Quaternionic Analysis of Conformal Maps and the Willmore Functional

TL;DR

This paper extends quaternionic analysis to weakly conformal maps and provides a new proof that the Willmore functional reaches a minimum on compact Riemann surfaces, including maps with ramification points.

Contribution

It introduces an extension of quaternionic analysis to weakly conformal maps, allowing a new proof of the Willmore functional's minimization that includes ramified conformal maps.

Findings

The Willmore functional attains a minimum on all smooth conformal maps from compact Riemann surfaces.

Extension of quaternionic analysis to square-integrable potentials and weakly conformal maps.

Development of a Darboux transformation generalizing the erivative.

Abstract

Quaternionic analysis, which describes conformal maps from Riemann surfaces into $\mathbb{R}^3$ or $\mathbb{R}^4$, is extended to weakly conformal maps. As a consequence we present a new proof that on any compact Riemann surface $X$ the Willmore functional, the integral of the square of the mean curvature, attains a minimum on the space of smooth conformal maps from $X$ to $\mathbb{R}^3$ or $\mathbb{R}^4$. This was first proven by Kuwert and Sch\"atzle under the assumption that the infimum of the Willmore functional is less than $8\pi$. In this case all conformal maps are unbranched, due to an estimate of Li and Yau. Rivi\`ere removed this restriction by allowing as limits conformal maps with ramification points. Our approach admits these weakly conformal maps from the very beginning, by extending the quaternionic function theory as developed by Pedit and Pinkall to square-integrable potentials. In Part~I we carry over the most important properties of holomorphic functions to quaternionic analysis with square-integrable potentials. We develop a differential operator, the Darboux transformation, that generalizes the $\partial$-derivative and with respect to which these holomorphic functions are infinitely many times differentiable. It transforms the Kodaira representation of a weakly conformal map into its Weierstra{\ss} representation. Parts~II and~III treat respectively the existence of a minima on the space of weakly conformal maps with ramification points and the regularity of this minima.