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Rumpa Masanta

Publications and source records attributed to Rumpa Masanta.

6 recordsLinked to original sources
arXiv2026

Rarity of $\boldsymbol{\mathcal{C}^{1,1}}$ solutions to the complex Monge--Ampère equation on weakly pseudoconvex domains

We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge--Ampère equation with $\mathcal{C}^\infty$-smooth data does not in general admit $\mathcal{C}^{1,1}$-smooth solutions. This working draft is a prelude to potential-theoretic solutions to some extension problems for mappings that were thought to rely on such $\mathcal{C}^{1,1}$-smooth solutions.

math.CV↗
arXiv2025

On some connections between Kobayashi geometry and pluripotential theory

In this paper, we explore some connections between Kobayashi geometry and the Dirichlet problem for the complex Monge--Ampère equation. Among the results we obtain through these connections are: $(i)$~a theorem on the continuous extension up to $\partial{D}$ of a proper holomorphic map $F: D\longrightarrow Ω$ between domains with $\dim_{\mathbb{C}}(D) < \dim_{\mathbb{C}}(Ω)$, and $(ii)$~a result that establishes the existence of bounded domains with ``nice'' boundary geometry on which Hölder regularity of the solutions to the complex Monge--Ampère equation fails. The first, a result in Kobayashi geometry, relies upon an auxiliary construction that involves solving the complex Monge--Ampère equation with Hölder estimates. The second result relies crucially on a bound for the Kobayashi metric.

math.CV↗
arXiv2025

Visibility domains relative to the Kobayashi distance in complex manifolds

In this paper, we extend the notion of visibility relative to the Kobayashi distance to domains in arbitrary complex manifolds. Visibility here refers to a property resembling visibility in the sense of Eberlein--O'Neill for Riemannian manifolds. Since it is difficult, in general, to determine whether domains are Cauchy-complete with respect to the Kobayashi distance, we do not assume so here. We provide many sufficient conditions for visibility. We establish a Wolff--Denjoy-type theorem in a very general setting as an application. We also explore some connections between visibility and Gromov hyperbolicity for Kobayashi hyperbolic domains in the above setting.

math.CV↗
arXiv2025

Taut visibility domains are not necessarily Kobayashi complete

We answer a question asked recently by Banik in the negative by showing that for each $n\geq 2$, there exists a taut visibility domain in $\mathbb{C}^n$ that is not Kobayashi complete. The domains that we produce are bounded and have boundaries that are very regular away from a single point.

math.CV↗
arXiv2023

Kobayashi complete domains in complex manifolds

In this paper, we give sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains in complex manifolds. The first result gives a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. This follows from our second result, which may be of independent interest, in a much more general setting. This extends a result of Gaussier to the setting of manifolds.

math.CV↗