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 \[ D_{\textit{g}}\psi=f(x)|\psi|_{\textit{g}}^{\frac2{m-1}}\psi \] on a closed spin manifold $(M,\textit{g})$ of dimension $m\geq2$ 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 $\widetilde M$ 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 B\"ar-Hijazi-Lott invariant.