Towards a Characterization of the Complete Rotationally Symmetric Minimal Surfaces with Plateau-Like Singularities

Abstract

The problem of finding and characterizing the surfaces in R3 which locally minimize area is known as Plateau's problem. Although the catenoid and the plane were proven in the 1700s to minimize area, there has been little further study of rotationally symmetric minimal surfaces. In this study, we investigate the complete rotationally symmetric solutions to Plateau's problem, revealing surprising depth due to singularities that may appear in a broad class of minimal surfaces. Our analysis is structured around the topology of the surface's generating graph, and we first consider surfaces of a simple topological type. For these surfaces, we prove new statements about complexity and shape, relating the number of singularities to the Hausdorff distance from a canonical example. We then consider more complicated structures, producing a novel surface with a handle (in particular, whose generating graph contains a 4-cycle). We finally provide direction for future study.