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 {yt, t ∈ [0, 1]} such that (yt)|t=0 is conformally equivalent to a minimal surface in S n+2 and (yt)|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 W0 ∈ R + , we construct complete minimal surfaces in H 4 with Willmore energy being equal to W0. An example of complete minimal Möbius strip in H 4 with Willmore energy 6 √ 5π 5≈ 10.733π 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".