Abstract
Let (M,g,σ) be an m-dimensional closed spin manifold, with a fixed Riemannian metric g and a fixed spin structure σ; let S(M) be the spinor bundle over M. The spinorial Yamabe-type problems address the solvability of the following equationDgψ=f(x)|ψ|g2m−1ψ,ψ:M→S(M),x∈M where Dg is the associated Dirac operator and f:M→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 M˜ into R3 with prescribed mean curvature f; meanwhile, for general dimensions and f≡constant≠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 (Sm,g) where the solution set of the spinorial Yamabe-type problem is not compact: 1). the geometric potential f is constant (say f≡1) with the background metric g being a Ck perturbation of the canonical round metric gSm, which is not conformally flat somewhere on Sm; 2). f is a perturbation from constant and is of class C2, while the background metric g≡gSm.
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