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

Publication date: 1 November 2020

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 i n [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} i n m a t h b b 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

f r a c 6 s q r t 5 p i 5 a p p r o x 10.733 p i

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".

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