AN EXTRINSIC RIGIDITY THEOREM FOR SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN A SPHERE

作者：Xu, Hong-We^*; Huang, Fei; Xiang, Fei

来源：Kodai Mathematical Journal, 2011, 34(1): 85-104.

DOI：10.2996/kmj/1301576764

• Submanifolds
• extrinsic rigidity
• mean curvature
• trace free second fundamental form

Let M be an n-dimensional closed submanifold with parallel mean curvature in Sn+p, (h) over tilde, it the trace free part of the second fundamental form, and (sigma) over tilde (u) = parallel to(h) over tilde (u, u)parallel to(2) for any unit vector u is an element of TM. We prove that there exists a positive constant C(n, p, H) (>= 1/3) such that if (sigma) over tilde (u) <= C(n. p, H), then either (sigma) over tilde (u) equivalent to 0 and M is a totally umbilical sphere, or (sigma) over tilde (u) equivalent to C(n, p, H). A geometrical classification of closed submanifolds with parallel mean curvature satisfying (sigma) over tilde (u) equivalent to C(n, p, H) is also given. Our main result is an extension of the Gauchman theorem [4].

• 单位
  浙江大学