# Free boundaries subject to topological constraints

## TL;DR

This paper explores the topological complexity of solutions to one-phase free boundary problems, reviewing prior work, proving a new topological removable singularities theorem, and discussing open problems related to multiply connected cases.

## Contribution

It introduces a new topological removable singularities theorem for free boundaries and discusses the topological classification of free boundary solutions.

## Key findings

- Proved a new topological removable singularities theorem.
- Reviewed the case of simply connected free boundaries in the plane.
- Outlined open problems for multiply connected free boundary cases.

## Abstract

We discuss the extent to which solutions to one-phase free boundary problems can be characterized according to their topological complexity. Our questions are motivated by fundamental work of Luis Caffarelli on free boundaries and by striking results of T. Colding and W. Minicozzi concerning finitely connected, embedded, minimal surfaces. We review our earlier work on the simplest case, one-phase free boundaries in the plane in which the positive phase is simply connected. We also prove a new, purely topological, effective removable singularities theorem for free boundaries. At the same time, we formulate some open problems concerning the multiply connected case and make connections with the theory of minimal surfaces and semilinear variational problems.