Ricci flow of manifolds with singularities at infinity

This proposal concerns geometric flows, which is a subject that lies at the interface of differential geometry, analysis, topology and the theory of nonlinear partial differential equations (PDEs). More specifically, we will consider Ricci flow, which is a way of taking a curved space, known as a Riemannian manifold, and deforming it in time to make it more uniform.

The importance of the field cannot be overstated. Ricci flow is famous for solving a string of major problems such as the 100 year old Poincaré conjecture, which had a $1,000,000 bounty attached to it, and Thurston's geometrisation conjecture, but the potential extent of its applications lies far beyond. Up until now, the theory has focussed almost exclusively on manifolds that are compact, or that have artificial constraints on their behaviour at infinity such as a uniform upper curvature bound or a positive uniform lower bound on the volume of every unit ball. This proposal is directed towards the next wave of applications. To realise these we must understand flows that are singular at infinity, and to do this we will need to advance the theory of nonlinear PDEs and understand better their interaction with geometry. We will require a collection of innovations, including new curvature estimates and a better understanding of the geometry at infinity of positively curved manifolds.

Even partial success along these lines will transform the applicability of the field. Progress will give us an understanding of the geometry and topology of open manifolds without artificial asymptotic constraints on their geometry. We give some illustrative examples of major open problems that would fall to the advances that we envisage, such as Yau's Uniformisation Conjecture, and describe a route to achieve them.

The proposal has some highly ambitious objectives. However, it also contains a collection of conjectures and problems, of varying difficulty, that push on many fronts against the central aim of understanding flows with unbounded curvature, and collapsing behaviour, at infinity. What is particularly exciting about this research direction is that only in the past few years have we been successful in developing the foundational theory to make this feasible. Thanks to the work of several international teams, including that of the PI and M. Simon in their resolution of the Anderson-Cheeger-Colding-Tian conjecture in 3D, we now have a clear idea of the required a priori estimates, which differ substantially from the scale-invariant estimates proved thus far, and we finally have a roadmap towards establishing them.

A large part of the immediate impact of this proposal is within other areas of pure mathematics. The research will have impact on the broad subject of geometric analysis since it will permit researchers to develop and apply Ricci flow in hitherto unreachable settings. However, in addition, there are some famous long-standing open problems centred on differential geometry that would be resolved in the event of sufficient progress. In this way the research will have significant impact on complex geometry, differential geometry, metric geometry and topology.

The technical advances that we envisage during this project will advance the entire field of geometric flows, which in turn has wide applications outside mathematics. For example, the twin geometric flow of Ricci flow is the mean curvature flow, which has seen applications traditionally in materials science, and more recently in a vast array of subjects from cell biology to droplet physics to relativity, with particularly intense activity in imaging. As we describe in the Pathways to Impact section, a previous member of the PI's group is applying the geometric flows he studied at Warwick to the analysis of dynamic MRI and dynamic PET data at the Commonwealth Scientific and Industrial Research Organisation with clinical data from Royal Brisbane and Women's Hospital and the Princess Alexandra Hospital, Brisbane. Ideas that will be developed in this project could have impact in such a direction over a period of a decade.

The proposed research will have diverse, positive impact for the UK. It will develop the UK infrastructure in geometric analysis, and the skills base for applying nonlinear PDEs to geometry, as called for at every opportunity by the two most recent EPSRC-commissioned International Reviews of Mathematics, and the EPSRC landscape document in mathematical analysis. It will add to the UK's postdoctoral training capacity in geometric analysis, contributing to a people pipeline for which demand is currently outstripping supply. In this way, it will contribute to the modernisation of the UK's research portfolio.

More generally, a healthy people pipeline consisting of mathematicians trained in fields that are highly relevant to many other disciplines, such as the field of geometric flows, is also essential to make the UK a fertile area for completely new industries to emerge. Companies such as Google have been able to continually transform mathematical ideas into spectacular economic success, and it is essential that we create an environment in the UK where such development can occur. This proposal will have a positive impact on this task. The more analytical sides of the proposal also represent an important training aspect. A large proportion of the UK's economy is made up of financial and industrial activity that is underpinned by nonlinear partial differential equations, and this proposal would have impact on training in that direction.