Willmore deformations between minimal surfaces in $H^{n+2}$ and $S^{n+2}$

Abstract

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 M\"{o}bius strip in $H^4$ with Willmore energy $\frac{6\sqrt{5}\pi}{5}\approx10.733\pi$ 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".