Fully nonlinear equations in geometry and general relativity

几何和广义相对论中的完全非线性方程

• 批准号: RGPIN-2020-06756
• 负责人: Lu, Siyuan
• 金额: $ 1.89万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=718079
• 关键词: Fully; nonlinear; equations; geometry; general

The mathematics I will explore in this research program are taken from various problems arising from geometry and general relativity. Since the universe of geometry and general relativity is highly nonlinear, I will study these problems from the point of fully nonlinear partial differential equations. I will mainly focus on the following three short-term objectives in the program. My first short-term objective is to study isometric embedding and quasi-local mass. This is the continuation of my previous work. One of the most important problems I am concerning is to establish isometric embedding into Schwarzschild manifold homologous to the horizon. The successful progress of this problem will lead to the localized Riemannian Penrose inequality, which is closely related to cosmic censorship conjecture in general relativity. My second short-term objective is to study the geometry of asymptotically hyperbolic manifolds. These manifolds come naturally from the study of general relativity with negative cosmological constant. It is also closely related to the study of null geometry in general relativity. One of the most important questions in this area is whether anti-de Sitter-Schwarzschild manifold is the ”best” model manifold in this class. To answer this question, I will first establish the rigidity of anti-de Sitter-Schwarzschild manifold among the class of static metrics. I will then study the capacity of asymptotically hyperbolic manifolds. The long time goal is to establish the Riemannian Penrose inequality in this setting. My third short-term objective concerns the fully nonlinear equations invariant under Mobius transformation in dimension 2. It is the natural extension of Yamabe problem in dimension 2. I will study the existence and uniqueness for such equations and their applications in geometry, particularly in conformal geometry. The successful progress in my projects will help us to understand the universe of geometry and general relativity. It will also shed lights on other areas of mathematics.

我将在这个研究项目中探索的数学是从几何和广义相对论中产生的各种问题中提取的。由于几何学和广义相对论的宇宙是高度非线性的，我将从完全非线性偏微分方程的角度来研究这些问题。我将主要关注以下三个短期目标。我的第一个短期目标是研究等距嵌入和准局部质量。这是我以前工作的延续。其中最重要的问题之一是建立等距嵌入到Schwarzschild流形同源的地平线。这个问题的成功进展将导致局部化的黎曼彭罗斯不等式，它与广义相对论中的宇宙监督猜想密切相关。我的第二个短期目标是研究渐近双曲流形的几何。这些流形自然来自于负宇宙学常数的广义相对论的研究。它也与广义相对论中零几何的研究密切相关。这一领域中最重要的问题之一是反德西特-史瓦西流形是否是这类流形中的“最佳”模型流形。为了回答这个问题，我将首先在静态度量类中建立反德西特-史瓦西流形的刚性。然后我将研究渐近双曲流形的容量。长期目标是在这种情况下建立黎曼彭罗斯不等式。我的第三个短期目标是关于在二维Mobius变换下完全非线性方程不变。它是Yamabe问题在2维空间的自然推广。我将研究这类方程的存在唯一性及其在几何中的应用，特别是在共形几何中的应用。我的项目的成功进展将有助于我们理解几何学和广义相对论的宇宙。它也将照亮数学的其他领域。