The goal of this project is to obtain further understanding of the structure of complete noncompact manifolds. The general theme is that understanding the solutions of some partial differential equations in terms of curvature can be used to obtain more information about the geometry and topology of the manifold. In this context, the PI will study bounds on the bottom of spectrum of the Laplacian in terms of the Ricci curvature and use the spectral information to study the structure of the manifold. The PI will also study geometric and analytic properties of complete gradient Ricci solitons, aiming at a better understanding of the singularities of the Ricci flow.
The proposed project will lead to a better understanding of the shape and structure of infinite geometric objects. These objects, called manifolds, appear frequently in science because physical laws can be modeled by certain partial differential equations on manifolds. Understanding the structure of manifolds is important in many fields, such as black holes and worm holes in physics, the structure of molecules in chemistry or the motion of liquid crystal in engineering. Partial differential equations have been a fundamental tool in geometry, for example the Ricci flow has been used to classify the shape of all three dimensional manifolds. Further advances in the study of geometric partial differential equations on manifolds would lead to new discoveries applicable to many areas of mathematics and science.
This project studied how curvature affects the structure of complete manifolds. From the several notions of curvature that can be defined on a manifold, Ricci curvature is one of the most intriguing and well studied. There has been much effort to understand the structure of manifolds that have Ricci curvature bounded below or, in particular, constant. Indeed, these are the well known Einstein manifolds, which have been the object of study for both mathematicians and physicists. The Ricci flow, proposed and studied by Hamilton since the early eighties, is a parabolic partial differential equation involving the Ricci curvature of a manifold. The Ricci flow is expected to start with a given Riemannian metric and flow it in time to an improved one, such as one of constant Ricci curvature. Indeed, Einstein manifolds are fixed points of this flow. The self similar solutions of the Ricci flow, called Ricci solitons, are natural generalizations of Einstein manifolds. Understanding Ricci solitons is an important step in the study of the Ricci flow, because they model the singularities of the flow. When the manifold is three dimensional, the classification of Ricci solitons was instrumental in the Hamilton-Perelman proof of the Poincare conjecture. A main goal of this project was to study and understand Ricci solitons in arbitrary dimension. Now there is a much larger variety of examples, so a complete classification is out of reach. It is more acceptable to try to understand how big is the space of Ricci solitons. This can be achieved in many ways, for example, by finding restrictions on the geometry and topology of solitons, or by showing that the space is compact in some sense. An outcome of the project proved that nontrivial Ricci solitons are connected at infinity, which means that they have only one end. This is a serious restriction on the structure at infinity of these spaces. In the process of proving this result, we have established some interesting connections with other fields. Such is the case of the smooth metric measure spaces and their associated Bakry-Emery curvature. This is a natural generalization of the Ricci curvature, in the presence of a density function on the manifold. Ricci solitons are smooth metric measure spaces with constant Bakry-Emery curvature, so in this sense they are again similar to Einstein manifolds. Another outcome of the project was that it provided significant evidence that Ricci solitons model smooth metric measure spaces with Bakry-Emery curvature bounded from below, so they are canonical spaces in this larger class of manifolds. Other efforts in studying the space of Ricci solitons have focused on proving a precompactness property. This means that we want to know whether any sequence of solitons admits a converging subsequence, and in what sense. This is a well studied topic for Einstein manifolds, and some of this work can be naturally generalized to Ricci solitons. Our point of view is that nontrivial Ricci solitons, that is, those who are not Einstein or do not contain Einstein factors, should be more rigid and special. Another important outcome of this project is that it proved that the Riemann curvature tensor of a Ricci soliton is controlled at infinity by its Ricci curvature. In particular, this provides more structure for the limit of sequences of Ricci solitons. Such a result is not true for Einstein manifolds. Ricci flow has also attracted the attention of physicists, being seen as the renormalization group flow in string theory. In this context, the Ricci solitons can be viewed as fixed points of the renormalization group flow. Other geometric flows, such as the mean curvature flow or the Yang-Mills flow, share many characteristics with the Ricci flow. They also have remarkable applications to other fields of mathematics and science, such as in general relativity or in computer visualization. More generally, understanding the geometry and topology of manifolds is important in physics, having impact in our understanding of the Universe, in engineering for digital signal processing, or in chemistry for the study of molecules.
