Singularity formation in Kahler geometry

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

This project focuses on special geometric structures, known as Kahler-Einstein metrics, which represent generalizations of solutions to Einstein's equations for gravity, as wells as structures called instantons, which generalize solutions to Maxwell's equations for electromagnetism. These structures play pivotal roles in modern theoretical physics, particularly in areas like string theory. More specifically, this project is mainly concerned with how these structures can develop defects, known as singularities, and to what extent the resulting object has a predictable shape, when viewed in the large. The successful resolution of these problems will substantially enhance our comprehension of the intricate interactions between various research directions in mathematics. This project will also have broader impacts, as it will give rise to numerous fascinating and interconnected questions offering ample research and training opportunities for graduate students keen on exploring this captivating field. The PI's research will be concentrated on several lines of investigation. Firstly, the PI plans to investigate the structure of complete Calabi-Yau metrics with Euclidean volume growth. The PI, along with J. Zhang, has already achieved a satisfactory structural theory under the assumption of quadratic curvature decay. However, the general scenario poses more significant challenges, necessitating the development of novel technical tools. Secondly, the PI is interested in understanding the collapsing phenomenon in canonical metrics. In this regard, previous collaborative work with R. Zhang has made substantial progress, specifically in the context of 4-dimensional hyperkähler metrics. Extending these findings to other settings holds paramount importance for their applications in geometry. Lastly, the PI will delve into related questions concerning Hermitian-Yang-Mills instantons. While parallel questions can be raised, answering them will require the incorporation of intriguing new techniques. 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.