Solutions of Spinorial Yamabe-type Problems on $S^m$: Perturbations and Applications

Abstract

This paper is part of a program to establish the existence theory for the conformally invariant Dirac equation \[ D_{\textit{g}}ψ=f(x)|ψ|_{\textit{g}}^{\frac2{m-1}}ψ\] on a closed spin manifold $(M,\text{g})$ of dimension $m\ge 2$ with a fixed spin structure, where $f:M\to \mathbb{R}$ is a given function. The study on such nonlinear equation is motivated by its important applications in Spin Geometry: when $m=2$, a solution corresponds to an isometric immersion of the universal covering $\tilde{M}$ into ${\mathbb{R}}^3$ with prescribed mean curvature $f$; meanwhile, for general dimensions and $f\equiv constant$, a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant.