Long-time behavior of the Hermitian-Yang-Mills flow on non-K\"ahler manifolds

TL;DR

This paper investigates the long-term evolution of the Hermitian-Yang-Mills flow on compact Hermitian manifolds, revealing eigenvalue convergence to geometric invariants and extending classical questions to non-Kähler settings.

Contribution

It establishes eigenvalue bounds, proves convergence to invariants in the Gauduchon case, and generalizes key results to non-Kähler manifolds.

Findings

Eigenvalues of mean curvature are monotonic along the flow.

Eigenvalues converge to invariants determined by the Harder-Narasimhan type.

Extension of Atiyah-Bott-Bando-Siu question to non-Kähler manifolds.

Abstract

In this paper, we study the long-time behavior of the Hermitian-Yang-Mills flow over compact Hermitian manifolds. We obtain the monotonicity of lower bound and upper bound of the eigenvalues of the mean curvature along the Hermitian-Yang-Mills flow. In the Gauduchon case, we show that the eigenvalues of the mean curvature converge to geometric invariants determined by the Harder-Narasimhan type. Furthermore, we generalize the Atiyah-Bott-Bando-Siu question to the non-K\"ahler case.