Submitted by ims_edit on Wed, 10/15/2025 - 11:19
preprint-id:
ims25-05
preprint-title:
On topology and singularities of quadrature domains.
preprint-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.
Submitted 25 September, 2025; originally announced September 2025.
preprint-year:
2025