arXiv subjects

Pritam Naskar

Publications and source records attributed to Pritam Naskar.

3 recordsLinked to original sources

arXiv2026

Local asymptotics near a smooth boundary point and estimates for a hyperbolic-type metric

In this paper, we investigate the local boundary behaviour of a recently developed hyperbolic-type metric $m_D$. First, employing a boundary-flattening technique and local behaviour of $m_D$-geodesics, we establish its asymptotic formula near any $C^1$-smooth boundary point. Next, we introduce a metric quantity analogous to the Nikolov--Andreev metric and show that $m_D$ is the inner metric associated with it. Finally, by establishing a sharp two-sided comparison inequality, we obtain an improved lower bound for the $m_D$-metric.

math.MG↗

arXiv2025

Modified Distance Ratio Metrics via Domain Diameter and their geometric implications

Let $D\subsetneq\mathbb{R}^n,~n\ge 2$, be a domain. In this manuscript, a new version of the Vuorinen's distance ratio metric $j_D$ [{\tt J. Analyse Math.} {\bf 45} (1985), 69--115], denoted by $\zeta_D$, and a version of Gehring-Osgood's distance ratio metric $j_D'$ [{\tt J. Analyse Math.} {\bf 36} (1979), 50--74], denoted by $\zeta_D'$, are introduced to better understand how quasihyperbolic geometry interacts with bounded uniform domains in $\mathbb{R}^n$. We show that the metric $m_D$, introduced in [{\tt arXiv:2505.10964v2}], is the inner metric of $\zeta_D$ and explore their relations to several well-known hyperbolic-type metrics. The paper includes ball inclusion properties of these metrics associated with the metric $m_D$ and other hyperbolic-type metrics. The distortion properties of them are also considered under several important classes of mappings. Furthermore, as an application, we demonstrate that uniform domains can be characterized in terms of metrics $\zeta_D$ and $m_D$.

math.MG↗

arXiv2025

A naive generalization of the hyperbolic and the quasihyperbolic metrics

Although the hyperbolic metric possesses many remarkable properties, it is not defined on arbitrary subdomains of $\mathbb{R}^n$ with $n \geq 2$. This article introduces a new hyperbolic-type metric that provides an alternative approach to this limitation. The proposed metric coincides with the hyperbolic metric on balls and half-spaces, and, quite unexpectedly, agrees with the quasihyperbolic metric in unbounded domains. We compute the density of this metric in several classical domains and discuss aspects of its curvature. Furthermore, we establish characterizations of uniform domains and John disks in terms of the newly defined metric. In addition, we investigate several geometric properties of the metric, including the existence of geodesics and the minimal length of non-trivial closed curves in multiply connected domains.

math.MG↗