RC-positivity and rigidity of harmonic maps into Riemannian manifolds

TL;DR

This paper proves that harmonic maps from compact Kähler manifolds with RC-positive curvature to Riemannian manifolds with non-positive complex sectional curvature must be constant, revealing rigidity properties of such maps.

Contribution

It establishes a new rigidity theorem for harmonic maps under RC-positivity and non-positive curvature conditions, extending understanding of harmonic map behavior.

Findings

Harmonic maps from RC-positive Kähler manifolds to non-positive curvature Riemannian manifolds are constant.

No non-constant harmonic maps exist from positively curved Kähler manifolds to non-positive curvature targets.

The result generalizes previous rigidity theorems in harmonic map theory.

Abstract

In this paper, we show that every harmonic map from a compact K\"ahler manifold with uniformly RC-positive curvature to a Riemannian manifold with non-positive complex sectional curvature is constant. In particular, there is no non-constant harmonic map from a compact K\"ahler manifold with positive holomorphic sectional curvature to a Riemannian manifold with non-positive complex sectional curvature.