Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415093 | Journal of Functional Analysis | 2014 | 32 Pages |
Abstract
Let Mn be a compact hypersurface with constant mean curvature H in Sn+1. Denote by S the squared norm of the second fundamental form of M. We prove that there exists an explicit positive constant γ(n) depending only on n such that if |H|â¤Î³(n) and β(n,H)â¤Sâ¤Î²(n,H)+n23, then Sâ¡Î²(n,H) and M is one of the following cases: (i) Sk(kn)ÃSnâk(nâkn), 1â¤kâ¤nâ1; (ii) S1(11+μ2)ÃSnâ1(μ1+μ2). Here β(n,H)=n+n32(nâ1)H2+n(nâ2)2(nâ1)n2H4+4(nâ1)H2 and μ=n|H|+n2H2+4(nâ1)2. For nâ{6,12,24}, we give examples to show that the assumption |H|â¤Î³(n) cannot be removed.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Hong-wei Xu, Zhi-yuan Xu,