SOLUTIONS OF SPINORIAL YAMABE-TYPE PROBLEMS ON Sm: PERTURBATIONS AND APPLICATIONS

This paper is part of a program to establish the existence theory for the conformally invariant Dirac equation Dg & psi; = f(x)I & psi;I 2 g & psi;m-1 on a closed spin manifold (M, g) of dimension m > 2 with a fixed spin structure, where f : M-+ 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 M  into R3 with prescribed mean curvature f; meanwhile, for general dimensions and f = constant, a solution provides an upper bound estimate for the Bar-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 (Sm, g): the geometric potential f is a perturbation from constant with g = gSm being the canonical round metric; and f = 1 with the metric g being a perturbation of gSm that is not conformally flat somewhere on Sm. 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 R3 with a common mean curvature are constructed, and furthermore, a strict inequality estimate for the Bar-Hijazi-Lott invariant on Sm, m > 4, is derived, which is the first result of this kind in the non-locally-conformally-flat setting.