# Free boundary minimal surfaces and overdetermined boundary value problems

## TL;DR

This paper explores the relationship between free boundary minimal surfaces in three-dimensional space and free boundary cones from a one-phase problem, proving that doubly connected minimal surfaces with free boundary are catenoids.

## Contribution

It establishes a novel connection between free boundary minimal surfaces and free boundary cones, and characterizes doubly connected minimal surfaces as catenoids.

## Key findings

- Doubly connected free boundary minimal surfaces in a ball are catenoids.
- A connection between free boundary minimal surfaces and free boundary cones is established.
- The paper provides new insights into the structure of free boundary minimal surfaces.

## Abstract

In this paper we establish a connection between free boundary minimal surfaces in a ball in $\mathbb{R}^3$ and free boundary cones arising in a one-phase problem. We prove that a doubly connected minimal surface with free boundary in a ball is a catenoid.