Delaunay type domains for an overdetermined elliptic problem in S^n x R and H^n x R

TL;DR

This paper constructs a family of symmetric, periodic domains in product manifolds S^n x R and H^n x R, where the first eigenfunction satisfies an overdetermined boundary condition, extending classical results to curved spaces.

Contribution

It introduces a new class of Delaunay type domains in curved product manifolds that bifurcate from cylinders, with explicit eigenfunction properties.

Findings

Existence of countably many Delaunay type domains in S^n x R and H^n x R.

Domains are rotationally symmetric and periodic in the R direction.

Domains converge to cylinders as the index j approaches zero.

Abstract

We prove the existence of a countable family of Delaunay type domains \Omega_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. The domains \Omega_j are rotationally symmetric and periodic with respect to the R-axis of the cylinder and as j converges to 0 the domain \Omega_j converges to the cylinder B^n x R.