On L∞ estimates for fully non-linear partial differential equations

• 成果类型: Article
• 署名作者: Guo, Bin; Phong, Duong H.
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2024.200.1.6
• 发表日期: 2024
• 页码: 365-398
• 关键词: complex monge-ampere; 2nd-order elliptic-equations; calabi-yau theorem; dirichlet problem; CURVATURE; Duality; METRICS

摘要

Sharp LO degrees estimates are obtained for general classes of fully non-linear PDE's on non-Kahler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the Kahler case. The key idea is still a comparison with an auxiliary Monge-Ampe`re equation, but this time on a ball with Dirichlet boundary conditions, so that it always admits a unique solution. The method applies not just to compact Hermitian manifolds, but also to the Dirichlet problem, to open manifolds with a positive lower bound on their injectivity radii, to (n - 1) form equations, and even to non-integrable almost-complex or symplectic manifolds. It is the first method applicable in any generality to large classes of non-linear equations, and it usually improves on other methods when they happen to be available for specific equations.