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

TL;DR

This paper proves that stable discrete minimal immersions or harmonic maps do not exist when mapping from finite weighted graphs into certain positively curved homogeneous spaces, highlighting limitations in stability for these geometric configurations.

Contribution

It establishes the nonexistence of stable discrete minimal and harmonic maps into specific homogeneous spaces with positive curvature, extending stability theory to discrete graph settings.

Findings

No stable discrete minimal immersions into certain homogeneous spaces.

No non-constant stable discrete harmonic maps into these spaces.

Results apply to Kähler C-spaces with positive holomorphic sectional 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.