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.

Loop spaces, Lie bialgebras and variations on Teichmüller space

A fundamental idea in “Grothendieck-Teichmüller theory” is that one can study the absolute Galois group of the rationals by studying the actions of the Galois group on the “Teichmüller tower”, i.e. the collection of (geometric) fundamental groups of all the moduli spaces $\mathcal{M}_{g,n}$ and the natural maps between them. This transforms a difficult arithmetic problem into a geometric problem.
The goal of this talk is to explain an extension of this idea where we replace one mysterious group (the absolute Galois group) with a family of mysterious groups (the higher genus Kashiwara-Vergne groups). The associated “KV tower” is a tower of fundamental groups of the free loop spaces on various punctured Reimann surfaces. We aim to describe how this variation on Teichmüller space turns a problem in Lie theory (the Kashiwara-Vernge problem) into a problem in topology.

Generalised spectral dimensions in non-perturbative quantum gravity

The crucial tasks of quantum gravity research is to find observables that show possible deviations from classical gravity. One such example is the scale dependent effective dimension of spacetime. It was first discovered in Causal Dynamical Triangulations (CDT) which is a model of non-perturbative quantum gravity. Subsequently, it has been discovered to be present in many different theories of quantum gravity. This seemingly universal phenomenon of scale-dependent effective dimensions in non-perturbative theories of quantum gravity has been shown to be a potential source of quantum gravity phenomenology. The scale-dependent effective dimension from quantum gravity has only been considered for scalar (and dual-scalar) fields. It is however possible that the non-manifold like structures, that are expected to appear near the Planck scale, have an effective dimension that depends on the type of field under consideration. To investigate this question, we have studied the spectral dimension associated to the Laplace-Beltrami operator generalised to k-form fields on spatial slices of the CDT spacetime. We have found that the two-form, tensor and dual scalar spectral dimensions exhibit a flow between two scales at which an effective dimension appears. However, the one-form and vector spectral dimensions show only a single effective dimension. The fact that the one-form and vector spectral dimension do not show a flow of the effective dimension can potentially be related to the absence of a dispersion relation for the electromagnetic field, but dynamically generated instead of as an assumption.
The goal of this talk is to explain an extension of this idea where we replace one mysterious group (the absolute Galois group) with a family of mysterious groups (the higher genus Kashiwara-Vergne groups). The associated “KV tower” is a tower of fundamental groups of the free loop spaces on various punctured Reimann surfaces. We aim to describe how this variation on Teichmüller space turns a problem in Lie theory (the Kashiwara-Vernge problem) into a problem in topology.

Complex manifolds of hyperbolic and non-hyperbolic-type

A general overview of the role played by curvature in the study of complex manifolds of hyperbolic-type and non-hyperbolic-type.

Curvature aspects of hyperbolicity in complex geometry

A compact complex manifold X is said to be Kobayashi hyperbolic if every holomorphic map from the complex plane to X is constant. An extension of a conjecture of Kobayashi predicts that all compact Kobayashi hyperbolic manifolds are projective and admit a Kähler--Einstein metric with negative Ricci curvature. We will discuss the curvature aspects of Kobayashi hyperbolic manifolds and their study via Gauduchon connections. We will also present a "positive analog" of the Kobayashi conjecture.

Separation of variables for spaces of constant curvature

I will discuss orthogonal separation of variables for spaces of constant curvature, with the emphasis on pseudo-Riemannian metrics. I will give a local description of all possible separating coordinates, and write the transformation  to flat coordinates. The problem was actively studied since at least hundred years: different aspects of separations were considered by Stäckel in the 19th century and by Eisenhart in the beginning of the 20th and in a series of papers and books of E. Kalnins, J. Kress and W. Miller, I will give a historical overview.
The new results are joint with A. Bolsinov and A. Konyaev and appeared within the Nijenhuis Geometry project and are a by-product of our classification of compatible inhomogeneous geometric Poisson brackets of degree 3+1; separation of variables is closely related to compatible pencils of Poisson brackets of degree 1. They are also closely related to geodesically equivalent metrics; all these will be mentioned if the time allows.

2 June 2023
10:10 am

Initial stability estimates for Ricci flow and three dimensional Ricci-pinched manifolds

In this talk we examine the Ricci flow of initial metric spaces which are Reifenberg and locally bi-Lipschitz to Euclidean space. We show that any two solutions starting from such an initial metric space, whose Ricci curvatures are uniformly bounded from below and whose curvatures are bounded by $c\cdot t^{-1}$, are exponentially in time  close to one another in the appropriate gauge. As an application, we show that smooth three dimensional, complete, uniformly Ricci-pinched Riemannian manifolds with bounded curvature are either compact or flat, thus confirming a conjecture of Hamilton and Lott. This is joint work with Alix Deruelle and Felix Schulze.