Nonexistence of stable discrete maps into some homogeneous spaces of nonnegative curvature

Abstract

We consider stabilities for the weighted length or energy functional of a discrete map from a finite weighted graph $(X,m_{E})$ into a smooth Riemannian manifold $(M,g)$. We prove the non-existence of a stable discrete minimal immersion or a non-constant stable discrete harmonic map from a finite weighted graph into certain homogeneous spaces, such as K\"ahler $C$-spaces of positive holomorphic sectional curvature and some simply-connected compact Riemannian symmetric spaces.