Non-isoparametric Serrin domains of $\mathbb{S}^3$ with connected toric boundary

TL;DR

This paper constructs new non-isoparametric Serrin domains in  with connected boundaries, showing that classical symmetry results do not hold in curved spaces by using bifurcation theory.

Contribution

It demonstrates the existence of non-radial Serrin domains in  with connected boundaries that are neither geodesic spheres nor Clifford tori, expanding understanding of geometric solutions.

Findings

Existence of small and large volume Serrin domains in 

Domains are non-isometric to classical symmetric solutions

Bifurcation method reveals non-radial solutions in curved space

Abstract

We investigate the overdetermined torsion problem $\begin{cases} -\Delta u = 1 & \text{in}\ \Omega\\ u=0 & \text{on}\ \partial \Omega\\ \frac{\partial u}{\partial \nu}=\text{const.} & \text{on}\ \partial \Omega, \end{cases}$ where $\Omega$ is a smooth Riemannian domain. Domains admitting a solution to this problem are called \textit{Serrin domains}, after the celebrated work of Serrin \cite{Se71}, where is proved that in $\mathbb{R}^n$ such domains are geodesic balls. In the present paper we establish the existence of two distinct types of Serrin domains of $\mathbb{S}^3$, respectively of small and large volume, each of whose boundary is connected and is neither isometric to a geodesic sphere nor to a Clifford torus. These domains arise as nontrivial perturbations of some classical symmetric solutions to the same problem. Our approach relies on an implicit construction based on the Crandall-Rabinowitz bifurcation theorem, which allows us to detect branches of non-radial solutions bifurcating from a family of radial ones. The resulting examples highlight new geometric configurations of the torsion problem in the three-dimensional sphere, providing another proof of the fact that the rigidity of Serrin-type results can fail in the presence of curvature.