Non-compactness results for the spinorial Yamabe-type problems with non-smooth geometric data

Let $(M,\mathit{\text{g}},\sigma )$ be an $m$-dimensional closed spin manifold, with a fixed Riemannian metric $\mathit{\text{g}}$ and a fixed spin structure $\sigma$; let $\mathbb{S}(M)$ be the spinor bundle over $M$. The spinorial Yamabe-type problems address the solvability of the following equation $$D_{\mathit{\text{g}}}\psi =f(x)\mathrm{\mid }\psi {\mathrm{\mid }}_{\mathit{\text{g}}}^{\frac{2}{m-1}}\psi ,\psi :M\to \mathbb{S}(M),x\in M$$ where $D_{\mathit{\text{g}}}$ is the associated Dirac operator and $f:M\to \mathbb{R}$ is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when $m=2$, a solution corresponds to a conformal isometric immersion of the universal covering $\tilde{M}$ into ${\mathbb{R}}^3$ with prescribed mean curvature $f$; meanwhile, for general dimensions and $f\equiv constant\mathrm{\ne }0$, a solution provides an upper bound estimate for the Bär-Hijazi-Lott invariant. The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold $(S^m,\mathit{\text{g}})$ where the solution set of the spinorial Yamabe-type problem is not compact: $1)\mathrm{.}$ the geometric potential $f$ is constant (say $f\equiv 1$) with the background metric $\mathit{\text{g}}$ being a $C^k$ perturbation of the canonical round metric ${\mathit{\text{g}}}_{S^m}$, which is not conformally flat somewhere on $S^m$; $2)\mathrm{.}$ $f$ is a perturbation from constant and is of class $C^2$, while the background metric $\mathit{\text{g}}\equiv {\mathit{\text{g}}}_{S^m}$.