The existence of constrained Willmore surfaces in R^3 and R^4

Date: Thursday 14th August 2025

Abstract

The Willmore energy of a immersion of a closed surface is the integral of the square of its mean curvature. This is a measure of how far a surface is from being a sphere. It is a conformal invariant. A constrained Willmore surface is a critical point of this energy functional under deformations that preserve the conformal class of the surface. By describing a surface in terms of "holomorphic" data, the Kodaira and Weierstrass representations, and formulating a a corresponding "weak" problem, we were able to take limits of sequences of immersions and prove the existence of minimizers in each conformal class.