arXiv subjects

John N. Treuer

Publications and source records attributed to John N. Treuer.

9 recordsLinked to original sources

arXiv2026

A new strong rigidity phenomenon for the Bergman metric

We establish a new local-to-global rigidity phenomenon for the Bergman metric. Namely, under natural geometric hypotheses, a local conformal identification of Bergman metrics determines the underlying complex manifold globally, up to the unavoidable ambiguity of removing Bergman-negligible subsets. More precisely, let $Ω\subseteq\mathbb C^n$ be a bounded domain with a complete Bergman metric, and suppose that the Bergman metric of a complex manifold $M$ is locally conformal, via a holomorphic map $f$, to that of $Ω$. We prove that the given local map $f$ extends to a biholomorphism $F\colon M\to D$ onto a subdomain $D\subseteqΩ$ in two complementary settings. If $M$ is Stein, then $Ω\setminus D$ is a closed pluripolar set. If $M$ is a bounded domain and $Ω$ satisfies a natural symmetry condition expressed in terms of its automorphism orbits, then $Ω\setminus D$ is Bergman-negligible. In particular, this applies when $Ω$ is a bounded homogeneous domain and yields a characterization, up to Bergman-negligible sets, of bounded domains with locally symmetric Bergman metrics. The latter answers a question raised by Loi--Palmieri and Zimmer. A key ingredient in the proof is a new Calabi-type extension theorem tailored to Bergman metrics.

math.CV↗

arXiv2026

Domains with Bergman metrics of constant curvature and Bergman-negligible subsets

Let $D$ be a bounded domain in $\mathbb{C}^n$. Suppose the holomorphic sectional curvature of its Bergman metric equals a negative constant $τ$. We show that $D$ is biholomorphic to a domain $Ω$ equal to the unit ball in $\mathbb{C}^n$ less a relatively closed set of measure zero, and that all $L^2$-holomorphic functions on $Ω$ extend to $L^2$-holomorphic functions on the ball. Consequently, $τ$ must equal the holomorphic sectional curvature of the unit ball. This generalizes a classical theorem of Lu. Some applications of the theorem, especially in extending classical work of Wong and Rosay, are also presented.

math.CV↗

arXiv2025

The Levi $q$-core and Property ($P_q$)

We introduce the Grassmannian $q$-core of a distribution of subspaces of the tangent bundle of a smooth manifold. This is a generalization of the concept of the core previously introduced by the first two authors. In the case where the distribution is the Levi null distribution of a smooth bounded pseudoconvex domain $Ω\subseteq \mathbb{C}^n$, we prove that for $1 \leq q \leq n$, the support of the Grassmannian $q$-core satisfies Property $(P_q)$ if and only if the boundary of $Ω$ satisfies Property $(P_q)$. This generalizes a previous result of the third author in the case $q=1$. The notion of the Grassmannian $q$-core offers a perspective on certain generalized stratifications appearing in a recent work of Zaitsev.

math.CV↗

arXiv2025

Rotational symmetries of domains and orthogonality relations

Let $Ω\subset \mathbb{C}^n$ be a domain whose Bergman space contains all holomorphic monomials. We derive sufficient conditions for $Ω$ to be Reinhardt, complete Reinhardt, circular or Hartogs in terms of the orthogonality relations of the monomials with respect to their $L^2$-inner products and their $L^2$-norms. More generally, we give sufficient conditions for $Ω$ to be invariant under a linear group action of an $r$-dimensional torus, where $r \in \{1,\ldots, n\}$.

math.CV↗

arXiv2024

Sharp decay rate for eigenfunctions of perturbed periodic Schrödinger operators

This paper investigates uniqueness results for perturbed periodic Schrödinger operators on $\mathbb{Z}^d$. Specifically, we consider operators of the form $H = -Δ+ V + v$, where $Δ$ is the discrete Laplacian, $V: \mathbb{Z}^d \rightarrow \mathbb{R}$ is a periodic potential, and $v: \mathbb{Z}^d \rightarrow \mathbb{C}$ represents a decaying impurity. We establish quantitative conditions under which the equation $-Δu + V u + v u = λu$, for $λ\in \mathbb{C}$, admits only the trivial solution $u \equiv 0$. Key applications include the absence of embedded eigenvalues for operators with impurities decaying faster than any exponential function and the determination of sharp decay rates for eigenfunctions. Our findings extend previous works by providing precise decay conditions for impurities and analyzing different spectral regimes of $λ$.

math.SP↗

arXiv2023

Modifications of the Levi core

We construct a family of subdistributions of the Levi core $\mathfrak{C}(\mathcal{N})$ called modified Levi cores $\{\mathcal{M}\mathfrak{C}_{\mathcal{A}}\}_{\mathcal{A}}$ indexed over closed distributions $\mathcal{A}$ that contain the Levi null distribution $\mathcal{N}$ and are contained in the complex tangent bundle $T^{1, 0}bΩ$ of a smooth bounded pseudoconvex domain $Ω$. We show that Catlin's Property ($P$) holds on $bΩ$ if and only if Property ($P$) holds on the support of one, and hence all, of the modified Levi cores. In $\mathbb{C}^2$, all of the modified Levi cores coincide. For a smooth bounded pseudoconvex complete Hartogs domain in $\mathbb{C}^2$ that satisfies Property ($P$), we show that its modified Levi core is trivial. This contrasts with $\mathfrak{C}(\mathcal{N})$, which can be nontrivial for such domains.

math.CV↗

arXiv2023

Sufficient condition for compactness of the $\overline{\partial}$-Neumann operator using the Levi core

On a smooth, bounded pseudoconvex domain $Ω$ in $\mathbb{C}^n$, to verify that Catlin's Property ($P$) holds for $bΩ$, it suffices to check that it holds on the set of D'Angelo infinite type boundary points. In this note, we consider the support of the Levi core, $S_{\mathfrak{C}(\mathcal{N})}$, a subset of the infinite type points, and show that Property ($P$) holds for $bΩ$ if and only if it holds for $S_{\mathfrak{C}(\mathcal{N})}$. Consequently, if Property ($P$) holds on $S_{\mathfrak{C}(\mathcal{N})}$, then the $\overline{\partial}$-Neumann operator $N_1$ is compact on $Ω$.

math.CV↗

arXiv2022

Rigidity theorems by capacities and kernels

For any open hyperbolic Riemann surface $X$, the Bergman kernel $K$, the logarithmic capacity $c_β$, and the analytic capacity $c_{B}$ satisfy the inequality chain $πK \geq c^2_β \geq c^2_B$; moreover, equality holds at a single point between any two of the three quantities if and only if $X$ is biholomorphic to a disk possibly less a relatively closed polar set. We extend the inequality chain by showing that $c_{B}^2 \geq πv^{-1}(X)$ on planar domains, where $v(\cdot)$ is the Euclidean volume, and characterize the extremal cases when equality holds at one point. Similar rigidity theorems concerning the Szegö kernel, the higher-order Bergman kernels, and the sublevel sets of the Green's function are also developed. Additionally, we explore rigidity phenomena related to the multi-dimensional Suita conjecture.

math.CV↗

arXiv2021

Sharp pointwise and uniform estimates for $\bar\partial$

We use weighted $L^2$-methods to obtain sharp pointwise estimates for the canonical solution to the equation $\bar\partial u=f$ on smoothly bounded strictly convex domains and the Cartan classical domain domains when $f$ is bounded in the Bergman metric $g$. We provide examples to show our pointwise estimates are sharp. In particular, we show that on the Cartan classical domains $Ω$ of rank $2$ the maximum blow up order is greater than $-\log δ_Ω(z)$, which was obtained for the unit ball case by Berndtsson. For example, for IV$(n)$ with $n \geq 3$, the maximum blow up order is $δ(z)^{1 -{n \over 2}}$ because of the contribution of the Bergman kernel. Additionally, we obtain uniform estimates for the canonical solutions on the polydiscs, strictly pseudoconvex domains and the Cartan classical domains under stronger conditions on $f$.

math.CV↗