Variational problems of total mean curvature of submanifolds in a sphere

Abstract:

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