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Geometric Analysis of Einstein Manifolds and Their Generalizations

爱因斯坦流形的几何分析及其推广

基本信息

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

项目摘要

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.

这个项目的目标是研究最初由物理产生的空间的几何结构。一个典型的例子是时空，它将物理空间的三维和时间的一维统一成一个四维系统，这样事件发生的地点和时间就可以在清晰有效的数学框架中进行研究。爱因斯坦的广义相对论指出，作为引力的来源，时空不是平坦的。时空如何弯曲可以通过曲率方程系统精确地确定，即爱因斯坦方程。从广义上讲，这个项目的中心是空间的曲率和几何之间的关系。后者，在我们的语境中，由局部和全局两个方面组成。局部几何是指空间在小尺度上的具体和刚性形状，而全局几何或拓扑是指在大尺度上连续变换不变的轮廓。空间的几何复杂性总是与爱因斯坦方程解的解析奇异行为相对应，这是一个基本原理。这个项目的核心部分是致力于开发新的工具和技术来理解爱因斯坦方程，它实质上反映了底层空间的新几何结构。除了探索微分几何前沿的开放和基本问题外，该项目还有助于建立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的对应关系。这个项目是关于爱因斯坦流形族坍缩到一个低维度量空间。与亚伦·纳伯一起，PI获得了坍缩爱因斯坦空间的规则和结构定理的新风格。PI将继续这个项目，探索奇异集的结构和对坍缩空间的气泡进行分类。除了研究一般塌缩理论外，PI还将与宋孙合作，在任何维度上构建大量新的塌缩爱因斯坦空间，这将预测新的现象，特别是高维几何。在高维情况下，极限奇异集的无序几何性质和缺乏有效的正则性理论将成为分析和构造过程中的主要困难。建设中的新工具和新技术比问题本身更有趣，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Kaehler结构的爱因斯坦4流形坍塌的问题。PI将与Gao Chen和Jeff Viaclovsky合作，解决涉及椭圆K3曲面的问题。本方向的第一部分将研究具有一般椭圆型纤曲的K3表面的度规表征。具体来说，PI和他的合作者将从几何上识别气泡，并定量描述每一类椭圆K3曲面的度量行为，这将从本质上有效地连接几何坍缩和代数退化。在第二部分中，PI与Hans-Joachim Hein， Song Sun和Jeff Viaclovsky合作，成功地在K3曲面上构造了坍缩至闭合区间的塌缩ricci平面度量族，这实际上给出了代数几何中极化K3曲面的II型复杂结构退化的度量几何描述。基于新的度量结构，PI和他的合作者将继续这个项目，以了解K3表面模空间的边界结构。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。