Tag - Geodesic flows

Gabriele Benedetti: Rigidity and Flexibility of Periodic Hamiltonian Flows

30th October, 2023

An old problem in classical mechanics is the existence of periodic flows within specific classes of Hamiltonian systems such as geodesic and magnetic flows, and central forces. In the last years, interest in this problem has been revitalized since recent research has unveiled a deep relationship between periodic Hamiltonian flows and systolic questions in symplectic and contact geometry. While only trivial examples of periodic flows among magnetic and central systems exist, Zoll and, later, Guillemin have shown that there are many exotic examples among geodesic flows on the two-sphere. Following Guillemin's approach, the goal of this talk is to show how the Nash-Moser implicit function theorem can be used to construct magnetic flows on the two-torus which are periodic for a single value of the energy.

Marco Mazzucchelli: Surfaces of Section, Anosov Reeb Flows, and the C2-Stability Conjecture for Geodesic Flows

3rd March, 2023

In this talk, based on joint work with Gonzalo Contreras, I will briefly sketch the proof of the existence of global surfaces of section for the Reeb flows of closed 3-manifolds satisfying a condition à la Kupka-Smale: non-degeneracy of the closed Reeb orbits, and transversality of the stable and unstable manifolds of the hyperbolic closed Reeb orbits. I will then present an application of this theorem to hyperbolic Reeb dynamics: a Reeb flow on a closed 3-manifold is Anosov if and only if the closure of the subspace of closed Reeb orbits is hyperbolic and the Kupka-Smale transversality condition holds. This result implies the validity of the C2 stability conjecture for Riemannian geodesic flows of closed surfaces: any such geodesic flow that is C2 structurally stable within the class of Riemannian geodesic flows must be Anosov.

Melanie Rupflin: Singularities of Teichmüller harmonic map flow

4th March, 2019

We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component.

Steve Zelditch: Local and global analysis of nodal sets

30th April, 2017

Nodal sets are zero sets of eigenfunctions of the Laplacian on a Riemannian manifold. Local analysis studies nodal sets in small balls, ignoring the global geometry. Global analysis exploits the dynamics of the geodesic flow to obtain information on nodal sets. First, I will describe the recent proof by Alexander Logunov of Yau's lower bound conjecture for hypersurface volumes of nodal sets. It is a local proof based mainly on the combinatorics of the Donnelly-Fefferman doubling exponent bounds. Second, I will describe recent results on numbers of nodal domains on surfaces of non-positive curvature. These results are based on the ergodicity of the geodesic flow.

Duong Phong: Supersymmetric string vacua with torsion and geometric flows

29th April, 2017

In 1986, a system of equations for compactifications of the heterotic string which preserve supersymmetry was proposed independently by C. Hull and A. Strominger. They are more complicated than the Calabi-Yau compactifications proposed earlier by P. Candelas, G. Horowitz, A. Strominger, and E. Witten, because they allow non-vanishing torsion and they incorporate terms which are quadratic in the curvature tensor. As such they are also particularly interesting from the point of view of both non-Kaehler geometry and the theory of non-linear partial differential equations. While the complete solution of such PDEs seems out of reach at the present time, we describe progress in developing a new general approach based on geometric flows which shares some features with the Ricci flow. In particular, this approach can recover the well-known non-perturbative solutions found in 2006 by J.X. Fu and S.T. Yau.

Masato Tsujii: Geodesic flow on negatively curved manifold and the semi-classical zeta function

5th December, 2013

We consider the one-parameter families of transfer operators for geodesic flows on negatively curved manifolds. We show that the spectra of the generators have some "band structure" parallel to the imaginary axis. As a special case of "semi-classical" transfer operator, we see that the eigenvalues concentrate around the imaginary axis with some gap on the both sides.