About the Degenerate Spectrum of the Tension Field for Mappings into a Symmetric Riemannian Manifold
Moussa Kourouma · Project Euclid (Cornell University) · 2017
Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds, where $(N,h)$ is symmetric, $v\\in W^{1,2}((M,g),(N,h))$, and $\\tau $ is the tension field for mappings from $(M,g)$ into $(N,h)$. We consider the nonlinear eigenvalue problem $\\tau (u)-\\lambda \\exp _{u}^{-1}v=0$, for $u$ $\\in W^{1,2}(M,N)$ such that $u_{\\left\\vert \\partial M\\right. }=v_{\\left\\vert \\partial M\\right.}$, and $\\lambda \\in \\mathbb{R}$. We prove, under some assumptions, that the set of all $\\lambda $, such that there exists a solution $(u,\\lambda )$ of this problem and a non trivial Jacobi field $V$ along $u$, is contained in $\\mathbb{R}_{+}$, is countable, and has no accumulation point in $\\mathbb{R}$. This result generalizes a well known one about the spectrum of the Laplace-Beltrami operator $\\Delta $ for functions from $(M,g)$ into $\\mathbb{R}$.