S. S. Chern posed a problem as follows. Consider the set of all closed minimal hypersurfaces in S<sup>n+1</sup>（1） with constant scalar curvature. Take the scalar curvature and the square of the length o...S. S. Chern posed a problem as follows. Consider the set of all closed minimal hypersurfaces in S<sup>n+1</sup>（1） with constant scalar curvature. Take the scalar curvature and the square of the length of the second fundamental form as a function on this set. Is the image of this function a discrete set of positive numbers?展开更多
设 M 是单位球面 S^(n+1)中的一个闭极小浸入超曲面,h 是 M 的第二基本形式,s 是 h 的模长的平方。根据 Simons 已得到的结果,若在 M 上有0≤s≤n,则 s=0或 n。本文讨论如下问题:s 是否有另一个较大的值?若有,这个值是什么?此问题收集到...设 M 是单位球面 S^(n+1)中的一个闭极小浸入超曲面,h 是 M 的第二基本形式,s 是 h 的模长的平方。根据 Simons 已得到的结果,若在 M 上有0≤s≤n,则 s=0或 n。本文讨论如下问题:s 是否有另一个较大的值?若有,这个值是什么?此问题收集到[7],我们得到定理设 M 是 S^(n+1)中的闭定向极小浸入超曲面,若 s 为大于 n 的常数,则s>n+(5-17^(1/2))/(3+17^(1/2))n>n+(n/9)展开更多
Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where...Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where 5" is the square of the length of the second fundamental form of M,‖·‖<sub>2</sub> denotes the L<sup>2</sup>-norm on M.In this paper,we generalize Hsu’s result to any compact surfaces in S<sup>3</sup> with constant mean curvature.展开更多
文摘S. S. Chern posed a problem as follows. Consider the set of all closed minimal hypersurfaces in S<sup>n+1</sup>（1） with constant scalar curvature. Take the scalar curvature and the square of the length of the second fundamental form as a function on this set. Is the image of this function a discrete set of positive numbers?
文摘设 M 是单位球面 S^(n+1)中的一个闭极小浸入超曲面,h 是 M 的第二基本形式,s 是 h 的模长的平方。根据 Simons 已得到的结果,若在 M 上有0≤s≤n,则 s=0或 n。本文讨论如下问题:s 是否有另一个较大的值?若有,这个值是什么?此问题收集到[7],我们得到定理设 M 是 S^(n+1)中的闭定向极小浸入超曲面,若 s 为大于 n 的常数,则s>n+(5-17^(1/2))/(3+17^(1/2))n>n+(n/9)
文摘Let M be a compact minimal surface in S<sup>3</sup>.Y.J.Hsu proved that if ‖S‖<sub>2</sub>≤2(2<sup>1/2</sup>π, then M is either the equatorial sphere or the Clifford torus,where 5" is the square of the length of the second fundamental form of M,‖·‖<sub>2</sub> denotes the L<sup>2</sup>-norm on M.In this paper,we generalize Hsu’s result to any compact surfaces in S<sup>3</sup> with constant mean curvature.
