Uniqueness for the Kähler-Yang-Mills equations

Abstract

A formula for the $\alpha$-K-energy functional for the Kähler-Yang-Mills (KYM) equations is provided in this paper. Using this formula and Chen's $\epsilon$-geodesic equation on the space of Kähler potentials, we prove that if a solution exists to the KYM equations for a simple vector bundle on a Kähler manifold with discrete automorphism group, then it is unique and the $\alpha$-K-energy is bounded from below. Inspired by the study of constant scalar curvature Kähler metrics, we introduce a coupled J-equation to study the $\alpha$-K energy functional.

24 pages. Comments are most welcome. No significant AI usage (only proof-reading and simplifying a couple of proofs by ChatGPT 5.6 Sol)