Abstract
The Clifford hypersurface is one of the simplest compact hypersurfaces in a unit sphere. We give two different characterizations of Clifford hypersurfaces among constant m-th order mean curvature hypersurfaces with two distinct principal curvatures. One is obtained by assuming embeddedness and by comparing two distinct principal curvatures. The proof uses the maximum principle to the two-point function, which was used in the proof of Lawson conjecture by Brendle (Acta Math. 211(2):177–190, 2013, [6]). The other is given by obtaining a sharp curvature integral inequality for hypersurfaces in a unit sphere with constant m-th order mean curvature and with two distinct principal curvatures, which generalizes Simons integral inequality (Simons, Ann. Math. (2) 88:62–105, 1968, [30]). This article is based on joint works (Min and Seo, Math. Res. Lett. 24(2):503–534, 2017, [18], Min and Seo, Monatsh. Math. 181(2):437–450, 2016, [19]) with Sung-Hong Min.
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