Higher genus maxfaces with arbitrarily many catenoid or planar ends

Abstract

Maximal surfaces in 3-dimensional Lorentz-Minkowski space arise as solutions to the variational problem of local area maximizing among the spacelike surfaces. These surfaces are zero mean curvature surfaces, and maximal surfaces with singularities are called generalized maximal surfaces. Maxfaces are a special class of these generalized maximal surfaces where singularities appear at points where the tangent plane contains a light-like vector. I will present the construction of a new family of maxfaces of high genus that are embedded outside a compact set and have arbitrarily many catenoid or planar ends using the node opening technique. The surfaces look like spacelike planes connected by small necks. Among the examples are maxfaces of the Costa-Hoffman-Meeks type. More specifically, the singular set form curves around the waists of the necks. In generic and some symmetric cases, all but finitely many singularities are cuspidal edges, and the non-cuspidal singularities are swallowtails evenly distributed along the singular curves. This work is conducted in collaboration with Dr. Hao Chen, Dr. Anu Dhochak, and Dr. Pradip Kumar.