Willmore deformations between minimal surfaces in ℍ^n+2 and 𝕊^n+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:Ũ→ S^n+2,t∈ [0,2π )} 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 Ũ 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≈ 10.733π 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”.