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

Abstract

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.