Lawson-Osserman constructed three types of non-parametric minimal cones of higher codimension based on Hopf fibrations between Euclidean spheres, which can been seen as Lipschitz solutions to the minimal surface equations which are not differentiable, thereby making sharp contrast to the regularity theorem for minimal hypersurfaces in Euclidean spaces. In this paper, the above constructions are generalized in a more general scheme. Once a mapping f can be written as the composition of a Riemannian submersion from a Euclidean sphere and an isometric minimal immersion into another Euclidean sphere, the graph of f yields a non-parametric minimal cone. Because the choices of the second component form huge moduli spaces, our constructions produce a constellation of uncountable many examples. For each such cone, there exists an entire minimal graph whose tangent cone at infinity is just the given one. Moreover, surprising phenomena on the existence, non-uniqueness and non-minimizing for the Dirichlet problem are discovered, due to the amusing spiral asymptotic behaviour of a particular autonomous system on the 2-plane.

海报