfaces with nonvanishing constant mean curvature, due to Alencar & do Carmo, to complete immersed CMC hypersurfaces, under the optimal bound for their umbilicity tensor. Our approach is inspired by the conformal method of Fischer-Colbrie and Catino, Mastrolia & Roncoroni.
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