Closed surfaces with bounds on their Willmore energy

TL;DR

This paper establishes bounds on the conformal type and metrics of closed surfaces in R^3 and R^4 with bounded Willmore energy, showing how energy constraints influence geometric and conformal properties.

Contribution

It provides new uniform estimates relating Willmore energy bounds to the conformal and metric structure of surfaces, extending to higher genus surfaces and different ambient dimensions.

Findings

Bound on the conformal type in terms of energy bounds.

Uniform equivalence of metrics depending only on energy gap.

Sharp energy bounds for genus p surfaces in R^3 and R^4.

Abstract

The Willmore energy of a closed surface in R^n is the integral of its squared mean curvature, and is invariant uner M\"obius transformations of R^n. We show that any torus in R^3 with energy at most $8 \pi-delta$ has a representative under the M\"obius action, for which the induced metric and a conformal metric of constant (zero) curvature are uniformly equivalent, with constants depending only on $delta>0$. An analogous estimate is also obtained for surfaces of fixed genus $p \geq 1$ in R^3 or R^4, assuming suitable energy bounds which are sharp for n=3. Moreover the conformal type is controlled in terms of the energy bounds.