Quaternionic Analysis of Conformal Maps and the Willmore Functional

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 Schätzle under the assumption that the infimum of the Willmore functional is less than $8π$. In this case all conformal maps are unbranched, due to an estimate of Li and Yau. Rivière 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 Weierstraß 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.