Metric geometry and analysis on Einstein manifolds

爱因斯坦流形的度量几何和分析

• 批准号: 2304818
• 负责人: Ruobing Zhang
• 金额: $ 22.23万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-01 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2304818&HistoricalAwards=false
• 关键词: Metric; geometry; analysis; Einstein; manifolds

Metric Riemannian geometry is a central subject in modern mathematics. The original concept dates back to Bernhard Riemann's famous Habilitation lecture "Ueber die hypothesen, welche der Geometrie zu Grunde liegen" (On the hypotheses which lie at the bases of geometry) delivered on 10 June 1854. The revolutionary creations in this lecture profoundly changed the global landscape of geometry. Specifically, Riemann proposed a novel strategy to generalize the geometry of surfaces to higher dimensions which he called Mannigfaltigkeiten (manifolds). A large variety of new notions and concepts were created: these include the notion of curvature which quantitatively measures how a space is curved, and the notion of geodesic which is a length-minimizing path connecting two points on a manifold. The studies of the metric structures of manifolds, what we now call metric Riemannian geometry, primarily focuses on the interplay between the global geometry of the underlying space and the metric structure, namely how the distance between two points can be realized or measured. This project is mainly concerned with the metric geometry of Einstein manifolds where the metric structures satisfy the Einstein equation in the theory of general relativity. The PI will integrate their research with training and mentorship at a variety of levels. This includes organizing summer workshops and mathematical retreats on Riemannian geometry; complex geometry and theoretical physics, and designing and developing new research oriented courses for undergraduate students. This project investigates the degenerations and quantitative behaviors of Einstein manifolds. In joint work with Song Sun, the PI has been working on the collapsing geometry of Einstein manifolds with special holonomy, leading to two major breakthroughs in the field: a complete classification of the Gromov-Hausdorff limits of the Einstein metrics on the K3 manifold, and a complete classification of asymptotic model geometries of gravitational instantons. The latter can be regarded as the bubble limits of the degenerating Einstein metrics on the K3 manifold. Building on this background, the PI will proceed to analogous questions in higher dimensions and investigate geometric structures for the degenerating Einstein metrics with generic holonomy in that setting. With a group of collaborators, the PI will also make advances in more refined geometry and moduli space problems regarding complete Calabi-Yau metrics. In a third direction, the PI will investigate the geometry and analysis of Poincare-Einstein spaces, which originated from the AdS/CFT correspondence in mathematical physics. The PI will focus on the singularity behaviors of degenerate operators on Poincare-Einstein manifolds, geometric finiteness and quantitative rigidity of Poincare-Einstein metrics, as well as regularity and degeneration theory of Poincare-Einstein metrics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

度量黎曼几何是现代数学的中心学科。最初的概念可以追溯到伯恩哈德·黎曼 (Bernhard Riemann) 于 1854 年 6 月 10 日发表的著名的训练讲座“Ueber die假设，welche der Geometrie zu Grunde liegen”（论几何基础上的假设）。这场讲座中的革命性创造深刻地改变了几何的全球景观。具体来说，黎曼提出了一种新颖的策略，将表面的几何形状推广到更高的维度，他称之为 Mannigfaltigkeiten（流形）。人们创造了各种各样的新概念：其中包括定量测量空间弯曲方式的曲率概念，以及连接流形上两点的长度最小化路径的测地线概念。对流形度量结构的研究，即我们现在所说的度量黎曼几何，主要关注基础空间的全局几何与度量结构之间的相互作用，即如何实现或测量两点之间的距离。该项目主要研究爱因斯坦流形的度量几何，其度量结构满足广义相对论中的爱因斯坦方程。 PI 将把他们的研究与各个级别的培训和指导结合起来。 这包括组织黎曼几何夏季研讨会和数学静修会；复杂几何和理论物理，并为本科生设计和开发新的研究型课程。该项目研究爱因斯坦流形的简并和定量行为。课题负责人与孙松合作，致力于具有特殊完整性的爱因斯坦流形的塌陷几何研究，取得了该领域的两项重大突破：K3流形上爱因斯坦度量的格罗莫夫-豪斯多夫极限的完整分类，以及引力瞬子渐近模型几何的完整分类。后者可以被视为 K3 流形上简并爱因斯坦度量的气泡极限。 在此背景下，PI 将继续研究更高维度的类似问题，并研究该环境中具有通用完整性的退化爱因斯坦度量的几何结构。与一组合作者一起，PI 还将在有关完整 Calabi-Yau 度量的更精细的几何和模空间问题上取得进展。第三个方向，PI将研究庞加莱-爱因斯坦空间的几何和分析，其起源于数学物理中的AdS/CFT对应关系。 PI 将重点关注简并算子在庞加莱-爱因斯坦流形上的奇点行为、庞加莱-爱因斯坦度量的几何有限性和定量刚性，以及庞加莱-爱因斯坦度量的正则性和简并理论。该奖项反映了 NSF 的法定使命，并已被通过使用基金会的智力优点和更广泛的影响审查标准进行评估，认为值得支持。