Conformal invariants of isometric embeddings of the smooth metrics of a surface
Streszczenie
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 $\tilde {n}$, and analyze the extrinsic quantities of these embeddings corresponding to variations of the metrics within a fixed conformal class. 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. We identify geometrically one class for which this invariant is the smallest, set the scale using its Einstein representative of minimal isometric embedding, and compute the invariant of an arbitrary class by using its Einstein representative of area equal to this scale. 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 $\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 $\mathbb {S}^2$, while when $k\geq 1$, the Lawson minimal surface $(\xi _{k,1},g_{\xi _{k,1}})$ fixes the scale, and we show that $\mathcal {W}_{f_g}(\Sigma ) \geq 4\, \mathrm{area}_{g_{\xi _{k,1}}} (\xi _{k,1})$, with the lower bound achieved by $f_g$ if and only if $f_g$ is conformally equivalent to $f_{g_{\xi _{k,1}}} : (\xi _{k,1}, g_{\xi _{k,1}}) \rightarrow (\mathbb {S}^3,\tilde {g}) \hookrightarrow (\mathbb {S}^{\tilde {n}},\tilde {g})$. 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.
