Plateau-Rayleigh instability of singular minimal surfaces

Abstract

We prove a Plateau-Rayleigh criterion of instability for singular minimal surfaces, providing explicit bounds on the amplitude and length of the surface. More generally, we study the stability of α-singular minimal hypersurfaces considered as hypersurfaces in weighted manifolds. If α < 0 and the hypersurface is a graph, then we prove that the hypersurface is stable. If α > 0 and the surface is cylindrical, we give numerical evidences of the instability of long cylindrical α-singular minimal surfaces.