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

TL;DR

This paper investigates the existence of solutions to a conformally invariant Dirac equation on closed spin manifolds, with applications in spin geometry including isometric immersions and invariant estimates.

Contribution

It advances the theory of spinorial Yamabe-type problems by establishing existence results for the Dirac equation with prescribed functions on manifolds.

Findings

Existence results for solutions to the Dirac equation on spin manifolds.

Connections between solutions and geometric properties like isometric immersions.

Upper bounds for the Bär-Hijazi-Lott invariant in certain cases.

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.