  We view all smooth metrics $g$ on a closed surface $\Sigma$ through their
Nash isometric embeddings $f_g: (\Sigma,g) \rightarrow (\mathbb{S}^{\tilde{n}},
\tilde{g})$ 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 $\Sigma$ 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 $\Sigma$ 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 $\Sigma$, 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 $\mathcal{W}_{f_g}(\Sigma) \geq
16\pi$, with the lower bound achieved if, and only if, $f_g$ is conformally
equivalent to this standard geodesic embedding of $\mathbb{S}^2$, and ${\rm
area}_g(\Sigma) \leq 4\pi$, while when $k\geq 1$, the Lawson minimal surface
$(\xi_{k,1},g_{\xi_{k,1}})$ fixes the scale, and we show that
$\mathcal{W}_{f_g}(\Sigma) \geq 4\, {\rm area}_{g_{\xi_{k,1}}} (\xi_{k,1})$,
with the lower bound achieved by $f_g$ if, and only if, $f_g$ is conformally
equivalent to $f_{g_{\xi_{k,1}}} : (\xi_{k,1}, g_{\xi_{k,1}}) \rightarrow
(\mathbb{S}^3,\tilde{g}) \hookrightarrow (\mathbb{S}^{\tilde{n}},\tilde{g})$,
and ${\rm area}_g(\Sigma)\leq {\rm area}_{g_{\xi_{k,1}}} (\xi_{k,1})$. For a
nonoriented $\Sigma$, we prove a likewise estimate from below for
$\mathcal{W}_{f_g}(\Sigma)$, and characterize conformally the surface that
realizes the optimal lower bound.
