On the stability of Killing cylinders in hyperbolic space
Let \(W\) be a bounded domain of the hyperbolic \(3\)-space \(\mathbb H^3\) with smooth boundary \(\partial W\), and \(\Sigma\) a surface in \(\mathbb H^3\) with non-empty boundary such that \(\mbox{int}(\Sigma)\subset\mbox{int}(W)\) and \(\partial\Sigma\subset\partial W\). A critical point of the area functional among all surfaces under these conditions that separate \(W\) in two domains of prescribed volumes is called a capillary surface. A Killing cylinder in \(\mathbb H^3\) is defined as the point-set obtained by the movement of a circle by hyperbolic translations of \(\mathbb H^3\), or as a surface of revolution generated by an equidistant line, or as the point-set of equidistant points from a given geodesic, these definitions all being equivalent.\N\NIn this paper, the authors study the stability of a Killing cylinder in \(\mathbb H^3\) when regarded as a capillary surface for the partitioning problem, and consider a variety of totally umbilical support surfaces, including horospheres, totally geodesic planes, equidistant surfaces and round spheres. In all of them, they explicitly compute the Morse index of the corresponding eigenvalue problem for the Jacobi operator, and also address the stability of compact pieces of Killing cylinders with Dirichlet boundary conditions when the boundary is formed by two fixed circles, exhibiting an analogous to the Plateau-Rayleigh instability criterion for Killing cylinders in the Euclidean space. Finally, they prove that the Delaunay surfaces can be obtained by bifurcating Killing cylinders supported on geodesic planes.
scientific article; zbMATH DE number 7880955
scientific article