On Lawson-Osserman Constructions

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.