Conformal embeddings of $S^2\to\mathbb{R}^3$ with prescribed mean curvature: A variational approach

TL;DR

This paper develops a variational approach to construct conformal immersions of the 2-sphere into three-dimensional space with prescribed mean curvature, inspired by spinorial methods related to the Yamabe problem.

Contribution

It introduces a novel variational framework combining spinorial techniques and the Weierstraß representation to solve the prescribed mean curvature problem for $S^2$.

Findings

Established existence results for conformal immersions with prescribed mean curvature.

Connected spinorial methods to classical geometric problems.

Extended the variational approach to a spinorial setting.

Abstract

Motivated by recent progress on a spinorial analogue of the Yamabe problem in the geometric literature, we study a conformally invariant spinor field equation on the $m$-sphere, $m\geq2$. Via variational methods and the spinorial Weierstra\ss\ representation, we study the problem of prescribing mean curvature for the immersion $S^2\to\mathbb{R}^3$.