Geometric Analysis of Einstein Manifolds and Their Generalizations

该项目的目标是研究最初由物理学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，作为引力源的时空是不平坦的。时空是如何弯曲的，可以由一个曲率方程组，即爱因斯坦方程精确地确定。广义地说，这个项目的中心是空间的曲率和几何之间的关系。后者，在我们的背景下，几何，包括局部和全局方面。局部几何是指在小尺度下空间的具体和刚性形状，而全局几何或拓扑则关注在大尺度下的轮廓，这些轮廓在连续变换下是不变的。空间的几何复杂性总是对应于爱因斯坦方程解的解析奇异性，这是一个基本原理。该项目的核心部分致力于开发新的工具和技术，以理解爱因斯坦方程，该方程反映了基本空间的新几何结构。除了追求微分几何前沿的开放和基本问题外，该项目还有助于建立新开发的几何结构与物理学科（如量子场论和弦理论）中的几何原理之间的对应关系。这个项目是关于一族爱因斯坦流形坍缩到一个低维度量空间。PI与Aaron Naber一起获得了一种新的正则性和坍缩爱因斯坦空间的结构定理。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。除了研究一般的坍缩理论外，PI还将与Song Sun合作，在任何维度上构建各种新的坍缩爱因斯坦空间，这将预测新的现象，特别是对于高维几何。在高维空间中，极限奇异集的几何性质的混乱和缺乏有效的正则性理论将构成分析和构造过程中的基本困难。新的工具和技术在建设中被期望比问题本身更有趣，这将产生许多问题和新的研究方向。在另一个研究方向中，PI将解决涉及具有Kaehler结构的爱因斯坦4-流形坍缩的问题。与高晨和杰夫Viaclovsky联合，PI将解决涉及椭圆K3曲面的问题。这个方向的第一部分将研究具有一般椭圆纤维化的K3曲面的度量特征。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曲面的度量行为，这将有效地将几何崩溃和代数退化联系起来。在第二部分中，PI与Hans-Joachim Hein，Song Sun和Jeff Viaclovsky一起，成功地在K3曲面上构造了一族坍缩到闭区间的坍缩Ricci平坦度量，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，具体目标是了解K3表面模量空间的边界结构。这个奖项反映了NSF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。

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.