Non-classical Solution to Hessian Equation from Cartan Isoparametric Cubic

A generalization of Cartan's theorem on isoparametric cubics

We give a generalization of the well-known result of E. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation $|\nabla f|^2=9|x|^4$ must be rotationally equivalent to either $x_n^3-3x_n(x_1^2+...+x_{n-1}^2)$, or to one of four exceptional Cartan cubic polynomials in dimensions $n=5,8,14,26$.

哈密顿稳态与特殊拉格朗日图的尖锐部分正则性

Sharp partial regularity of Hamiltonian stationary and special Lagrangian graphs

该研究证明了任意光滑近凯勒流形中哈密顿稳态拉格朗日 Lipschitz 子流形的弱解在豪斯多夫维数至多为 $n-5$ 的奇异集外光滑，并构造反例说明该维数估计对特殊拉格朗日图是尖锐的。

Cartan-Hadamard流形上Allen–Cahn方程的Busemann剖面的刚性与存在性

Rigidity and existence of Busemann profiles for the Allen--Cahn equation on Cartan--Hadamard manifolds

该研究在Cartan-Hadamard流形上探讨Allen–Cahn方程的Busemann剖面，证明了相关刚性原理，明确了双稳不平衡势下Busemann前沿存在的条件，并在特定曲面上验证了结果的尖锐性。