NSF Award
1745517
An Einstein manifold is a Riemannian manifold of constant Ricci curvature. This means that we have a space permeated by a physical field governed by a certain system of nonlinear partial differential equations, closely related to but subtly different from how the gravitational field in Einstein's theory of General Relativity permeates spacetime. Independent of their origin in physics, Einstein manifolds are beautiful mathematical objects in their own right whose study links together many disparate areas of mathematics. The space underlying an Einstein manifold usually has dimension 4 or higher, which makes it difficult to visualize such structures. However, Einstein manifolds in dimension 4 share many basic features with minimal surfaces in dimension 2, which have been a staple of mathematical research and popularization for decades.<br/><br/>It is a key feature of nonlinear partial differential equations that solutions may form singularities and break down at points of infinite physical field strength. An "end" of an Einstein manifold is a neighborhood of such a singularity in space. The aim of this project is to advance our understanding of these ends in two complementary ways: by constructing new examples with unexpected properties, and by proving classification theorems to narrow down the range of possible phenomena. On the examples side, this will include the construction of Einstein metrics with isolated singularities not homeomorphic to their tangent cones, Einstein metrics with complex hyperbolic cusp ends, Einstein metrics interpolating between different Ricci-flat cones, and Einstein manifolds of unbounded topology. On the classification side, the main goal is to resolve open questions in the theory of gravitational instantons (complete hyper-Kahler manifolds of real dimension 4), by further developing a compactification technique recently introduced by the PI and his coauthors.
