Variational Problems of Surfaces in a Sphere

Abstract

Let $x:M \to \mathbb{S}{^{n + p}}(1)$ be an n-dimensional submanifold immersed in an (n + p)-dimensional unit sphere $\mathbb{S}{^{n + p}}(1)$. In this paper, we study n-dimensional submanifolds immersed in $\mathbb{S}{^{n + p}}(1)$ which are critical points of the functional ${\cal S}(x) = \int_M {{S^{{n \over 2}}}} dv$, where S is the squared length of the second fundamental form of the immersion x. When $x:M \to \mathbb{S}{^{2 + p}}(1)$ is a surface in $\mathbb{S}{^{2 + p}}(1)$, the functional ${\cal S}(x) = \int_M {{S^{{n \over 2}}}} dv$ represents double volume of image of Gaussian map. For the critical surface of ${\cal S}(x)$, we get a relationship between the integral of an extrinsic quantity of the surface and its Euler characteristic. Furthermore, we establish a rigidity theorem for the critical surface of ${\cal S}(x)$.