Conformal invariants of isometric embeddings of the smooth metrics on a surface

We view all smooth metrics $g$ on a closed surface $\mathrm{\Sigma }$ through their Nash isometric embeddings $f_g:(\mathrm{\Sigma },g)\to ({\mathbb{S}}^{\tilde{n}},\tilde{g})$ into a standard sphere of large, but fixed, dimension $\tilde{n}$. We define the Willmore functional ${\mathcal{W}}_{f_g}$ over this space of metrics on $\mathrm{\Sigma }$ in terms of the extrinsic quantities of $f_g$. Its infimum over metrics in a conformal class is an invariant of the class varying differentiably with it. If $\mathrm{\Sigma }$ is oriented of genus $k$, when $k=0$, we use the gap theorem of Simons to show that there is a unique conformal class of metrics on $\mathrm{\Sigma }$, whose invariant $16\pi$ is the value for the standard totally geodesic embedding of ${\mathbb{S}}^2\hookrightarrow {\mathbb{S}}^{\tilde{n}}$, and we have that ${\mathcal{W}}_{f_g}(\mathrm{\Sigma })\ge 16\pi$, with the lower bound achieved if, and only if, $f_g$ is conformally equivalent to this standard geodesic embedding of ${\mathbb{S}}^2$, and ${\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}_g(\mathrm{\Sigma })\le 4\pi$, while when $k\ge 1$, the Lawson minimal surface $({\xi }_{k,1},g_{{\xi }_{k,1}})$ fixes the scale, and we show that ${\mathcal{W}}_{f_g}(\mathrm{\Sigma })\ge 4{\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}_{g_{{\xi }_{k,1}}}({\xi }_{k,1})$, with the lower bound achieved by $f_g$ if, and only if, $f_g$ is conformally equivalent to $f_{g_{{\xi }_{k,1}}}:({\xi }_{k,1},g_{{\xi }_{k,1}})\to ({\mathbb{S}}^3,\tilde{g})\hookrightarrow ({\mathbb{S}}^{\tilde{n}},\tilde{g})$, and ${\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}_g(\mathrm{\Sigma })\le {\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}}_{g_{{\xi }_{k,1}}}({\xi }_{k,1})$. For a nonoriented $\mathrm{\Sigma }$, we prove a likewise estimate from below for ${\mathcal{W}}_{f_g}(\mathrm{\Sigma })$, and characterize conformally the surface that realizes the optimal lower bound.