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Variational Problems of Surfaces in a Sphere
Acta Mathematica Sinica, English Series ( IF 0.8 ) Pub Date : 2020-11-24 , DOI: 10.1007/s10114-020-9441-y
Bang Chao Yin

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 $${\cal S}(x)$$ .

中文翻译:

球面的变分问题

令 $$x:M \to \mathbb{S}{^{n + p}}(1)$$ 是一个浸入 (n + p) 维单位球体 $$\mathbb{S 中的 n 维子流形}{^{n + p}}(1)$$ 。在本文中,我们研究了沉浸在 $$\mathbb{S}{^{n + p}}(1)$$ 中的 n 维子流形,它们是泛函 $${\cal S}(x) = \int_M {{S^{{n \over 2}}}} dv$$ ,其中 S 是浸没 x 的第二基本形式的平方长度。当 $$x:M \to \mathbb{S}{^{2 + p}}(1)$$ 是 $$\mathbb{S}{^{2 + p}}(1)$$ 中的曲面,泛函 $${\cal S}(x) = \int_M {{S^{{n \over 2}}}} dv$$ 表示高斯图图像的两倍体积。对于$${\cal S}(x)$$ 的临界面,我们得到了表面的一个外在量的积分与其欧拉特性之间的关系。此外,
更新日期:2020-11-24
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