Finite index solutions to the Bernoulli problem in three dimensions are axially symmetric

Abstract

We show that every entire solution to the Bernoulli (or one-phase) free boundary problem with finite Morse index in ${\mathbb{R}}^3$ is axially symmetric. In fact, we additionally prove that the same result would follow in any dimension $4\le n\le 6$ in which stable entire solutions are shown to be flat.