论文标题
格林的功能和复杂的Monge-ampère方程
Green's functions and complex Monge-Ampère equations
论文作者
论文摘要
均匀的$ l^1 $，在紧凑型Kähler歧管上获得了绿色功能的下限。与Riemannian歧管的Cheng-Li经典定理不同，下限不直接取决于RICCI曲率，而仅取决于体积形式及其某些衍生物的整体界限。特别是，获得了绿色功能在kähler歧管上的统一下限，这仅取决于标量曲率的下限，以及$ q> 1 $的音量形式的$ l^q $ norm。证明依赖于辅助蒙格 - 安培的方程式，从根本上是非线性的。绿色功能的下限意味着$ C^1 $和$ c^2 $的估计值，用于复杂的monge-ampère方程，对右侧函数的依赖性较大。
Uniform $L^1$ and lower bounds are obtained for the Green's function on compact Kähler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only on integral bounds for the volume form and certain of its derivatives. In particular, a uniform lower bound for the Green's function on Kähler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an $L^q$ norm for the volume form for some $q>1$. The proof relies on auxiliary Monge-Ampère equations, and is fundamentally non-linear. The lower bounds for the Green's function imply in turn $C^1$ and $C^2$ estimates for complex Monge-Ampère equations with a sharper dependence on the function on the right hand side.
