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

TL;DR

The paper studies the overdetermined torsion problem $- abla^2 u = 1$ in a domain $\Omega$ within the 3-sphere $\mathbb{S}^3$, seeking domains whose boundary supports a constant normal derivative. Building on Serrin’s classical rigidity in Euclidean space, the authors construct two branches of Serrin domains in $\mathbb{S}^3$ with connected boundary that are neither geodesic spheres nor Clifford tori, by perturbing a symmetric, isoparametric family $\Omega_\lambda$ and applying Crandall–Rabinowitz bifurcation. The analysis reduces to a spectral study of the linearization $\mathbb{L}_\lambda$ of the boundary operator, identifying critical modes via zeros of $\sigma_n(\lambda)$ and proving transversality at those points. The resulting nonisoparametric Serrin domains demonstrate that Serrin-type rigidity can fail in curved ambient spaces, highlighting new geometric configurations for the torsion problem on $\mathbb{S}^3$ and extending previous bifurcation-based constructions in related settings.

Abstract

We investigate the overdetermined torsion problem $\begin{cases} -Δu = 1 & \text{in}\ Ω\\ u=0 & \text{on}\ \partial Ω\\ \frac{\partial u}{\partial ν}=\text{const.} & \text{on}\ \partial Ω, \end{cases}$ where $Ω$ 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.

Theorems & Definitions (33)

• Definition 1.1
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5: Crandall-Rabinowitz
• Lemma 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• Proposition 3.1: Properties of $L^*$