On Jacobi Fields Along Eigenmappings of the Tension Field for Mappings into a Symmetric Riemannian Manifold

Abstract

We prove that the mean value (for some measure $\mu =\chi dx$ with $\chi \geq 0,dx=$ Riemannian measure) of the squared norm of the gradient of the unitary direction of a Jacobi field along an eigenmapping $v$ (associated to an eigenvalue $\lambda \geq 0$) of the tension field, for mappings from a compact Riemannian manifold $(M,g)$ into a symmetric Riemannian manifold $(N,h)$ of positive sectional curvature, is smaller than $c\lambda $, where $c>0$ depends only on the diameter and upper and lower curvature bounds of $(N,h)$. For negative $\lambda $, we prove that there is no nonvanishing Jacobi field along the eigenmappings, under the same assumptions on $(M,g)$ and $(N,h)$.