Convex functions on Grassmannian manifolds and Lawson-Osserman problem

New area-minimizing Lawson-Osserman cones

It has been 40 years since Lawson and Osserman introduced the three minimal cones associated with Dirichlet problems in their 1977 Acta paper [LO77]. The first cone was shown area-minimizing by Harvey and Lawson in the celebrated paper [HL82]. In this paper, we confirm that the other two are also area-minimizing. In fact, we show that every Lawson-Osserman cone of type (n, p, 2) constructed in [XYZ] is area-minimizing.

On instability of Type (II) Lawson-Osserman Cones

We obtain the instability of Type (II) Lawson-Osserman cones in Euclidean spaces, and thus provide a family of (uncountably many) unstable solutions with singularity to the Dirichlet problem for minimal graphs of high codimension versus smooth unstable ones by Lawson-Osserman through a min-max technique. To our knowledge, these are the first examples of non-smooth unstable minimal graphs and unlikely detectible through the mean curvature flow or min-max theory.

Lawson-Osserman锥的可积性及其应用

Integrability of Lawson-Osserman Cone and its Applications

本文刻画了Lawson-Osserman锥C在R^7中链接M的Jacobi算子非正特征值对应的所有特征函数，证明了C是可积的，并应用于证明M在S^6中作为极小子流形的刚性以及推导R^7中渐近于C的极小子流形的最优衰减阶。

Self-expanders to the mean curvature flow based on the generalized Lawson-Osserman cone

We derive the equation of self-similar solutions to mean curvature flow based on the generalized Lawson-Osserman cone and prove the existence of self-expanders by modifying the theory of equilibria in the autonomous system. In particular, those self-expanders are unique if a local assumption is given.

On Lawson-Osserman Constructions

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-p…

关于Lawson-Osserman猜想

On the Lawson-Osserman conjecture

本文证明了二维平面情形下面积泛函外部变分的Lipschitz临界点必为光滑映射，从而解决了Lawson和Osserman于1977年提出的猜想。