Lipschitz solutions to the minimal surface system with prescribed isolated singularities

The bridge principle is the idea that you can join compact minimal submanifolds along their boundaries to produce an “approximately minimal” submanifold, called the approximate solution, and apply a small normal perturbation to make the new configuration minimal. It dates back to Lévy (1948) and Courant (1950), and was originally proposed to study non-uniqueness in the Plateau problem. The first general bridge theorem was proved by Smale in 1987 for smooth submanifolds using PDE techniques. He later extended his method to minimal hypercones (1989) and, finally, minimal submanifolds with isolated conical singularities (1993) to produce new singular examples of minimal submanifolds. 

In this talk, we demonstrate how Smale’s bridge principle can be used to build new examples of solutions to the minimal surface system with isolated singularities using only calculus and elementary PDE methods. When applied to the Lawson-Osserman cone, we obtain strictly stable examples with the lowest possible dimension having any finite number of prescribed singularities. We will also discuss potential applications.