Variational problems of total mean curvature of submanifolds in a sphere

Abstract

Let H \mathbf {H} be the mean curvature vector of an n n -dimensional submanifold in a Riemannian manifold. The functional H = ∫ ‖ H ‖ n \mathcal {H}=\int \|\mathbf {H}\|^{n} is called the total mean curvature functional. In this paper, we present the first variational formula of H \mathcal {H} and then, for a critical surface of H \mathcal {H} in the ( 2 + p 2+p )-dimensional unit sphere S 2 + p \mathbb {S}^{2+p} , we establish the relationship between the integral of an extrinsic quantity of the surfaces and its Euler characteristic number.