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

We prove that if $f_g:(\mathrm{\Sigma },g)\to (\text{mb}{S}^{2+p},\mathrm{tg})$ is a smooth minimal isometric embedding of a Riemannian surface $(\mathrm{\Sigma },g)$, and $[0,1]\ni t\to g_t$ is a path of area preserving conformal deformations of $g$ on $f_g(\mathrm{\Sigma })$, then there exists a path of conformal diffeomorphism $F_t:(\text{mb}{S}^{2+p},F_t^{\ast }\mathrm{tg})\to (\text{mb}{S}^{2+p},\mathrm{tg})$ that starts at ${\text{BOne}}_{\text{mb}{S}^{2+p}}$, set theoretically fixes $f_g(\mathrm{\Sigma })$ for all $t$, and it is such that $F_t^{\ast }\tilde{g}{\mid }_{f_g(M)}=g_t$ with $f_{g_t}:(\mathrm{\Sigma },g_t)\to (\text{mb}{S}^{2+p},\mathrm{tg})$ a path of minimal embedding deformations of the initial $f_g$. We apply this result to the Lawson surface $(\mathrm{\Sigma },g)=({\xi }_{k\mathrm{/}m,m},g_{{\xi }_{k\mathrm{/}m,m}})$, $m\mathrm{\mid }k>1$, to conclude that if $a={\mu }_{g_{{\xi }_{k\mathrm{/}m,m}}}(\mathrm{\Sigma })$, and $[0,1]\ni t\to g_t$ is a path of area $a$ metrics conformal deformations of $g_{k\mathrm{/}m,m}$ to a metric $g_a$ of scalar curvature $4\pi \chi (\mathrm{\Sigma })\mathrm{/}a$, then $f_{g_{{\xi }_{k\mathrm{/}m,m}}}:({\xi }_{k\mathrm{/}m,m},g_{{\xi }_{k\mathrm{/}m,m}})\to (\text{mb}{S}^3,\mathrm{tg})$ 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 $(\text{mb}{S}^3,\mathrm{tg})$ as minimal surfaces. If ${\sigma }^2(\mathrm{\Sigma }):={\sup }_{[g]\in \text{mc}C(\mathrm{\Sigma })}(4\pi \chi (\mathrm{\Sigma }){)}^2\mathrm{/}(\frac{1}{4}{\inf }_{g\in [g]}\text{mc}{W}_{f_g}(\mathrm{\Sigma }))$, $\text{mc}{W}_{f_g}(\mathrm{\Sigma })$ the Willmore energy of $f_g$ and $\text{mc}C(\mathrm{\Sigma })$ the space of classes, then $(4\pi \chi (\mathrm{\Sigma }){)}^2\mathrm{/}(\frac{1}{4}\text{mc}{W}_{f_g}(\mathrm{\Sigma }))\le {\sigma }^2(\mathrm{\Sigma })=(4\pi \chi (\mathrm{\Sigma }){)}^2\mathrm{/}(\frac{1}{4}\text{mc}{W}_{f_{g_{{\xi }_{k,1}}}}(\mathrm{\Sigma }))$, and we describe the $f_g$s for which the equality is achieved.