Proves global existence and smooth convergence of the prescribed Hermitian-Yang-Mills flow to a metric satisfying Λ_ω(√-1 R^h) = P for slope-stable holomorphic bundles, with an application to the tangent bundle on Fano manifolds.
RC-positivity, comparison theorems and prescribed Hermitian-Yang-Mills tensors II
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abstract
In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem for Higgs bundles over compact complex manifolds. Let $ (E,\theta) $ be a Higgs bundle over a compact Hermitian manifold $(M,\omega_g) $. Suppose that there exists a smooth Hermitian metric $ h_0 $ on $E$ such that the Hermitian-Yang-Mills tensor $ \Lambda_{\omega_g}\left(\sqrt{-1} R^{D^{h_0}}\right) $ of the Higgs connection is positive definite. Then for any Hermitian positive definite tensor $ P\in \Gamma\left(M,E^*\otimes \bar E^*\right) $, there exists a unique smooth Hermitian metric $ h $ on $E$ such that $$\Lambda_{\omega_g} \left(\sqrt{-1} R^{D^h}\right)=P.$$ We also establish quantitative Chern number inequalities for Higgs bundles.
fields
math.DG 4years
2026 4verdicts
UNVERDICTED 4representative citing papers
Introduces the prescribed Hermitian-Yang-Mills flow and proves its long-time convergence to a solution of Λ_ω_g(√R^h) = P for general prescribed P using a parabolic comparison principle for uniform C^0 bounds.
An explicit iteration on Hermitian metrics on a stable bundle over a compact Gauduchon manifold converges smoothly to the unique Hermitian-Einstein metric.
Dynamical construction of Hermitian-Einstein metrics on stable bundles using HYM iteration, extended to Higgs bundles, plus a new heat-flow proof for twisted prescribed HYM equations.
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The prescribed Hermitian-Yang-Mills flow II
Proves global existence and smooth convergence of the prescribed Hermitian-Yang-Mills flow to a metric satisfying Λ_ω(√-1 R^h) = P for slope-stable holomorphic bundles, with an application to the tangent bundle on Fano manifolds.
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The prescribed Hermitian-Yang-Mills flow I
Introduces the prescribed Hermitian-Yang-Mills flow and proves its long-time convergence to a solution of Λ_ω_g(√R^h) = P for general prescribed P using a parabolic comparison principle for uniform C^0 bounds.
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Iterative construction of Hermitian-Einstein metrics on stable bundles
An explicit iteration on Hermitian metrics on a stable bundle over a compact Gauduchon manifold converges smoothly to the unique Hermitian-Einstein metric.
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The Hermitian-Yang-Mills Iteration on Stable Bundles
Dynamical construction of Hermitian-Einstein metrics on stable bundles using HYM iteration, extended to Higgs bundles, plus a new heat-flow proof for twisted prescribed HYM equations.