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Xiaokui Yang

Publications and source records attributed to Xiaokui Yang.

At least 19 recordsLinked to original sources
arXiv2026

Iterative construction of Hermitian-Einstein metrics on stable bundles

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

math.DG↗
arXiv2026

Human and AI collaboration for pulmonary nodule segmentation

Medical expert annotators are scarce, and blind reliance on artificial intelligence (AI) can be misleading, motivating approaches in which humans, particularly junior medical trainees or even non-medical personnel, collaborate with AI to achieve robust medical segmentation. Although the Segment Anything Model (SAM) shows promise for general-purpose image segmentation, its performance in human-AI collaboration for specialized medical tasks has not been thoroughly evaluated. Here we present Hi-Seg, a human-in-the-loop segmentation framework for pulmonary nodules built on SAM. Humans iteratively refine prompts through trial-and-error learning and semantic reasoning, progressively guiding SAM toward higher-quality masks. Using chest CT scans from 1,179 patients across 12 centers, we conducted the first large-scale external validation of collaborative human-SAM segmentation. Across all annotator groups, Hi-Seg achieved a mean Dice score of almost 85%, outperforming five state-of-the-art deep learning models by 10-22% and 13 SAM variants by 1-29%. Hi-Seg improved segmentation accuracy while reducing annotation time for medical annotators, and briefly trained non-medical annotators achieved performance comparable to that of the junior medical student. These findings suggest that human-in-the-loop segmentation can reduce clinician workload, enable scalable crowdsourced annotation, and transform clinical workflows by facilitating the safe and efficient integration of foundation models into routine clinical practice.

cs.CV↗
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 K\"ahler manifold $(M,\omega_g)$. Suppose that for every proper coherent subsheaf $F\subset E$, the following inequality holds: $$ deg_{\omega_g}(F)<deg_{\omega_g}(E). $$ Then, for any initial Hermitian metric $h_0$ on $E$ and any positive-definite Hermitian tensor $P\in \Gamma(M,E^*\otimes \overline E^*)$, the prescribed Hermitian-Yang-Mills flow $$ \ \frac{\partial h}{\partial t} = -\Lambda_{\omega_g}\left(\sqrt{-1}\, R^h\right) + P, $$ 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 $$ \Lambda_{\omega_g}\left(\sqrt{-1}\, R^{h_\infty}\right) = P. $$ As an application, we establish that on a Fano manifold $M$, for any Hermitian metric form $\omega$ and any positive-definite Hermitian tensor $P\in\Gamma(M,T^{*1,0}M\otimes T^{*0,1}M)$, there exists a unique Hermitian metric tensor $h$ on $T^{1,0}M$ such that $$ \Lambda_\omega\left(\sqrt R^h\right)=P.$$ This may be viewed as an analogue of the Calabi-Yau theorem for Fano manifolds.

math.DG↗
arXiv2026

Existence of twisted Hermitian-Einstein metrics on unstable vector bundles

In this paper, we demonstrate that twisted Hermitian-Einstein metrics on holomorphic vector bundles exist without obstruction. More precisely, for an arbitrary holomorphic vector bundle $E$ over a compact K\"ahler manifold $(M,\omega_g)$, we prove that the twisted Hermitian-Einstein equation $$\Lambda_{\omega_g}\left(\sqrt{-1}R^h\right) = \lambda h + P$$ admits a unique smooth solution $h$, provided that $P\in\Gamma(M,E^*\otimes\bar{E}^*)$ is positive-definite and $\lambda<\lambda_E^-$. The constant $\lambda_E^-$ is intrinsically associated with the stability constant of $E$. This result extends the classical Donaldson-Uhlenbeck-Yau (DUY) theorem for stable bundles and, in the limit $P\rightarrow0$, gives a new proof of the DUY theorem. As an application, we obtain an intrinsic Chern number inequality for unstable vector bundles: $$\int_M \left((r-1)c_1(E)^2 - 2rc_2(E)\right) \wedge \omega_g^{n-2} \leq \Bigl\lfloor \frac{r^2}{4} \Bigr\rfloor \frac{(\lambda_E^+-\lambda_E^-)^2}{4\pi^2 n^2} \int_M \omega_g^n.$$

math.DG↗
arXiv2026

Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors II

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.

math.DG↗
arXiv2026

Existence of Hermitian metrics with prescribed Hermitian-Yang-Mills tensors I

In this paper, we solve the prescribed Hermitian-Yang-Mills tensor problem. Let $ E $ be a holomorphic vector bundle over a compact K\"ahler 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^{h_0}\right) $ is positive-definite. Then for any positive-definite Hermitian tensor $ P\in \Gamma\left(M,E^*\otimes \overline E^*\right) $, there exists a unique smooth Hermitian metric $ h $ on $E$ such that $$\Lambda_{\omega_g} \left(\sqrt{-1} R^h\right)=P.$$ The proof is based on a new comparison theorem for Hermitian-Yang-Mills tensors. Inspired by these results, we have also derived quantitative Chern number inequalities that apply to both holomorphic vector bundles and compact K\"ahler manifolds.

math.DG↗
arXiv2025

Weitzenb\"ock-Bochner-Kodaira formulas with quadratic curvature terms

In this paper we establish new Bochner-Kodaira formulas with quadratic curvature terms on compact K\"ahler manifolds: for any $\eta\in \Omega^{p,q}(M)$, $$ \left\langle\Delta_{\overline \partial} \eta,\eta\right\rangle =\left\langle \Delta_{{\overline\partial}_F} \eta,\eta\right\rangle +\frac{1}{4}\left\langle \left(\mathcal {R} \otimes \mathrm{Id}_{\Lambda^{p+1,q-1}T^*M}\right)(\mathbb T_\eta),\mathbb T_\eta \right\rangle. $$ This linearized curvature term yields new vanishing theorems and provides estimates for Hodge numbers under exceptionally weak curvature conditions. Furthermore, we derive Weitzenb\"ock formulas with quadratic curvature terms on both Riemannian and K\"ahler manifolds.

math.DG↗
arXiv2025

Chern number identities on compact complex surfaces and applications

In this paper, we establish Chern number identities on compact complex surfaces. As an application, we prove that if $(M,g)$ is a compact Riemannian four-manifold with constant scalar curvature and admits a compatible complex structure $J$ such that the complexified Ricci curvature is a non-positive $(1,1)$ form, then $M$ is a K\"ahler surface.

math.DG↗
arXiv2025

Comparison theorems in Hermitian geometry I

This paper develops second variational formulas and index forms in the context of Hermitian geometry. Building upon these analytical foundations, we establish results analogous to classical theorems in Riemannian geometry, including Myers' theorem, Laplacian comparison theorems and volume comparison theorems.

math.DG↗
arXiv2025

First eigenvalue estimates on complete K\"ahler manifolds

Let $ (M,\omega_g) $ be a complete K\"ahler manifold of complex dimension $n$. We prove that if the holomorphic sectional curvature satisfies $\mathrm{HSC} \geq 2 $, then the first eigenvalue $\lambda_1$ of the Laplacian on $(M,\omega_g)$ satisfies $$ \lambda_1 \geq \frac{320(n-1)+576}{81(n-1)+144}.$$ This result is established through a new Bochner-Kodaira type identity specifically developed for holomorphic sectional curvature.

math.DG↗
arXiv2025

Conformal extremal metrics and constant scalar curvature

Let $M$ be a compact complex manifold of dimension $n\geq 2$. We prove that for any Hermitian metric $\omega$ on $M$, there exists a unique smooth function $f$ (up to additive constants) such that the conformal metric $\omega_g =e^f \omega$ solves the fourth-order nonlinear PDE $$\square_g^*(s_g|s_g|^{n-2})=0,$$ where $s_g$ is the Chern scalar curvature of $\omega_g$, and $\square_g^*$ denotes the formal adjoint of the complex Laplacian $\square_g=\mathrm{tr}_{\omega_g}\sqrt{-1}\partial\bar\partial$ with respect to $\omega_g$. This equation arises as the Euler-Lagrange equation of the $n$-Calabi functional $$C_{n}(\omega_g)=\int |s_g|^n\frac{\omega_g^n}{n!}$$ within the conformal class of $\omega_g$. Moreover, we show that the critical metric $\omega_g$ minimizes the $n$-Calabi functional within the conformal class $[\omega]$. In particular, if $\omega_g$ is a Gauduchon metric, then $\omega_g$ has constant Chern scalar curvature.

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↗
arXiv2021

Algebraic fibre spaces with strictly nef relative anti-log canonical divisor

Let $(X,Δ)$ be a projective klt pair, and $f:X\to Y$ a fibration to a smooth projective variety $Y$ with strictly nef relative anti-log canonical divisor $-(K_{X/Y}+Δ)$. We prove that $f$ is a locally constant fibration with rationally connected fibres, and the base $Y$ is a canonically polarized hyperbolic projective manifold. In particular, when $Y$ is a single point, we establish that $X$ is rationally connected. Moreover, when $\dim X=3$ and $-(K_X+Δ)$ is strictly nef, we prove that $-(K_X+Δ)$ is ample, which confirms the singular version of a conjecture of Campana-Peternell for threefolds.

math.AG↗