Metric geometry and analysis on Einstein manifolds

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·黎曼在1854年6月10日发表的著名的《论几何基础上的假设》（Ueber die hypothesis, welche der Geometrie zu Grunde liegen）。这个讲座中革命性的创造深刻地改变了全球几何的格局。具体来说，Riemann提出了一种将曲面几何推广到更高维度的新策略，他称之为Mannigfaltigkeiten（流形）。各种各样的新概念和概念被创造出来：其中包括曲率的概念，它定量地测量空间是如何弯曲的，以及测地线的概念，它是连接流形上两点的长度最小路径。流形度量结构的研究，即我们现在所说的度量黎曼几何，主要关注底层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。本课题主要研究爱因斯坦流形的度规几何，其中度规结构满足广义相对论中的爱因斯坦方程。PI将把他们的研究与不同层次的培训和指导结合起来。这包括组织夏季工作坊和黎曼几何数学静修；复杂几何和理论物理，以及为本科生设计和开发新的研究型课程。本课题研究爱因斯坦流形的退化和数量行为。在与宋孙的合作中，PI一直在研究具有特殊完整度的爱因斯坦流形的坍缩几何，导致了该领域的两个重大突破：K3流形上爱因斯坦测度的Gromov-Hausdorff极限的完全分类，以及引力瞬子渐近模型几何的完全分类。后者可以看作是退化的爱因斯坦度量在K3流形上的泡极限。在此背景下，PI将继续研究高维的类似问题，并研究在这种情况下具有一般完整度的退化爱因斯坦度量的几何结构。与一组合作者一起，PI还将在更精细的几何和模空间问题上取得进展，这些问题涉及完整的Calabi-Yau度量。在第三个方向上，PI将研究庞加莱-爱因斯坦空间的几何和分析，它起源于数学物理中的AdS/CFT对应。PI将重点研究简并算子在庞加莱-爱因斯坦流形上的奇异行为，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退化理论。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.