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arXiv · 2606.21062

The prescribed Hermitian-Yang-Mills flow I

Zhiyao Xiong·Xaokui Yang·Shing-Tung Yau
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Abstract

In this paper, we introduce a broad class of flows, including the prescribed Hermitian-Yang-Mills flow: $$\frac{\partial h}{\partial t}=-Λ_{ω_g}\left(\sqrt{-1} R^h\right)+P$$ where $P\inΓ(M,E^*\otimes\bar{E}^*)$ is a prescribed Hermitian tensor associated with a holomorphic vector bundle $E$ over a Kähler (or Hermitian) manifold $(M,ω_g)$. We establish the long-time convergence of the flow to a limiting metric $h_{\infty}$ and use it to solve the prescribed Hermitian-Yang-Mills tensor equation $$Λ_{ω_g}\left(\sqrt{-1} R^{h_\infty}\right)=P, $$ for a general class of prescribed Hermitian tensors $P$. The crucial uniform $C^0$-estimate of $\{h(t)\}$ along the flow is obtained via a parabolic comparison principle.

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Zhiyao Xiong, Xaokui Yang, Shing-Tung Yau. 2026-09-05. The prescribed Hermitian-Yang-Mills flow I. https://arxiv.org/abs/2606.21062

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