Geometry and Analysis

The aim of the talk is to present recent developments of high frequency analysis for sub-elliptic operators and in sub-Riemannian geometry. I will start with discussing why these questions are closely related to many aspects of harmonic analysis. 

Magnetic monopoles were introduced by Dirac in 1931 to explain the quantisation of electric charges. In his model, they are singular solutions to an extension of Maxwell's equations allowing non-zero magnetic charges. An alternative model, developed by 't Hooft and Polyakov in the 1970s, is given, after a certain simplification, by smooth solutions to a different set of equations, the Bogomolny equations, whose moduli space of solutions has connected components Mk indexed by positive integers k (the "total magnetic charge"). These moduli spaces, which are non-compact manifolds, have an interpretation in terms of rational self-maps of CP1 due to Donaldson and their stable homotopy types may be described in terms of braid groups by a result of F. Cohen, R. Cohen, Mann and Milgram. A partial compactification of Mk has recently been constructed by Kottke and Singer, whose boundary strata may be called "ideal" or "asymptotic" monopole moduli spaces. I will describe joint work with Ulrike Tillmann in which we prove the existence of stability patterns in the homology of these spaces.

Is there a curvature identity that holds on any Riemannian manifold? Through the Chern-Gauss-Bonnet theorem, we can derive curvature identities that apply to 4-dimensional or 6-dimensional Riemannian manifolds. As an application of curvature identities, we prove Lichnerowicz’s conjecture in 4 dimensions under a slightly more general setting. Furthermore, we explore weakly Einstein manifolds, which arise as a generalization of 4-dimensional Einstein manifolds through the application of curvature identities. We also investigate the existence and non-existence of weakly Einstein metrics on certain Lie groups in recent studies, and propose a conjecture based on these results. (This is joint work with Y. Euh, S. Kim and Nikolayevsky.)

Nearly Kähler manifolds are Riemannian 6-manifolds admitting real Killing spinors. They are the cross-sections of Riemannian cones with holonomy G2. Like the Einstein equation, the nearly Kähler condition has a variational interpretation in terms of volume functionals, first introduced by Hitchin in 2001.

The existence problem for nearly Kähler manifolds is poorly understood, and the only currently known inhomogeneous examples were found in 2017 by Foscolo and Haskins using cohomogeneity one methods. For one of their examples, we establish non-trivial bounds on the coindex of the Hitchin-type and Einstein functionals. We do this by analysing the eigenvalue problem for the Laplacian on coclosed primitive (1,1)-forms under a cohomogeneity-one symmetry assumption.

Constant mean curvature (CMC) surfaces are special geometric variational objects, closely related to minimal surfaces. The key properties of a CMC surface are its area, mean curvature, genus, and index. The index of a CMC surface measures its stability: the index counts how many ways one can perturb the surface to decrease the area while keeping the enclosed volume constant. In this talk we discuss relationships between these key properties. In particular we present recent joint work with Ben Sharp, where we bound the index of CMC surfaces linearly from above by genus and the correct scale-invariant quantity involving mean curvature and area.