Research

I am interested in solving differential equations that arise in differential geometry, as well as mathematical analysis generally. I quite often use symmetry reductions in my quest to study solutions of these equations. Descriptions of my papers are given below, alongside their respective links (if available). 

ArXiv preprints

• Scalar curvature along Ebin geodesics (with Christoph Böhm and Brian Clarke, to appear in Crelle). Here we are interested in the behaviour of scalar curvature along curves of Riemannian metrics. We show that, in case the curve is an Ebin geodesic on the space of metrics with fixed volume form on a compact manifold (dimension at least five), there is an overwhelming tendancy for scalar curvature to converge uniformly to minus infinity. This observation generalises classical results about the Einstein-Hilbert functional on homogeneous metrics obtained by Böhm, Wang and Ziller.

• Local solvability of the Poisson equation for closed G_2 structures (with Artem Pulemotov). In a neighbourhood of a point, we show that it is always possible to find a closed G_2 structure on a seven-dimensional manifold with Hodge Laplacian coinciding with a prescribed closed three-form that is either positive or negative. 

• Ancient Ricci pancakes of bounded girth (with Theodora Bourni, Ramiro Lafuente and Mat Langford). We construct an O(2)xO(n-1)-invariant ancient Ricci flow on S^n for n at least 3. As t tends to -infinity, this Ricci flow tends  to a flat cylinder S^1x R^(n-1) in the 'waist' region of the sphere, and tends  to the product of a cigar soliton on R^2 with flat R^(n-2) in the 'tip' region. In particular, this ancient Ricci flow is collapsing. 

• Rotationally-invariant Ricci flow with surgery (with Max Hallgren and Yongjia Zhang, to appear in Transactions of the American Mathematical Society). We obtain existence and uniqueness of weak solutions of Ricci flow in the special case that the initial Riemannian metric is rotationally-invariant. This extends work of Kleiner-Lott and Bamler-Kleiner to higher-dimensional manifolds. 

• SU(2)-invariant steady gradient Ricci solitons on four-manifolds (to appear in Communications in Analysis and Geometry). A number of interesting four-dimensional steady Ricci solitons have been constructed which are foliated by homogeneous Berger spheres on SU(2)/Z_n, and are completed by inserting a copy of S^2 into the topology. We provide a simpler construction of these U(2)-invariant examples, and also produce a new SU(2)-invariant soliton for n=4. 

Published articles

• The prescribed cross curvature problem on the three-sphere (with Artem Pulemotov). The cross curvature tensor, which was introduced by Ben Chow and Richard Hamilton in 2004, is a symmetric (0,2) tensor field which  describes how close a three-dimensional Riemannian manifold is to having constant sectional curvature. Hamilton later conjectured that on S^3, any positive-definite symmetric (0,2) tensor field would be the cross curvature of exactly one Riemannian metric. We provide a wealth of examples demonstrating that the existence component is likely to be true, but we also find a counter-example to the uniqueness component of this conjecture.

• SO(2)xSO(3)-invariant Ricci solitons and ancient flows on S^4. The best general compactness result for shrinking Ricci solitons on compact four-dimensional manifolds is by Haslhoffer-Muller, who show that an appropriately-normalised sequence of such solitons converges in the orbifold sense, provided we have a uniform lower bound on the Pereman entropy. In this paper, we upgrade this result to obtain uniform Riemann curvature bounds on S^4 in the special case that our shrinking solitons are invariant under the cohomogeneity one action of SO(2)xSO(3). This provides strong evidence to suggest that the only available Ricci solitons in this context are the canonical ones. However, our analysis provides a sequence of `almost' Ricci solitons that are reflective of the existence of a `pancake' ancient Ricci flow solution. We  discuss the existence of this ancient Ricci flow solution, which would be an SO(2)xSO(3)-invariant analogue of the well-known `Perelman sausage' ancient flow. 

• Deducing flux from single point temperature history when relative spatial variation of flux is prescribed (with David Buttsworth). We discuss the problem of determining how much heat is passing through the surface of a body, when some information is known about ambient heat flow. The fundamental mathematical model for this situation is that of the heat equation with Neumann conditions. However, in our situation, we only know the Neumann conditions up to an unknown time-varying constant, so the boundary conditions are equipped with a Dirichet measurement at a single point on the boundary. The resulting boundary value problem is usually well-posed, and can be solved with a combination of Neumann heat kernel methods, and Laplace transforms. 

• Prescribing Ricci curvature on a product of spheres (with Anusha Krishnan). We examine  the global solvability of the prescribed Ricci curvature problem for certain cohomogeneity one metrics on products of spheres. Our work provides the first global existence result for solutions of the prescribed Ricci curvature problem on closed cohomogeneity one manifolds since a result of Richard Hamilton in 1984. 

• Local stability of Einstein metrics under the Ricci iteration (with Max Hallgren). We establish a local stability result for the Ricci iteration, which can be thought of as a discrete analogue of the Ricci flow. Of course, our main tool is the inverse function theorem, but this can only be applied once we 'quotient out' by the diffeomorphism invariance of the Ricci curvature. This is achieved using techniques similar to those used in the proof of Ebin's Slice Theorem. 

• The prescribed Ricci curvature problem for homogeneous metrics (with Artem Pulemotov). This is a survey article, included as a chapter of `Differential Geometry in the large' which was published as part of the London Mathematical Society's lecture note series. 

• On the Ricci iteration for homogeneous metrics on spheres and projective spaces  (with Artem Pulemotov, Yanir Rubinstein and Wolfgang Ziller). We completely determine the long-time behaviour of the Ricci iteration for all homogeneous metrics on spheres and projective spaces, except for those Sp(n+1)-invariant metrics on the sphere of dimension 4n+3. In this most challenging case, we obtain some partial results, as well as a new result for the prescribed Ricci curvature equation. 

• Cohomogeneity-one quasi-Einstein metrics. We obtain some fairly broad existence results for solutions to the Dirichlet problem for cohomogeneity one quasi-Einstein metrics. We also establish an optimal rate of blow-up for solutions of these ordinary differential equations near singularities. 

• The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds. We use Leray-Schauder degree theory to find existence results for solutions to the Dirichlet problem for cohomogeneity one Einstein metrics.

• The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups. We examine the global solvability of the prescribed Ricci curvature problem for left-invariant metrics on the three-dimensional unimodular Lie groups. This is one of the first papers to examine the global problem on non-compact manifolds.