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In this paper we show that locally there exists a Willmore deformationbetween minimal surfaces in $S^{n+2}$ and minimal surfaces in $H^{n+2}$, i.e.,there exists a smooth family of Willmore surfaces $\{y_t,t\in[0,1]\}$ 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 $H^{n+2}$. Forsome cases the deformations are global. Consider the Willmore deformations ofthe Veronese two-sphere and its generalizations in $S^4$, for any positivenumber $W_0\in\mathbb R^+$, we construct complete minimal surfaces in $H^4$with Willmore energy being equal to $W_0$. An example of complete minimalM\"{o}bius strip in $H^4$ with Willmore energy$\frac{6\sqrt{5}\pi}{5}\approx10.733\pi$ is also presented. We also show thatall isotropic minimal surfaces in $S^4$ admit Jacobi fields different fromKilling fields, i.e., they are not "isolated".

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