Geometric Analysis of Einstein Manifolds and Their Generalizations

爱因斯坦流形的几何分析及其推广

• 批准号: 2212818
• 负责人: Ruobing Zhang
• 金额: $ 14.16万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-10-01 至 2023-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2212818&HistoricalAwards=false
• 关键词: Geometric; Analysis; Einstein; Manifolds; Their

The goal of this project is to study geometric structures of the spaces originally arising from physics. A prototypical example is the spacetime which unifies the three dimensions of physical space and the one dimension of time into a four-dimensional system such that where and when events occur can be studied in clean and effective framework of mathematics. It is indicated by Einstein's general relativity theory that, as a source of gravitation, the spacetime is not flat. How the spacetime is curved can be precisely determined by a system of curvature equations, namely, the Einstein equations. Broadly speaking, this project centers on the relationship between the curvature of a space and the geometry on it. The latter, geometry in our context, consists of both local and global aspects. The local geometry refers to the concrete and rigid shape of a space at small scales, while the global geometry or topology focuses on the profiles at large scales which are invariant under continuous transformations. It is a fundamental principle that geometric complications of a space always correspond to the analytic singularity behaviors of the solution to the Einstein equation. The central part of this project is dedicated to the development of new tools and techniques in understanding the Einstein equation, which reflects substantially new geometric structures of the underlying space. In addition to pursuing open and fundamental problems at the forefront in differential geometry, this project also contributes to establishing correspondence between the new developed geometric structures and the conjectural principles in physical disciplines such as quantum field theory and string theory. This project is concerned with a family of Einstein manifolds collapsing to a lower dimensional metric space. Together with Aaron Naber, the PI obtained a new flavor of regularity and structure theorem for collapsing Einstein spaces. The PI will continue this project to explore the structure of the singular sets and classifying bubbles for collapsing spaces. Besides studying general collapsing theory, joint with Song Sun, the PI will construct a large variety of new collapsed Einstein spaces in any dimension, which will predict new phenomena especially for higher dimensional geometries. In higher dimensions, the wild geometric nature of the limiting singular set and the lack of effective regularity theory would constitute essential difficulties in analysis and in the construction procedure. The new tools and techniques in the construction are expected more interesting than the problem itself, which will generate many problems and new directions to study. In another line of investigation, the PI will address issues involving collapsing Einstein 4-manifolds with Kaehler structures. Joint with Gao Chen and Jeff Viaclovsky, the PI will address issues involving elliptic K3 surfaces. The first part of this direction would study the metric characterizations of K3 surfaces with generic elliptic fibrations. Specifically, the PI and his collaborators will geometrically identify the bubbles and quantitatively describe the metric behaviors in each class of elliptic K3 surfaces, which would essentially connect the geometric collapsing and algebraic degeneration in an effective way. In the second part, with Hans-Joachim Hein, Song Sun and Jeff Viaclovsky, the PI have managed to construct a family of collapsed Ricci-flat metrics on K3 surfaces which collapse to a closed interval, which in effect gives a metric-geometric description for the Type II complex structures degeneration of polarized K3 surfaces in algebraic geometry. Based on the new metric constructions, the PI and his collaborators will continue this program with a specific goal to understand the boundary structure of the moduli space of the K3 surface.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.

该项目的目标是研究最初源于物理学的空间几何结构。一个典型的例子是时空，它将物理空间的三个维度和时间的一维统一成一个四维系统，这样就可以在干净有效的数学框架中研究事件发生的地点和时间。爱因斯坦的广义相对论表明，作为引力源的时空并不是平坦的。时空如何弯曲可以通过曲率方程组（即爱因斯坦方程）精确确定。从广义上讲，该项目以空间曲率与其几何形状之间的关系为中心。后者，即我们上下文中的几何，由局部和全局两个方面组成。局部几何是指小尺度空间的具体而刚性的形状，而全局几何或拓扑则关注大尺度的轮廓，这些轮廓在连续变换下保持不变。一个基本原理是，空间的几何复杂性总是对应于爱因斯坦方程解的解析奇点行为。该项目的核心部分致力于开发新的工具和技术来理解爱因斯坦方程，该方程反映了底层空间的全新几何结构。除了追求微分几何前沿的开放和基本问题外，该项目还有助于在新发展的几何结构与量子场论和弦论等物理学科的猜想原理之间建立对应关系。该项目涉及一系列坍缩到较低维度量空间的爱因斯坦流形。与 Aaron Naber 一起，PI 获得了爱因斯坦空间坍缩的规律性和结构定理的新风格。 PI 将继续这个项目，探索奇异集的结构并对塌陷空间的气泡进行分类。除了与孙松一起研究一般坍缩理论外，PI还将在任何维度上构造大量新的坍缩爱因斯坦空间，这将预测新现象，特别是对于高维几何。在更高维度中，极限奇异集的狂野几何性质和缺乏有效的正则理论将构成分析和构造过程中的本质困难。构建中的新工具和技术预计比问题本身更有趣，这将产生许多问题和新的研究方向。在另一项调查中，PI 将解决涉及凯勒结构的爱因斯坦 4 流形塌缩的问题。 PI 将与高晨和 Jeff Viaclovsky 合作解决涉及椭圆 K3 曲面的问题。该方向的第一部分将研究具有通用椭圆纤维振动的 K3 表面的度量特征。具体来说，PI和他的合作者将从几何上识别气泡并定量描述每类椭圆K3曲面中的度量行为，这将从本质上以有效的方式连接几何崩溃和代数退化。在第二部分中，PI 与 Hans-Joachim Hein、Song Sun 和 Jeff Viaclovsky 一起成功地在 K3 曲面上构建了一系列折叠 Ricci 平坦度量，这些度量折叠为闭区间，这实际上给出了度量几何描述用于代数几何中极化 K3 表面的 II 型复杂结构退化。基于新的度量结构，PI 和他的合作者将继续该项目，其具体目标是了解 K3 表面模空间的边界结构。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准。

项目成果

Hodge theory on ALG ∗ manifolds

ALG 的 Hodge 理论 — 流形

• DOI: 10.1515/crelle-2023-0016
• 发表时间: 2023-05
• 期刊: Journal für die reine und angewandte Mathematik (Crelles Journal
• 作者: Chen, Gao;Viaclovsky, Jeff;Zhang, Ruobing
• 通讯作者: Zhang, Ruobing

Nilpotent structures and collapsing Ricci-flat metrics on the K3 surface

K3 表面上的幂零结构和塌陷 Ricci 平坦度量

• DOI: 10.1090/jams/978
• 发表时间: 2022-01
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: Hein, Hans;Sun, Song;Viaclovsky, Jeff;Zhang, Ruobing
• 通讯作者: Zhang, Ruobing