On Lawson-Osserman Constructions

TL;DR

This paper generalizes Lawson-Osserman minimal cone constructions using harmonic submersions and minimal immersions, producing uncountably many non-parametric minimal cones with new insights into their solutions and properties.

Contribution

It introduces a broad scheme for constructing minimal cones from sphere mappings combining harmonic submersions and minimal immersions, expanding the class of known examples.

Findings

Uncountably many new minimal cones constructed

Existence of entire minimal graphs with given tangent cones

New phenomena in solutions to the Dirichlet problem for minimal surfaces

Abstract

Lawson-Osserman constructed three types of non-parametric minimal cones of high codimensions based on Hopf maps between spheres, which correspond to Lipschitz but non-differentiable solutions to the minimal surface equations, thereby making sharp contrast to the regularity theorem for minimal graphs of codimension 1. In this paper, we develop the constructions in a more general scheme. Once a mapping f between unit spheres is composited of a harmonic Riemannian submersion and a homothetic (i.e., up to a constant factor, isometric) minimal immersion, certain twisted graph of f can yield a non-parametric minimal cone. Because the choices of the second component usually form a huge moduli space, our constructions produce a constellation of uncountably many examples. For each such cone, there exists an entire minimal graph whose tangent cone at infinity is just the given one. Moreover, new phenomena on the existence, non-uniqueness and non-minimizing of solutions to the related Dirichlet problem are discovered.