Quantization of the Willmore Energy in Riemannian Manifolds

TL;DR

This paper proves that the Willmore energy of spheres and surfaces of arbitrary genus in Riemannian manifolds quantizes under certain boundedness and convergence conditions, extending previous results in geometric analysis.

Contribution

It establishes the energy quantization for Willmore surfaces of arbitrary genus under weak convergence and boundedness assumptions, generalizing known results for spheres.

Findings

Energy quantization holds for Willmore spheres with bounded energy and area.

The result extends to arbitrary genus surfaces under weak convergence and moduli space compactness.

Provides a framework for understanding energy distribution in geometric variational problems.

Abstract

We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area are uniformly bounded. The analogous energy quantization result holds for Willmore surfaces of arbitrary genus, under the additional assumptions that the immersion maps weakly converge to a limiting (possibly branched, weak immersion) map from the same surface, and that the conformal structures stay in a compact domain of the moduli space.