Geometric Analysis of Einstein Manifolds and Their Generalizations

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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.