The total mean curvature of submanifolds in a Euclidean space.

Abstract

LetM be an n-dimensional submanifold in a Euclidean spaceEn+p of dimension n+p. Denote by R the normalized scalar curvature and byH the mean curvature of M. Ōtsuki [O] introduced a kind of curvatures, λ1 ≥ λ2 ≥ · · · ≥ λp for M 2 in E2+p, and showed that they can be used to study the geometry of surfaces in higher-dimensional Euclidean space. Shiohama [S] proved that a complete oriented surface inE2+p with λα = 0 (1 ≤ α ≤ p) is a cylinder. Chen [C1] classified compact oriented surfaces in E2+p with λp ≥ 0. In higher-dimensional cases, Chen [C3] introduced the notion of αth scalar curvatures, λ1 ≥ λ2 ≥ · · · ≥ λp for M in En+p, and found a relationship between the αth scalar curvatures and the scalar curvature. When n = 2, it reduces to that introduced by Ōtsuki [O]. Chen [C3] also proved that a closed submanifold M (n ≥ 3) in En+p with ∫ Mn (λ1) n/2 dV = cn and λα = 0 (2 ≤ α ≤ p) is an n-sphere, where cn is the volume of the unit n-sphere and dV denotes the volume element of M. In this paper, we give a further description of the behavior of the αth scalar curvatures and obtain some applications of them. In Section 2, we first prove that λα ≤ 0 (2 ≤ α ≤ p) for any submanifold M in En+p. Then we prove an inequality involving the integral of λ1 for closed M in En+p with R ≥ 0. Suppose that M is closed in En+p. The total mean curvature of M is defined to be the integral ∫ Mn H dV. An interesting and outstanding problem is to find the best possible lower bound of this integral in terms of the geometric or topologic invariants of M. A special case of this problem is the famous Willmore’s conjecture. There have been many results obtained on this problem. In Section 3 we give an estimate of the total mean curvature for closed submanifolds in En+p with R ≥ 0. The main result of this paper is the following theorem.