The PI proposes to construct a Riemannian metric on the Gromoll-Meyer sphere with positive sectional curvature. This is a long standing open problem and is now especially relevant as Brendle-Schoen have shown that no exotic sphere admits a metric with positive complex sectional curvatures. As a separate specific project the PI also proposes to understand when quasi-Einstein and gradient soliton metrics are forced to be Einstein metrics. These questions go back to the earliest works on Einstein metrics, but have received new attention with Perel'man classification of certain gradient solitons in three dimensions. Understanding quasi-Einstein metrics is also important in general relativity as they occur as solutions to the Einstein field equations.
In general terms the PI wishes to investigate what types of geometries are possible on specific topological objects. Two objects such as a doughnut and a tea cup are geometrically very different but topologically similar. The goal is to find the nicest possible geometries for specific topological objects. This is of interest to mathematicians, physicists, computer scientists,medical scientists and many other people as it is becoming increasingly clear that our flat Euclidean picture is not always the correct or even nicest model to use.
The goal of the project was to study the special nature of solutions to Einstein's field equations: Are such solutions unique if one assumes they have certain special properties? The PI showed that under certain symmetry assumptions, that are present in many known solutions, that such solutions are unique. The natural symmetry conditions are the same that one sees in a model of the solar stystem where the sun has a gravitational field that is the same in all directions. Are such solutions stable? This is a classic question of whether they are "peak" or "pits". That is to say, are solutions placed at a peak where a small change will make them fall away from their special position, or are they in a pit where a small change will make them comes back to the origial position. The PI established pit stability for significant new classes of solutions. Do homogeneous solutions have wholes or other features that might create tunnnels in the universe. The homogeneous assumption is that the solution should look the same everywhere. This does not preclude in general that it might have wholes that are doughnut shaped. Some progress on this very old and classical problem was made, but many unexplored possibilities still remain.
