arXiv subjects

Shreedhar Bhat

Publications and source records attributed to Shreedhar Bhat.

6 recordsLinked to original sources

arXiv2026

Strongly Pseudoconvex Domains with Finite Dimensional $p$-Bergman Spaces

We show that for every $n\ge2$, $p>2$, and positive integer $m$, there exists a finite-volume pseudoconvex domain $D\subset\mathbb C^n$ with real-analytic strongly pseudoconvex boundary whose $p$-Bergman space $A^p(D)$ has dimension $m$. These domains admit smooth strictly plurisubharmonic exhaustions and satisfy $H^\infty(D)=\mathbb C$ and $\dim A^s(D)=\infty$ for every $0<s\le2$. Moreover, for every finite $s\ge p$, we have $\dim A^s(D)=\lceil pm/s\rceil$. As an application, we show that the Bergman projection fails to map $\mathcal{C}_c^\infty(D)$ into $A^p(D)$ and admits no bounded extension to $L^s(D)$ or $L^{s/(s-1)}(D)$ for any finite $s\ge p$.

math.CV↗

arXiv2026

Abundance of Bergman metrics with constant positive holomorphic sectional curvature

An outstanding open question, which has attracted renewed attention following the pioneering work of Huang--Li--Treuer, is whether, for a given positive integer $m$, there exists a complex manifold whose Bergman metric is locally isometric to $m$ times the Fubini--Study metric. Previously, this question had only been resolved in the case $m=1$. In this paper, we construct, for any pair of positive integers $(m,n)$ with $n \geq 2$, an $\mathbb{R}$-parameter (hence uncountable) family of Reinhardt domains in $\mathbb{C}^n$ whose Bergman metrics are all locally isometric to $m$ times the Fubini--Study metric. Moreover, we show that the domains in this family are mutually Bergman inequivalent. This not only answers the folklore question, but also suggests that a reasonable classification of the geometry of such complex manifolds is infeasible. We also note such examples cannot exist in dimension one. The results complete the remaining open case in the study of complex manifolds whose Bergman space separates points and whose Bergman metric has constant holomorphic sectional curvature. Our approach differs from existing methods in the literature. We reduce the construction to a mapping problem and apply a Brouwer fixed point argument to establish the existence of the desired domains.

math.CV↗

arXiv2026

Stability of the Monomial Basis Kernel of Reinhardt domains

On a pseudoconvex Reinhardt domain $Ω\subset\mathbb{C}^n$ the $p$-Bergman space $A^p(Ω)$ admits a canonical basis of monomials indexed by a subset $S_p(Ω)\subset\mathbb{Z}^n$. The corresponding $p$-Monomial Basis Kernel (or $p$-MBK) is defined by a series involving these monomials and their norms. This article records stability properties of the $p$-MBK and of the index set $S_p(Ω)$ with respect to the parameter $p$. First, under mild hypotheses, the $p$-MBK depends continuously on $p\in[1,\infty)$, and a Ramadanov-type theorem holds for $p$-MBK for an increasing sequence of pseudoconvex Reinhardt domains. Second, for certain special classes of monomial polyhedra, we explicitly compute the index set and the associated Threshold exponents. Finally, these explicit models are used to illustrate structural properties of the index sets under finite unions, intersections, and products.

math.CV↗

arXiv2025

On the dimension of the $p$-Bergman spaces

The investigation of the dimension of Bergman spaces has long been a central topic in several complex variables, uncovering profound connections with potential theory and function theory since the pioneering work of Carleson, Wiegerinck, and others in the 1960s. We investigate the dimension of $p$-Bergman spaces associated with pseudoconvex domains in $\mathbb{C}^n$. By constructing $L^p$-versions of the extension theorems of Ohsawa and Ohsawa-Takegoshi, we establish several geometric and potential-theoretic criteria that ensure the spaces are infinite-dimensional. Sufficient conditions for the infinite dimensionality of $p$-Bergman spaces of complete N-circled fibered Hartogs domains, balanced domains, and weighted $p$-Fock spaces are obtained by applying the mentioned $L^p$-analogs of extension theorems and generalizing a sufficient condition of Jucha.

math.CV↗

arXiv2024

Duality and Interpolation of Bergman Spaces

This paper explores the dual space corresponding to p-Bergman space and examines the essential condition for the dual space to be a q-Bergman space. The investigation involves a detailed examination of the interpolation space of a Banach couple. Additionally, we draw comparisons between the `duality', `integrability' and `regularity' properties of a domain.

math.CV↗

arXiv2023

$p$-Skwarczyński distance

We introduce a new distance on a domain $Ω\subset \mathbb{C}^n$ using the `minimizer' functions on $A^p(Ω)$. We discuss its invariance, completeness, and other aspects related to it.

math.CV↗