Minimally embedded Riemann surfaces in $\mb{S}^3$ and the conformal deformation of their metrics

Abstract

We prove that if $f_g: (\Sigma,g) \rightarrow (\mb{S}^{2+p},\tg)$ is a smooth minimal isometric embedding of a Riemannian surface $(\Sigma,g)$, and $[0,1]\ni t \rightarrow g_t$ is a path of area preserving conformal deformations of $g$ on $f_g(\Sigma)$, then there exists a path of conformal diffeomorphism $F_t: (\mb{S}^{2+p}, F_t^*\tg) \rightarrow (\mb{S}^{2+p},\tg)$ that starts at $\BOne_{\mb{S}^{2+p}}$, set theoretically fixes $f_g(\Sigma)$ for all $t$, and it is such that $F^{*}_t \tilde{g}\mid_{f_g(M)}=g_t$ with $f_{g_t}: (\Sigma,g_t) \rightarrow (\mb{S}^{2+p},\tg)$ a path of minimal embedding deformations of the initial $f_g$. We apply this result to the Lawson surface $( \Sigma,g)=(\xi_{k/m,m}, g_{\xi_{k/m,m}})$, $m|k>1$, to conclude that if $a=\mu_{g_{\xi_{k/m,m}}}(\Sigma)$, and $[0,1]\ni t \rightarrow g_t$ is a path of area $a$ metrics conformal deformations of $g_{k/m,m}$ to a metric $g_a$ of scalar curvature $4\pi \chi(\Sigma)/a$, then $f_{g_{\xi_{k/m,m}}}: (\xi_{k/m,m},g_{\xi_{k/m,m}}) \rightarrow (\mb{S}^3, \tg)$ has associated minimal isometric conformal deformations $f_{g_t}$ to the isometric embedding $f_{g_a}$ of $g_a$, in sharp contrast with the situation of the standard sphere $\xi_{0,1}$ and Clifford torus $(\xi_{1,1}$, which are the only orientable Riemannian surfaces of genus $0$ and $1$ isometrically embedded into $(\mb{S}^3,\tg)$ as minimal surfaces. If $\sigma^2(\Sigma):=\sup_{[g]\in \mc{C}(\Sigma)}(4\pi \chi( \Sigma))^2/\left(\frac{1}{4}\inf_{g\in [g]}\mc{W}_{f_g}(\Sigma)\right)$, $\mc{W}_{f_g}(\Sigma)$ the Willmore energy of $f_g$ and $\mc{C}( \Sigma)$ the space of classes, then $(4\pi \chi(\Sigma))^2/\left( \frac{1} {4}\mc{W}_{f_g}(\Sigma) \right) \leq \sigma^2(\Sigma)=(4\pi \chi( \Sigma))^2/\left(\frac{1}{4}\mc{W}_{f_{g_{\xi_{k,1}}}}(\Sigma) \right)$, and we describe the $f_g$s for which the equality is achieved.