Abstracts of the Arbeitsgruppenseminar

Prof. Bernd Ammann und Mitarbeiter, Zimmer 119

Cipriana Anghel-Stan, Universität Göttingen

On the spectrum of the Dirac operator on degenerating Riemannian surfaces

Nov 4, 2024, 14:15

We study the behavior of the spectrum of the Dirac operator on degenerating families of compact Riemannian surfaces, when the length of a simple closed geodesic shrinks to zero, under the hypothesis that the spin structure along the pinched geodesic is non-trivial.
The difficulty of the problem stems from the non-compactness of the limit surface, which has finite area and two cusps. The main idea in this investigation is to construct an adapted pseudodifferential calculus, in the spirit of the celebrated \emph{b}-algebra of Melrose, which includes both the family of Dirac operators on the family of compact surfaces and the Dirac operator on the limit non-compact surface, together with their resolvents.

Hannes Lehmann

The conformal covariance of the twisted Dirac operator

Nov 18, 2024, 14:15

In this talk we will generalize some results from Oussama Hijazi’s article ”SPECTRAL PROPERTIES OF THE DIRAC OPERATOR” regarding the conformal covariance of the Dirac operator. Our generalization covers the case of twisted Dirac operators.

Moritz Wahl

Elastic Curves with Variable Bending Stiffness

Jan 13, 2025, 14:15

We study stationary points of the bending energy of curves γ:[a,b]→ℝⁿ subject to constraints on the arc-length and total torsion while simultaneously allowing for a variable bending stiffness along the arc-length of the curve. Physically, this can be understood as a model for an elastic wire with isotropic cross-section of varying thickness. We derive the corresponding Euler-Lagrange equations for variations that are compactly supported away from the end points thus obtaining characterizations for elastic curves with variable bending stiffness. Moreover, we provide a collection of alternative characterizations, e.g., in terms of the curvature function. Adding to numerous known results relating elastic curves to dynamics, we establish connections between elastic curves with variable bending stiffness and damped pendulums and the flow of vortex filaments with finite thickness.

Tian Xu

Conformal deformation of a Riemannian metric via an Einstein-Dirac parabolic flow

Jan 20, 2025, 8:15 via Zoom

We introduce a new parabolic flow deforming a Riemannian metric on a spin manifold by following a constrained gradient flow of the total scalar curvature. This flow is built out of the well-known Dirac-Einstein functional. We prove the local well-posedness of the flow.
Zoom link: https://uni-regensburg.zoom-x.de/j/65611902454, Meeting ID 656 1190 2454

Jonathan Glöckle

Spacetime rigidity via MOTS

Jan 20, 2025, 14:15

This talk is devoted to an initial data rigidity theorem by Eichmair, Galloway and Mendes. The goal is to obtain an improvement of their conclusions by constructing a lightlike parallel vector field in the rigid situation. This is motivated by an analogous rigidity theorem of mine in the spin case. As an application, we obtain a spacetime rigidity result for certain dominant energy condition spacetimes with toroidal Cauchy hypersurface. I will try and introduce all the relevant notions about spacetimes and initial data sets so that the talk is accessible with basic knowledge of differential geometry.

Hannes Lehmann

Solving a conformally covariant Dirac equation via branched coverings

Feb 3, 2025, 14:15

Starting from a solution of the conformally covariant Dirac equation Dψ = |ψ|²ψ on a Riemannian spin surface M, we construct solutions of the same equation on another surface N by pulling them back along a branched covering N→ M. A motivation to study solutions of this equation is that they translate locally into branched conformally immersed surfaces of constant mean curvature in ℝ³.

Back to the plan of the Arbeitsgruppenseminar

Bernd Ammann, 01.02.2025
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