arXiv subjects
Publications and source records attributed to Annapurna Banik.
We provide a necessary and sufficient condition for the existence of Hölder continuous solutions to the complex Monge--Ampère equation on bounded domains in $\mathbb{C}^n$. This condition is motivated by a paper by S.-Y. Li. We also prove a result on the regularity of solutions to the complex Monge--Ampère equation on general $B$-regular domains in $\mathbb{C}^n$.
We give an explicit lower bound, in terms of the distance from the boundary, for the Kobayashi metric of a certain class of bounded pseudoconvex domains in $\mathbb{C}^n$ with $\mathcal{C}^2$-smooth boundary using the regularity theory for the complex Monge--Ampere equation. Using such an estimate, among other tools, we construct a family of unbounded Kobayashi hyperbolic domains in $\mathbb{C}^n$ having a certain negative-curvature-type property with respect to the Kobayashi distance. As an application, we prove a Picard-type extension theorem for the latter domains.
The earliest examples of visibility domains, given by Bharali--Zimmer, are pseudoconvex. In fact, all known examples of visibility domains are pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi-complete, visibility domains.
We prove a couple of results on local continuous extension of proper holomorphic maps $F:D \rightarrow Ω$, $D, Ω\varsubsetneq \mathbb{C}^n$, making local assumptions on $\partial{D}$ and $\partialΩ$. The first result allows us to have much lower regularity, for the patches of $\partial{D}, \partialΩ$ that are relevant, than in earlier results. The second result (and a result closely related to it) is in the spirit of a result by Forstneric--Rosay. However, our assumptions allow $\partialΩ$ to contain boundary points of infinite type.