论文标题
$ h^{n+2} $和$ s^{n+2} $之间的最小表面之间的Willmore变形
Willmore deformations between minimal surfaces in $H^{n+2}$ and $S^{n+2}$
论文作者
论文摘要
在本文中，我们表明，在本地存在$ s^{n+2} $的最小表面之间存在一个较小的变形，而最小表面则是$ h^{n+2} $，即，存在一个光滑的willmore表面$ \ {y___t，t \ in [0,1]等效于$ s^{n+2} $和$（y_t）| _ {t = 1} $在$ h^{n+2} $中的最小表面上等效。在某些情况下，变形是全局的。考虑一下Veronese两杆的Willmore变形及其在$ s^4 $中的概括，对于任何正数$ W_0 \ in \ Mathbb r^+$，我们在$ H^4 $中构建完整的最小表面，而Willmore Energy等于$ W_0 $。 $ h^4 $中的完整最小Möbius带有Willmore Energy $ \ frac {6 \ sqrt {5}π} {5} {5} \lot10.7333π$。我们还表明，$ s^4 $中的所有各向同性最小表面都接纳了与杀死场不同的雅各比田地，即它们不是“隔离的”。
In this paper we show that locally there exists a Willmore deformation between minimal surfaces in $S^{n+2}$ and minimal surfaces in $H^{n+2}$, i.e., there exists a smooth family of Willmore surfaces $\{y_t,t\in[0,1]\}$ such that $(y_t)|_{t=0}$ is conformally equivalent to a minimal surface in $S^{n+2}$ and $(y_t)|_{t=1}$ is conformally equivalent to a minimal surface in $H^{n+2}$. For some cases the deformations are global. Consider the Willmore deformations of the Veronese two-sphere and its generalizations in $S^4$, for any positive number $W_0\in\mathbb R^+$, we construct complete minimal surfaces in $H^4$ with Willmore energy being equal to $W_0$. An example of complete minimal Möbius strip in $H^4$ with Willmore energy $\frac{6\sqrt{5}π}{5}\approx10.733π$ is also presented. We also show that all isotropic minimal surfaces in $S^4$ admit Jacobi fields different from Killing fields, i.e., they are not "isolated".
