A general gap theorem for submanifolds with parallel mean curvature in Rn+p

作者：Xu, Hong-Wei^*; Gu, Juan-Ru

来源：Communications in Analysis and Geometry, 2007, 15(1): 175-193.

• MINIMAL SUBMANIFOLDS
• RIGIDITY THEOREM
• RIEMANNIAN-MANIFOLDS
• SPHERE
• HYPERSURFACES
• CONSTANT
• SOBOLEV

Let M-n (n >= 3) be an n-dimensional complete submanifold with parallel mean curvature in Rn+P. Denote by H and S the mean curvature and the squared length of the second fundamental form of M, respectively. By using a direct method, we prove that if integral(M)(S - nH(2))(n/2)dM < C(n), where C(n) is an explicit positive constant depending only on n, then S = nH(2), i.e., M-n is a totally umbilical submanifold. In particular, if H = 0, then M = R-n; if H not equal 0, then M = S-n (1/H). It improves the gap theorems proposed by Lei Ni and Xu.

• 出版日期2007-1
• 单位浙江大学