AI 中文总结
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。
AI 中文摘要
我们证明了任意光滑近凯勒流形中哈密顿稳态拉格朗日 Lipschitz 子流形的尖锐部分正则性结果：对应方程的每个弱解在豪斯多夫维数至多为 $n-5$ 的相对闭奇异集之外是光滑的。我们通过构造一个非零二次齐次粘性解来证明该估计是最优的，该解 $U\in C^{1,1}(\mathbb{R}^5)\setminus C^2(\mathbb{R}^5)$ 满足零相位特殊拉格朗日方程，其在 $\mathbb{S}^4$ 上的水平集是嘉当等参叶状结构的叶。它的梯度图是一个非平坦的校准锥，在顶点外是实解析的。这也给出了特殊拉格朗日方程的第一个 $C^{1,1}$ 但非 $C^2$ 的解，表明在特殊拉格朗日图的情形下，相同的维数估计是尖锐的。
英文摘要
We prove a sharp partial regularity result for Hamiltonian stationary Lagrangian Lipschitz submanifolds in arbitrary smooth almost Kähler manifolds: every weak solution of the corresponding equation is smooth away from a relatively closed singular set of Hausdorff dimension at most $n-5$. We show that the estimate is optimal by constructing a nonzero two-homogeneous viscosity solution \[ U\in C^{1,1}(\mathbb{R}^5)\setminus C^2(\mathbb{R}^5) \] of the phase-zero special Lagrangian equation, whose level sets on $\mathbb{S}^4$ are the leaves of Cartan's isoparametric foliation. Its gradient graph is a non-flat calibrated cone, real analytic away from the vertex. This also gives the first $C^{1,1}$ but non-$C^2$ solution of the special Lagrangian equation, and shows that the same dimensional estimate is sharp in the case of special Lagrangian graphs.
Comments33 pages, comments welcome!
