About the degenerate spectrum of the tension field for mappings into a symmetric Riemannian manifold (Q2012165)

Let \((M, g)\) and \((N, h)\) be Riemannian manifolds, and \(u: M \to N\) a smooth mapping. In [Afr. Mat., Sér. III 13, 87--109 (2002; Zbl 1027.58013)], \textit{J. Jost} together with the author introduced the following functional \[ E_\lambda(u) = \frac{1}{2} \left[ \int_M \| du\|^2 dv_g - \lambda \int_M d_N^2(u(x), w(x))\, dv_g\right] \] for some fixed \(\lambda \in {\mathbb R}\) and \(w\in C^1(M, N)\). Here \(d_N\) denotes the Riemannian distance function on \((N, h)\). And they proved that the Euler-Lagrange equation for \(E_\lambda\) is given by \[ L_\lambda(u):= \tau(u) - \lambda \exp_u^{-1}w = 0, \] where \(\tau(u)\) is the tension field of \(u\). A solution \(u\) of \(L_\lambda(u)=0\) is called an eigenmapping of the tension field \(\tau\) associated to the eigenvalue \(\lambda\). They showed that the spectrum of \(\tau\) in this sense may be continuous and the set of eigenvalues and eigenmappings may bifurcate even when \((N, h)\) has nonpositive sectional curvature. They also studied the second variation for \(E_\lambda(u)\), and from it, one can call \(V \in \Gamma(u^{-1}TN))\) a Jacobi field along a solution \(u\) when \[ \nabla_V L_\lambda(u):= \nabla_{e_i}\nabla_{e_i}V - \nabla_{{\nabla_{e_i}e_i}} V + R^N(V, du(e_i))du(e_i) - \lambda \nabla_V \exp^{-1}w = 0. \] In this paper, the author studies some qualitative properties on the solutions of the equations \(L_\lambda(u)=0\) and \(\nabla_V L_\lambda(u)=0\). The author proves, under some assumptions, that the set of all \(\lambda\) such that there exists a solution \((u, \lambda)\) of this nonlinear eigenvalue problem and a nontrivial Jacobi field \(V\) along \(u\) is countable and has no accumulation points in \({\mathbb R}\). Moreover it is shown that such a set of \(\lambda\) is contained in \({\mathbb R}^+\).