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Zhiyao Xiong

Publications and source records attributed to Zhiyao Xiong.

6 recordsLinked to original sources
arXiv2026

The prescribed Hermitian-Yang-Mills flow I

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.

math.DG↗
arXiv2026

Iterative construction of Hermitian-Einstein metrics on stable bundles

Let $E$ be a stable holomorphic vector bundle over a compact Kähler (or Gauduchon) manifold $(M,ω_g)$. We show that for any real number $μ>0$ and any initial Hermitian metric $h_0$ on $E$, there exists a unique iteration sequence $\{h_m\}$ satisfying $$ Λ_{ω_g}\left(\sqrt{-1}R^{h_{m+1}}\right) =(λ_E-μ)h_{m+1}+μh_m, $$ and $\{h_m\}$ converges smoothly to a Hermitian-Einstein metric $h_\infty$ on $E$ satisfying $$ Λ_{ω_g}\left(\sqrt{-1}R^{h_{\infty}}\right) =λ_Eh_\infty, $$ where $λ_E\in \mathbb R$ is the stability constant. A key feature of this proof is that it is independent of Donaldson's variational framework and applies to non-Kähler manifolds.

math.DG↗
arXiv2026

The prescribed Hermitian-Yang-Mills flow II

We prove an analogue of the classical Donaldson-Uhlenbeck-Yau theorem by using the prescribed Hermitian-Yang-Mills flow. Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(M,ω_g)$. Suppose that for every proper coherent subsheaf $F\subset E$, the following inequality holds: $$ deg_{ω_g}(F)<deg_{ω_g}(E). $$ Then, for any initial Hermitian metric $h_0$ on $E$ and any positive-definite Hermitian tensor $P\in Γ(M,E^*\otimes \overline E^*)$, the prescribed Hermitian-Yang-Mills flow $$ \ \frac{\partial h}{\partial t} = -Λ_{ω_g}\left(\sqrt{-1}\, R^h\right) + P, $$ admits a global smooth solution on $[0,\infty)$. Moreover, as $t\rightarrow\infty$, the flow converges smoothly to a Hermitian metric $h_\infty$ on $E$ satisfying $$ Λ_{ω_g}\left(\sqrt{-1}\, R^{h_\infty}\right) = P. $$ As an application, we establish that on a Fano manifold $M$, for any Hermitian metric form $ω$ and any positive-definite Hermitian tensor $P\inΓ(M,T^{*1,0}M\otimes T^{*0,1}M)$, there exists a unique Hermitian metric tensor $h$ on $T^{1,0}M$ such that $$ Λ_ω\left(\sqrt R^h\right)=P.$$ This may be viewed as an analogue of the Calabi-Yau theorem for Fano manifolds.

math.DG↗
arXiv2024

RC-positivity, Schwarz's lemma and comparison theorems

It is well-known that the classical Schwarz lemma yields an explicit comparison of two Hermitian metrics with uniform constant negative curvature bounds through holomorphic maps between complex manifolds. In this paper, we establish Schwarz lemmas for holomorphic bundle maps between abstract Hermitian holomorphic vector bundles with various positive curvature bounds. As applications, we prove Schwarz lemmas for holomorphic maps between complex manifolds whose curvature tensors are described by the notion ``RC-positivity''. In particular, new diameter and volume comparison theorems are obtained by using Schwarz lemmas.

math.DG↗