On the Local Version of the Chern Conjecture: CMC Hypersurfaces with Constant Scalar Curvature in $S^{n 1}$
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Authors: S. C. DE ALMEIDA, F. G. B. BRITO, M. SCHERFNER AND S. WEISS
DOI: 10.46793/KgJMat2001.101A
Abstract:
After nearly 50 years of research the Chern conjecture for isoparametric hypersurfaces in spheres is still an unsolved and important problem and in particular its local version is of great interest, since here one loses the power of Stokes’ Theorem as a method for proving it. Here we present a related result for CMC hypersurfaces in ????n+1 with constant scalar curvature and three distinct principal curvatures.
Keywords:
Constant mean and scalar curvature, isoparametric hypersurfaces, Chern conjecture.
References:
[1] S. Chang, On closed hypersurfaces of constant scalar curvatures and mean curvatures in Sn+1, Pacific J. Math. 165 (1994), 67–76.
[2] T. Otsuki, Minimal hypersurfaces in a Riemannian manifold of constant curvature, Amer. J. Math. 92 (1970), 145–173.
[3] M. Scherfner, S.-T. Yau and S. Weiss, A review of the Chern conjecture for isoparametric hypersurfaces in Spheres, in: Advances in Geometric Analysis, Advanced Lectures in Mathematics (ALM) 21, International Press, Somerville, 2012, 175–187.
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