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

TL;DR

This paper demonstrates the existence of smooth Willmore deformations connecting minimal surfaces in spheres and hyperbolic spaces, constructing explicit examples with prescribed Willmore energies and analyzing their stability properties.

Contribution

It establishes local and some global Willmore deformations between minimal surfaces in $S^{n+2}$ and $H^{n+2}$, including explicit constructions in $H^4$ and stability analysis of isotropic minimal surfaces.

Findings

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

Construction of complete minimal surfaces in $H^4$ with prescribed Willmore energy

Identification of non-isolated isotropic minimal surfaces in $S^4$

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".