Dr Ben Sharp
I completed my PhD at the University of Warwick in 2012 under the supervision of Professor Peter Topping. I have since held postdoctoral positions at Imperial College London and the Scuola Normale Superiore di Pisa before returning to the UK as a Warwick Zeeman lecturer. I have been a lecturer in Leeds since January 2018.
Many problems in differential geometry and physics are phrased in terms of solutions to some variational problem, and most likely a system of partial differential equations. The main focus of my research is concerned with the analytic aspect of these problems.
The prototype object of study is a harmonic map between Riemannian manifolds - which is a "critical point" of the Dirichlet energy, and solves a non-linear, second-order elliptic PDE. Whilst the problem can be phrased in a straightforward manner, getting a handle on general existence and regularity results for such objects is tough and involves some challenging PDE and geometric measure theory. Some of my recent work is related to the regularity theory of harmonic maps.
From a purely PDE perspective, a fundamental tool (in harmonic maps) is a so-called "Wente" estimate, with links to Calderon-Zygmund estimates, Hardy spaces and more widely harmonic analysis. A more recent interest of mine is in trying to understand the optimality of these estimates. Surprisingly there is an intimate relationship between optimal Wente estimates (a purely 'analytical' problem) and surfaces of constant mean curvature in three-dimensions. Surfaces of constant mean curvature arise naturally in the form of soap bubbles, but equally are fundamental objects in geometry and physics.
Other interests of mine are in minimal surfaces and Willmore surfaces. Minimal surfaces are a subset of constant mean curvature surfaces, but in this case the mean curvature is zero meaning that they minimise area in small regions. They have maintained a prominent role in pure mathematics since the 18th century (starting with Euler and Lagrange), and have been used to prove theorems in both theoretical physics and geometry. My recent interests here are in understanding the relationship between the Morse index of a minimal surface (a measure of how far they are from being local minimisers) and their geometric-analytic properties.