半稳定 $J$-方程的 $J$-零轨迹与弱解的局部正则性
The $J$-Null Locus and Local Regularity of Weak Solutions of the Semistable $J$-Equation
- University of Iowa（爱荷华大学）
- East China Normal University（华东师范大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明半稳定$J$-方程的数值$J$-零轨迹与Murakami弱解的奇异轨迹重合，并利用正则化定理和相对先验估计建立弱解在零轨迹外的局部光滑性。
AI中文摘要：
对于半稳定 $J$-方程，我们证明了数值 $J$-零轨迹与Murakami弱解的环境 $C^2$-奇异轨迹重合。因此，该奇异轨迹是一个具有有限多个不可约分量的真解析子集，并且弱解在 $J$-零轨迹之外是局部光滑的。证明依赖于两个解析要素。首先，我们建立了奇异 $J$-子解的正则化定理，表明Demailly的全局正则化保持定量的严格锥条件。因此，具有给定解析极点的奇异严格子解可以在其极点集之外被替换为光滑的严格子解。其次，我们推导了适应于这些子解的相对先验估计：由行列式控制得到的相对 $L^\infty$-估计，以及在正则轨迹的紧子集上产生一致 $C^2$-界的加权二阶估计。
英文摘要：
For the semistable $J$-equation, we prove that the numerical $J$-null locus coincides with the ambient $C^2$-singular locus of Murakami's weak solution. Consequently, this singular locus is a proper analytic subset with finitely many irreducible components, and the weak solution is locally smooth outside the $J$-null locus. The proof relies on two analytic ingredients. First, we establish a regularization theorem for singular $J$-subsolutions, showing that Demailly's global regularization preserves quantitative strict cone conditions. Thus, singular strict subsolutions with prescribed analytic poles can be replaced by smooth strict subsolutions away from their pole sets. Second, we derive relative a priori estimates adapted to these subsolutions: a relative $L^\infty$-estimate from determinant control and a weighted second order estimate yielding uniform $C^2$-bounds on compact subsets of the regular locus.
