Complex manifolds are higher-dimensional surfaces that are defined using the complex numbers. A "metric" on such a space is an object that endows that space with a shape or more precisely, with Ricci curvature. The study of these metrics lies at the nexus of complex geometry and geometric analysis. This project is concerned with the construction of complete Kahler metrics with prescribed Ricci curvature on non-compact Kahler manifolds; more specifically, non-compact Calabi-Yau manifolds and Kahler-Ricci solitons. Applications of such manifolds permeate throughout physics and mathematics. Indeed, Calabi-Yau manifolds of complex dimension three have become prominent in string theory where they supposedly model the six additional real dimensions of space-time that we do not see, whereas Kahler-Ricci solitons provide models for the formation of singularities in an important evolution equation, namely the Kahler-Ricci flow. Efforts will be made to construct new examples of these manifolds and to formulate their existence in terms of algebraic geometric criteria. This will enhance our understanding of the geometry of non-compact Kahler manifolds and will involve applications of techniques from analysis, in particular partial differential equations, and algebraic geometry.In more technical terms, the proposed research project comprises two parts. The goal of the first part is to construct more examples of non-compact complete Calabi-Yau manifolds with Euclidean volume growth and with a singular non-flat tangent cone at infinity, developing the work of Li, the PI and Rochon, and Szekelyhidi. Such manifolds have been used as building blocks for G2-manifolds and may shed light on a conjecture of Yau asserting the compactifiability of a complete Calabi-Yau manifold with finite topology. Moreover, these manifolds have applications in physics such as in the AdS/CFT correspondence and string theory. The second part of the proposal deals with studying complete non-compact gradient Kahler-Ricci solitons. The objectives of this part are threefold.(1) To glue the expanding gradient Kahler-Ricci solitons that the PI has constructed with Deruelle together on a compact Kahler manifold to construct a solution of the Kahler-Ricci flow with singular initial data. This ties in with the analytic Minimal Model Program using the Kahler-Ricci flow proposed by Song and Tian and builds on the work of Gianniotis and Schulze. (2) To construct examples of complete expanding gradient Kahler-Ricci solitons with a singular tangent cone at infinity with Euclidean volume growth of which there are currently no known examples. This builds on the work of the PI and Deruelle.(3) To study the existence and uniqueness of non-compact complete shrinking gradient Kahler-Ricci solitons for a fixed holomorphic vector field. This is a natural extension of work by the PI, Deruelle, and Sun.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.
