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

On symmetric Willmore surfaces in spheres II: The orientation reversing case

“…Usually dressing action concerns elements in the loop group. In [73], we realized that one can use the most simple elements, i.e., elements in K C \ K since such elements also induce a non-trivial dressing action on harmonic maps into G/K. In fact, by use of K C -dressing actions on the conformal Gauss map of a Willmore surface, we built an interesting Willmore deformation of minimal surfaces in S n+2 and in H n+2 .…”

“…Applying this deformation to the Veronese surface in S 4 and its generalization, one can derive complete minimal surfaces in H 4 with any positive Willmore energy. We refer to [73] for the explicit expressions of these examples. 10.…”

“…(Theorem 1.1 of[73]) Let y : U → S n+2 be a minimal surface from a simply connected open subset U ⊂ M . There exists a family of Willmore surfaces y t : U ′ ⊂ U → S n+2 , t ∈ [0, π], such that y t | t=0 = y and y t | t=π/2 is conformally equivalent to a minimal surface in H n+2 .…”