刘维尔定理与Evans-Krylov估计
Liouville theorems and Evans-Krylov estimates
AI总结：
本课程围绕“刘维尔定理与Schauder型正则性估计等价”的经典思路，介绍奇异参考度量下复Monge-Ampère方程正则性的最新研究，包含经典估计回顾、新证明及奇异背景研究，配套习题与问题清单。
AI中文摘要：
分析学中一个至少可追溯至Simon（1997年）工作的经典思路是：椭圆型或抛物型偏微分方程（PDE）解的刘维尔定理与Schauder型正则性估计等价。本课程的目标是阐述这一思路在奇异参考度量下复Monge-Ampère方程正则性研究中的若干最新进展。我们将先快速回顾Calabi-Aubin-Yau的经典$C^2$与$C^3$估计，随后给出欧氏球上Evans-Krylov $C^{2,α}$估计的一种新证明。在此基础上，我们将讨论柱面、锥面等奇异背景的情形，介绍Hein、Tosatti、Lee与Klemmensen的近期研究成果。本课程的讨论并不全面，且预设学习者已掌握包括Aubin-Yau定理在内的Kähler几何知识。课程包含五道附解答的习题以及一份问题清单。
英文摘要：
A classical idea in analysis going back at least to work of Simon (1997) is that Liouville theorems for solutions to elliptic or parabolic PDEs are equivalent to Schauder type regularity estimates. The goal of this course is to describe some recent developments of this idea concerning the regularity of the complex Monge-Ampère equation with respect to singular reference metrics. We will start with a quick look at the classical $C^2$ and $C^3$ estimates of Calabi-Aubin-Yau and then present a new proof of the Evans-Krylov $C^{2,α}$ estimate on a Euclidean ball. Based on this we will consider the case of singular backgrounds such as cylinders and cones, discussing some recent work by Hein, Tosatti, Lee and Klemmensen. Our discussion is far from complete and knowledge of Kähler geometry including the Aubin-Yau theorems is assumed. The course includes five exercises with solutions and a problem list.
