On topology and singularities of quadrature domains

Rashmita; Sabyasachi Mukherjee

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arXiv · 2509.21468

On topology and singularities of quadrature domains

Rashmita·Sabyasachi Mukherjee

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Abstract

We prove a linear upper bound for the number of singular points on the boundary of a quadrature domain, improving a previously known quadratic bound due to Gustafsson \cite{Gus88}. This linear upper bound on the number of boundary double points also strengthens the bound on the connectivity (i.e., the number of complementary components) of a quadrature domain given by Lee and Makarov \cite{LM16}. Our proofs use conformal dynamics and hyperbolic geometry arguments. Finally, we introduce a new dynamical method to construct multiply connected quadrature domains.

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Rashmita, Sabyasachi Mukherjee. 2025-09-25. On topology and singularities of quadrature domains. https://arxiv.org/abs/2509.21468

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