An explicit iteration on Hermitian metrics on a stable bundle over a compact Gauduchon manifold converges smoothly to the unique Hermitian-Einstein metric.
The prescribed Hermitian-Yang-Mills flow I
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abstract
In this paper, we introduce a broad class of flows, including the prescribed Hermitian-Yang-Mills flow: $$\frac{\partial h}{\partial t}=-\Lambda_{\omega_g}\left(\sqrt R^h\right)+P$$ where $P\in\Gamma(M,E^*\otimes\bar{E}^*)$ is a prescribed Hermitian tensor associated with a holomorphic vector bundle $E$ over a K\"ahler (or Hermitian) manifold $(M,\omega_g)$. 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 $$\Lambda_{\omega_g}\left(\sqrt R^{h_\infty}\right)=P, $$ 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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math.DG 1years
2026 1verdicts
UNVERDICTED 1representative citing papers
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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.