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arXiv · 2304.02807

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

Takeshi Isobe·Tian Xu
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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.

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Takeshi Isobe, Tian Xu. 2023-04-06. Solutions of Spinorial Yamabe-type Problems on $S^m$: Perturbations and Applications. https://arxiv.org/abs/2304.02807

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