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Weixia Zhu

Publications and source records attributed to Weixia Zhu.

7 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

Degree-Three Rational Sphere Maps: Sharp Denominator Region and Gram Normal Forms

We study degree-three rational sphere maps in two complex variables. After a standard normalization, the denominator of such a map takes the form \[ g_σ(z)=1+σ_1 z_1^2+σ_2 z_2^2, \qquad σ_1,σ_2\geq 0. \] A basic question is: which pairs $(σ_1,σ_2)$ can actually occur as the denominator of a degree-three rational sphere map? The first main result of the paper gives a complete answer: such a denominator occurs if and only if \[ 0\leq σ_1,σ_2<1, \qquad \sqrt{1-σ_1^2}+\sqrt{1-σ_2^2}>1. \] Our approach converts the sphere-mapping condition into a finite-dimensional Gram-matrix positivity problem. Furthermore, for each admissible parameter $ σ=(σ_1,σ_2), $ we determine all possible minimal target dimensions in which the corresponding denominator $g_σ$ can be realized. We also give a Gram-matrix normal form for maps with a fixed denominator and compute, for each admissible $σ$, the dimension of the moduli space of equivalence classes of rational sphere maps realizing $g_σ$. Finally, we extend the Gram-matrix method to arbitrary source dimension and obtain a general sufficient condition for the existence of degree-three rational sphere maps.

math.CV↗
arXiv2026

Semi-classical heat kernel asymptotics on complex manifolds with boundary

Let $M$ be a relatively compact open subset of a complex manifold $M'$ with smooth boundary $X$ and let $L$ be a holomorphic line bundle over $M'$. Assuming that condition $Z(q)$ holds, we establish the semi-classical asymptotic behavior of $e^{-\frac{t}{k}\Box^{q}_k}$ near the boundary $X$ as $k\to\infty$, where $\Box^{q}_k$ is the $\bar{\partial}$-Neumann Laplacian acting on $(0,q)$-forms on $M$ with values in $L^k$. Our results extend the seminal work of Bismut to complex manifolds with boundary. As applications of our results, we provide a heat kernel-based proof of the holomorphic Morse inequalities for complex manifolds with boundary and derive a semi-classical Weyl law for the $\bar{\partial}$-Neumann Laplacian.

math.CV↗
arXiv2024

Heat kernel asymptotics for Kodaira Laplacians of high power of line bundle over complex manifolds

This paper presents a simple method to prove the heat kernel asymptotics for the Kodaira Laplacian with respect to the high power of a holomorphic Hermitian line bundle $(L,h^L)$ over a possibly non-compact Hermitian manifold $(M,ω)$. As a consequence, we give a direct proof of the holomorphic Morse inequalities on covering manifolds. Furthermore, we generalize it to the vector bundle via the $L^2$ Le Potier isomorphism and provide an algebraic version of the holomorphic Morse inequalities. The approach used in this work employs a scaling technique and is applicable to $M$ regardless of its compactness.

math.DG↗
arXiv2021

Heat kernel asymptotics for Kohn Laplacians on CR manifolds

Let $X$ be an abstract orientable not necessarily compact CR manifold of dimension $2n+1$, $n\geq1$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Suppose that condition $Y(q)$ holds at each point of $X$, we establish asymptotics of the heat kernel of Kohn Laplacian with values in $L^k$. As an application, we give a heat kernel proof of Morse inequalities on compact CR manifolds. When $X$ admits a transversal CR $\mathbb R$-action, we also establish asymptotics of the $\mathbb R$-equivariant heat kernel of Kohn Laplacian with values in $L^k$. As an application, we get $\mathbb R$-equivariant Morse inequalities on compact CR manifolds with transversal CR $\mathbb R$-action.

math.CV↗
arXiv2019

Spectral Stability of the $\bar\partial-$Neumann Laplacian: Domain Perturbations

We study spectral stability of the $\bar\partial$-Neumann Laplacian on a bounded domain in $\mathbb{C}^n$ when the underlying domain is perturbed. In particular, we establish upper semi-continuity properties for the variational eigenvalues of the $\bar\partial$-Neumann Laplacian on bounded pseudoconvex domains in $\mathbb{C}^n$, lower semi-continuity properties on pseudoconvex domains that satisfy property ($P$), and quantitative estimates on smooth bounded pseudoconvex domains of finite D'Angelo type in $\mathbb{C}^n$.

math.CV↗