Localized Curvature Domination and Rigidity of Harmonic Maps

TL;DR

This paper proves a new localized rigidity theorem for harmonic maps between Riemannian manifolds, replacing global curvature conditions with image-dependent bounds, leading to conditions under which the map must be constant or totally geodesic.

Contribution

It introduces a localized curvature domination condition for harmonic maps, extending classical rigidity results to image-dependent curvature bounds.

Findings

Harmonic maps are constant if domain Ricci curvature dominates localized curvature bounds.

At the critical threshold, harmonic maps have parallel differentials and totally geodesic images.

The theorem generalizes Yano-Ishihara-type rigidity using localized curvature conditions.

Abstract

We establish a localized Bochner-type rigidity theorem for harmonic maps between Riemannian manifolds. Let $f : (M,g) \to (\overline{M},\overline{g})$ be a harmonic map from a compact manifold. Instead of assuming a global nonpositivity condition on the sectional curvature of the target, we impose a curvature bound localized to the image $f(M)$, expressed via the maximal sectional curvature encountered along the image. We prove that if the minimal Ricci curvature of the domain dominates this image-dependent curvature bound in a sharp quantitative pinching inequality involving the maximal energy density of $f$, then the map is constant. At the critical threshold, we obtain a homothetic classification: the differential is parallel and the image is totally geodesic. The result replaces global curvature sign assumptions with an image-dependent curvature domination principle and yields a localized analogue of Yano-Ishihara-type rigidity.