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

Let (M, g, sigma) be an m-dimensional closed spin manifold, with a fixed Riemannian metric g and a fixed spin structure sigma; let S(M) be the spinor bundle over M . The spinorial Yamabetype problems address the solvability of the following equation D-g psi = f (x)vertical bar psi vertical bar(2/m-1)(g) psi, psi : M -> S(M), x is an element of M where D-g 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) over tilde into R-3 with prescribed mean curvature f; meanwhile, for general dimensions and f equivalent to constant not equal 0, a solution provides an upper bound estimate for the Bar-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, 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 gbeing a Ckperturbation of the canonical round metric gS(m), which is not conformally flat somewhere on S-m; 2). fis a perturbation from constant and is of class C-2, while the background metric g= g(Sm). (c) 2024 Elsevier Inc. All rights reserved.