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Debaprasanna Kar

Publications and source records attributed to Debaprasanna Kar.

6 recordsLinked to original sources
arXiv2026

Strong Localization of the Kobayashi-Eisenman Volume Element and Its Boundary Asymptotics

We establish a quantitative version of strong localization of the Kobayashi-Eisenman volume element and the quotient invariant near plurisubharmonic peak points of domains in $\mathbb{C}^n$. As an application of this strong localization result, we derive the non-tangential asymptotic limit of the Kobayashi-Eisenman volume element at exponentially flat infinite type boundary points of domains in $\mathbb{C}^{n+1}$.

math.CV↗
arXiv2024

Weighted boundary limits of the Kobayashi--Fuks metric on h-extendible domains

We study the boundary behavior of the Kobayashi--Fuks metric on the class of h-extendible domains. Here, we derive the non-tangential boundary asymptotics of the Kobayashi--Fuks metric and its Riemannian volume element by the help of some maximal domain functions and then using their stability results on h-extendible local models.

math.CV↗
arXiv2024

Geometric estimates and comparability of Eisenman volume elements with the Bergman kernel on (C-)convex domains

We establish geometric upper and lower estimates for the Carathéodory and Kobayashi-Eisenman volume elements on the class of non-degenerate convex domains, as well as on the more general class of non-degenerate $\mathbb{C}$-convex domains. As a consequence, we obtain explicit universal lower bounds for the quotient invariant both on non-degenerate convex and $\mathbb{C}$-convex domains. Here the bounds we derive, for the above mentioned classes in $\mathbb{C}^{n}$, only depend on the dimension $n$ for a fixed $n\geq 2$. Finally, it is shown that the Bergman kernel is comparable with these volume elements up to small/large constants depending only on $n$.

math.CV↗
arXiv2023

Existence of geodesic spirals for the Kobayashi--Fuks metric on planar domains

In this note, we discuss the following problem: Given a smoothly bounded strongly pseudoconvex domain $D$ in $\mathbb{C}^n$, can we guarantee the existence of geodesics for the Kobayashi--Fuks metric which ``spiral around" in the interior of $D$? We find an affirmative answer to the above question for $n=1$ when $D$ is not simply connected.

math.CV↗
arXiv2022

Some remarks on the Kobayashi--Fuks metric on strongly pseudoconvex domains

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly bounded above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ is the Bergman kernel for $D$ on the diagonal and $g_{B, D}$ is the Riemannian volume element of the Bergman metric on $D$, is the potential for a Kähler metric on $D$ known as the Kobayashi--Fuks metric. In this note we study the localization of this metric near holomorphic peak points and also show that this metric shares several properties with the Bergman metric on strongly pseudoconvex domains.

math.CV↗