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 \[ D_g\psi =f(x)|\psi |_g^{\frac 2{m-1}}\psi \] on a closed spin manifold $(M,g)$ of dimension $m\geq 2$ 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 $\widetilde M$ 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 Bär-Hijazi-Lott invariant.

Comparing with the existing issues, the aim of this paper is to establish multiple existence results in a new geometric context, which have not been considered in the previous literature. Precisely, in order to examine the dependence of solutions of the aforementioned nonlinear Dirac equations on geometrical data, concrete analyses are made for two specific models on the manifold $(S^m,g)$: the geometric potential $f$ is a perturbation from constant with $g=g_{S^m}$ being the canonical round metric; and $f\equiv 1$ with the metric $g$ being a perturbation of $g_{S^m}$ that is not conformally flat somewhere on $S^m$. The proof is variational: solutions of these problems are found as critical points of their corresponding energy functionals. The emphasis is that the solutions are always degenerate: they appear as critical manifolds of positive dimension. This is very different from most situations in elliptic PDEs and classical critical point theory. As corollaries of the existence results, multiple distinct embedded spheres in $\mathbb {R}^3$ with a common mean curvature are constructed, and furthermore, a strict inequality estimate for the Bär-Hijazi-Lott invariant on $S^m$, $m\geq 4$, is derived, which is the first result of this kind in the non-locally-conformally-flat setting.