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

Abstract

We investigate the overdetermined torsion problem $\{\begin{array}{ll}-\Delta u=1& \text{in}\Omega \\ u=0& \text{on}\partial \Omega \\ \frac{\partial u}{\partial \nu }=\text{const.}& \text{on}\partial \Omega ,\end{array}$ 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.