Continuous analytic capacity and holomorphic motions

TL;DR

This paper constructs a specific compact set demonstrating that continuous analytic capacity can vary discontinuously under holomorphic motions, providing new insights and proofs in complex analysis.

Contribution

It presents a counterexample showing discontinuity of analytic capacity under holomorphic motions and offers a new proof of related variation results.

Findings

Continuous analytic capacity can be discontinuous under holomorphic motions.

Extremal functions for continuous analytic capacity may not exist.

Provides a new proof of existing variation results in analytic capacity.

Abstract

We construct a compact set whose continuous analytic capacity does not vary continuously under a certain holomorphic motion, thereby answering a question of Paul Gauthier. Our example is inspired by holomorphic dynamics and relies on the works of Bishop--Carleson--Garnett--Jones and Browder--Wermer relating tangent points of Jordan curves, harmonic measure and Dirichlet algebras. Our approach also provides a new proof of a result of Ransford, Younsi and Ai on the variation of analytic capacity under holomorphic motions. In addition, we show that extremal functions for continuous analytic capacity may not exist.