Variational Problems of Surfaces in a Sphere

• 作者: Bang Chao YIN
• 来源期刊: 数学学报（英文版）
• 年，卷（期）: 2021,（4）
• 页码范围: 657-665
• 页数: 9页
• 语种: 英文

摘要

Let x : M →Sn+p(1) be an n-dimensional submanifold immersed in an (n+p)-dimensional unit sphere Sn+p(1). In this paper,we study n-dimensional submanifolds immersed in Sn+p(1) which are critical points of the functional S(x) =∫MSn/2dv, where S is the squared length of the second fundamental form of the immersion x. When x : M →S2+p(1) is a surface in S2+p(1), the functional S(x) =∫MSn/2dv represents double volume of image of Gaussian map. For the critical surface of 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 S(x).