Localized Curvature Domination and Rigidity of Harmonic Maps

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.