Willmore deformations between minimal surfaces in Hn+2 and Sn+2

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 : U ~ → S n + 2 , t ∈ [ 0 , 2 π ) } such that ( y t ) | t = 0 is conformally equivalent to a minimal surface in S n + 2 and ( y t ) | t = π / 2 is conformally equivalent to a minimal surface in H n + 2 . Here U ~ is a simply connected open subset of the surface M . For some cases the deformations are global. By the Willmore deformations of the Veronese two-sphere and its generalizations in S 4 , for any positive number W 0 ∈ 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 6 5 π 5 ≈ 10.733 π 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”.