2022/04/26 by Bin Guo, Guo, Bin, D. H. Phong +1
Mathematics · #Geometry and complex manifolds #Algebraic Geometry and Number Theory #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2204.12549
Sharp L^∞ estimates are obtained for general classes of fully non-linear PDE's on non-Kähler manifolds, complementing the theory developed earlier by the authors in joint work with F. Tong for the Kähler case. The key idea is still a comparison with an auxiliary Monge-Ampère 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.