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Xiaojun Huang

Publications and source records attributed to Xiaojun Huang.

At least 19 recordsLinked to original sources
arXiv2026

A Hopf Lemma for Holomorphic Maps into Hyperquadrics

More than twenty years ago, Baouendi and the first author proved that a holomorphic map between hyperquadrics of the same signature is either totally degenerate or has a nonvanishing normal derivative for its normal component. This established a CR analogue of the classical Hopf lemma in arbitrary codimension, in the absence of pseudoconvexity. They further conjectured that the same Hopf-type property holds for holomorphic maps between Levi-nondegenerate hypersurfaces of the same signature. In this paper, we provide a counterexample to this conjecture in full generality. We also prove the conjecture when the target hypersurface is a hyperquadric of any codimension, arguably the most important case for applications.

math.CV↗
arXiv2026

Localization of Bergman Kernels and the Cheng-Yau Conjecture on Real Analytic Pseudoconvex Domains

In this paper, we establish the localization of Bergman kernels for unbounded pseudoconvex domains near boundary points of finite D'Angelo type. This result was proved by Engliš more than twenty years ago for bounded pseudoconvex domains and had remained as an open question in the unbounded setting. Related foundational work was carried out by Fefferman, Kerzman, Boutet de Monvel--Sjöstrand, Boas, Bell, and others. Combining our localization theorem with an extension theorem of Mir--Zaitsev, we prove that the Bergman metric of a bounded pseudoconvex domain with real-analytic boundary is Einstein if and only if the domain is biholomorphic to the unit ball. This result advances a longstanding conjecture of Cheng and Yau. A key step is to show that the Bergman metric of a smooth, possibly unbounded, pseudoconvex domain cannot be Kähler--Einstein if its boundary contains a non-strongly pseudoconvex \(h\)-extendible point. We further prove that a bounded weakly pseudoconvex real-analytic domain with a Kähler--Einstein Bergman metric must possess a weakly pseudoconvex \(h\)-extendible boundary point, thereby reducing the problem to the \(h\)-extendible setting. This paper draws deeply on, and reveals essential connections among many subfields of microlocal analysis, several complex variables, and complex geometry.

math.CV↗
arXiv2026

On Point Separation of Bergman Space on Stein Manifolds with Constant Holomorphic Sectional Curvature

We prove that the Bergman space of a Stein manifold separates points whenever its Bergman metric is well defined and has non-positive constant holomorphic sectional curvature. We construct examples of Stein manifolds whose Bergman metric is well defined and has positive constant holomorphic sectional curvature, while their Bergman spaces do not separate points. We also construct examples of Stein manifolds whose Bergman metric is well defined and has constant scalar curvature, which can be negative, zero, or positive, yet whose Bergman spaces do not separate points. Combined with previously established results in [HuLi1], this shows that a Stein manifold cannot admit a well-defined flat Bergman metric, and that it admits a well-defined Bergman metric with negative constant holomorphic sectional curvature if and only if it is biholomorphic to the unit ball of the same dimension, possibly with a pluripolar set removed. Our proof is based on Hörmander's $L^2$ estimates for $\overline\partial$- equations; the curvature condition, together with Calabi's rigidity and extension theorems, is used to construct the required bounded strictly plurisubharmonic functions. The construction of Stein manifolds with positive constant holomorphic sectional curvature for their Bergman metric is based on classical hyperelliptic Riemann surface theory and its higher-dimensional generalizations.

math.CV↗
arXiv2026

Ultralow shot noise limited giant passive resonant gyroscope for Earth rotation measurement

Optical gyroscopes directly measure the Earth's rotation and are promising instruments for real-time geophysical observations and Earth orientation parameter (EOP) determination requiring both high precision and high temporal resolution. Large-scale ring laser gyroscopes (RLGs) currently reach rotational resolutions around $10^{-11}\,\mathrm{(rad/s)/\sqrt{Hz}}$, but their quantum noise limits make it challenging to meet the requirements of future high-temporal-resolution EOP measurements. Passive resonant gyroscopes (PRGs), on the other hand, offer a potentially lower photon shot noise limit and more flexible power scaling, even if their demonstrated rotational resolutions are still about two orders of magnitude below those of leading RLGs. Here we demonstrate a $64\,\mathrm{m^{2}}$ giant passive resonant gyroscope HUST-2, and develop with an extremely low shot noise level. We experimentally obtain a shot noise limited of $5.7(1)\times10^{-13}\,\mathrm{(rad/s)/\sqrt{Hz}}$ at $1\,\mathrm{mW}$ incident optical power, following the characteristic $1/\sqrt{P}$ scaling. Through systematic suppression of dominant technical noise sources, HUST-2 further achieves a measured rotational resolution of $3\times10^{-11}\,\mathrm{(rad/s)/\sqrt{Hz}}$, bringing PRGs into the performance regime of leading large-scale RLGs for the first time. The gap between the present demonstrated rotational resolution and the shot noise limit indicates nearly two orders of magnitude further improvement potential. Reaching this limit would enable high-precision length-of-day (LOD) measurements with $10$-$100\,\mathrm{s}$ temporal resolution and lays the foundation for future large-scale gyroscope networks dedicated to real-time EOP determination.

physics.optics↗
arXiv2026

A note on csc Bergman metric

In this note, we show that if the Bergman metric of a pseudoconvex domain in $\mathbb C^n$($n\geq 3$) has constant scalar curvature, then every strongly pseudoconvex boundary point of the domain is spherical.

math.CV↗
arXiv2024

Bounding smooth Levi-flat hypersurfaces in a Stein manifold

This paper is concerned with the problem of constructing a smooth Levi-flat hypersurface locally or globally attached to a real codimension two submanifold in $\mathbb C^{n+1}$, or more generally in a Stein manifold, with elliptic CR singularities, a research direction originated from a fundamental and classical paper of E. Bishop. Earlier works along these lines include those by many prominent mathematicians working both on complex analysis and geometry. We prove that a compact smooth (or, real analytic) real codimension two submanifold $M$, that is contained in the boundary of a smoothly bounded strongly pseudoconvex domain, with a natural and necessary condition called CR non-minimal condition at CR points and with two elliptic CR singular points bounds a smooth-up-to-boundary (real analytic-up-to-boundary, respectively) Levi-flat hypersurface $\widehat{M}$. This answers a well-known question left open from the work of Dolbeault-Tomassini-Zaitsev, or a generalized version of a problem already asked by Bishop in 1965. Our study here reveals an intricate interaction of several complex analysis with other fields such as symplectic geometry and foliation theory.

math.CV↗
arXiv2024

Bergman metrics as pull-backs of the Fubini-Study metric

Domains and more generally complex manifolds whose Bergman metrics have constant holomorphic sectional curvature are characterized. Our approach is to treat the Bergman metrics as the pull-back by the Bergman-Bochner maps of the Fubini-Study metric of the complex projective space of infinite dimension. Several new domains with surprising curvature properties for their Bergman metrics are constructed. A new conjecture is also formulated at the end of the paper.

math.CV↗
arXiv2024

A Platform for All-optical Thomson/ Compton Scattering with Versatile Parameters

A dual-beam platform for all-optical electron-photon scattering, or Thomson/Compton scattering, with adjustable collision-angle and parameter tuning ability has been developed, which, in principle, can be used for the verification of strong-field quantum electrodynamics effects. Combining this platform with a 200 TW Ti:Sapphire laser system, we demonstrated the generation of inverse Compton scattering X/gamma-rays with tunable energies from tens of keV to MeV. The polarization of X/gamma radiation was manipulated by controlling the polarization of scattering laser. In the near future, by combining this experimental platform with multi-PW laser facilities, it is proposed to experimentally generate X/gamma radiation with orbital angular momentum for the nuclear isomer excitation, and more importantly, to explore the regime transition from nonlinear Thomson scattering to nonlinear Compton scattering, eventually to demonstrate the verification of theories on extremely strong field quantum electrodynamics effects.

hep-ex↗
arXiv2024

A note on weak mean equicontinuity and strong mean sensitivity

In this paper, we study the weak mean metric and give some properties by replacing the Besicovitch pseudometric with weak mean metric in the definition of mean equicontinuity and mean sensitivity. We study an opposite side of weak mean equicontinuity, strong mean sensitivity and we obtain a version of Auslander-Yorke dichotomies: minimal topological dynamical systems are either weak mean equicontinuous or strong mean sensitive, and transitive topological dynamical systemss are either almost weak mean equicontinuous or strong mean sensitive. Furthermore, motivated by the localized idea of sensitivity, we introduce some notions of new version sensitive tuples and study the properties of these sensitive tuples, we show that a transitive dynamical system is strong mean sensitive if and only if it admits a strong mean sensitive tuple. Finally, We introduce the notions of weakly mean equicontinuity of a topological dynamical system respect to a given continuous function $f$, and we show that a topological dynamical system is weakly mean equicontinuity then it is weakly mean equicontinuity with respect to every continuous function.

math.DS↗
arXiv2024

A note on weak Banach mean equicoontinuity

Consider a topological dynamical system $(X, T)$ endowed with the metric $d$. We introduce a novel function as $\overline{BF}(x, y) = \limsup_{n-m \rightarrow +\infty} \inf_{σ\in S_{n,m}} \frac{1}{n-m} \sum_{k=m}^{n-1} d\left(T^{k} x, T^{σ(k)} y\right)$, where the permutation group $S_{n,m}$ is utilized. It is demonstrated that $BF(x, y)$ exists when $x, y \in X$ are uniformly generic points. Leveraging this function, we introduce the concept of weak Banach mean equicontinuity and establish that the dynamical system $(X, T)$ exhibits weak Banach mean equicontinuity if and only if the uniform time averages $f_B^{*}(x) = \lim_{n-m \rightarrow +\infty} \frac{1}{n-m} \sum_{k=m}^{n-1} f\left(T^{k} x\right)$ are continuous for all $f \in C(X)$. Finally, we demonstrate that in the case of a transitive system, the equivalence between weak Banach mean equicontinuity and weak mean equicontinuity is established.

math.DS↗
arXiv2023

Regular types and order of vanishing along a set of non-integrable vector fields

This paper has two parts. We first survey recent efforts on the Bloom conjecture which still remains open in the case of complex dimension at least 4. Bloom's conjecture concerns the equivalence of three regular types. There is a more general important notion, called the singular D'Angelo type (or simply, D'Angelo type). While the finite D'Angelo type condition is the right one for the study of local subelliptic estimates for Kohn's $\overline{\partial}$-Neumann problem, regular types are important as their finiteness gives the global regularity up to the boundary of solutions of Kohn's $\overline{\partial}$-Neumann problem. In the second part of the paper, we provide a proof of a seemingly elementary but a truly fundamental property (Theorem 2.2 or its CR version Theorem 2.5) on the vanishing order of smooth functions along a system of non-integrable vector fields. A special case, Corollary 2.6, of Theorem 2.5 had already appeared in a paper of D'Angelo. A main goal in this part is to provide proofs for these results for the purpose of future references. Our arguments are based on a deep normalization theorem for a system of non-integrable vector fields due to Helffer-Nourrigat, as well as its late generalization in Boauendi-Rothschild.

math.CV↗
arXiv2020

Complex geodesics and complex Monge-Ampère equations with boundary singularity

We study complex geodesics and complex Monge-Ampère equations on bounded strongly linearly convex domains in $\mathbb C^n$. More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge-Ampère equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in $\mathbb C^n$.

math.CV↗
arXiv2019

The systems with almost Banach mean equicontinuity for abelian group actions

In this paper, we give the concept of Banach-mean equicontinuity and prove that three concepts, Bnanach-, Weyl- and Besicovitch-mean equicontinuity of a dynamic system with abelian group action are equivalent. Furthermore, we obtain that the topological entropy of a transitive, almost Banach-mean equicontinuous dynamical system with abelain group action is zero. As an application with our main result, we show that the topological entropy of the Banach-mean equicontinuous system under the action of an abelian groups is zero.

math.DS↗
arXiv2019

Bergman-Einstein metrics, hyperbolic metrics and Stein spaces with spherical boundaries

In this new version, we give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in $\mathbb{C}^n, n\geq 2,$ is Kähler-Einstein if and only if the domain is biholomorphic to the ball. We establish versions of various classical theorems that are used in the solution for Stein spaces. Among other things, we construct a hyperbolic metric over a Stein space with spherical boundary. We also prove the Q. K. Lu type uniformization theorem for Stein spaces with isolated normal singularities.

math.CV↗