A new gap for complete hypersurfaces with constant mean curvature in space forms

2018/10/31 by Gu, Juanru, Lei, Li, Xu, Hongwei
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1810.13080

Abstract

Let M be an n-dimensional closed hypersurface with constant mean curvature and constant scalar curvature in an unit sphere. Denote by H and S the mean curvature and the squared length of the second fundamental form respectively. We prove that if S > α(n, H), where n≥ 4 and H≠ 0, then S > α(n, H) + B_n(n H²)/(n - 1). Here α(n, H) = n + (n³)/(2 (n - 1)) H² - (n (n - 2))/(2 (n - 1))√(n² H⁴ + 4 (n - 1) H²), B_n=(1)/(5) for 4≤ n ≤ 20, and B_n=(49)/(250) for n>20. Moreover, we obtain a gap theorem for complete hypersurfaces with constant mean curvature and constant scalar curvature in space forms.

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