Differential Geometry Seminars 2023

Scattering for wave equations with sources close to the light cone

For the classical wave equation there is an isometry from the space of initial data to the space of radiation fields. I will review an improved result which allows us to construct global solutions to the wave equation with detailed asymptotics from knowledge of the radiation field, when the latter is known to decay. For wave equations with sources near the light cone, the radiation field has a logarithmic singularity, and solutions have non-trivial decay properties in the interior of the light cone. I will discuss the scattering problem in this setting, and focus on the role of homogeneous solutions in the interior, and exterior of the light cone, and their role in the construction of matching approximate solutions. The sources we consider have applications to semi-linear wave equations satisfying the weak null condition as they appear in the study of Einstein's equations in harmonic coordinates. This is joint work with Hans Lindblad.

Arbitrarily high order concentration-compactness for curvature flow

We extend Struwe and Kuwert-Schätzle's concentration-compactness method for the analysis of geometric evolution equations to flows of arbitrarily high order, with the geometric polyharmonic heat flow (GPHF) of surfaces, a generalisation of surface diffusion flow, as exemplar. For the (GPHF) we apply the technique to deduce localised energy and interior estimates, a concentration-compactness alternative, pointwise curvature estimates, a gap theorem, and study the blowup at a singular time. This gives general information on the behaviour of the flow for any initial data. Applying this for initial data satisfying $||A^o||_2^2 < \varepsilon$ where $\varepsilon$ is a universal constant, we perform global analysis to obtain exponentially fast full convergence of the flow in the smooth topology to a standard round sphere.The talk will focus mostly on the concentration compactness duality and how it translates from its initial appearance in elliptic setting in the literature to the parabolic setting of flows and finally to our geometric polyharmonic heat flow.

Cohomogeneity One Ancient Ricci Flows

The construction and classification of ancient solutions to the Ricci flow is a popular subject in geometric analysis which has arisen out of the need to provide useful models of finite-time singularities of Ricci flow. In this talk, I will discuss classical examples of ancient Ricci flows which are rotationally-invariant, as well as some more recent constructions of ancient flows on spheres which are invariant under a product of two orthogonal groups.