Vorträge

Donnerstag, 27.11.2025: On the Perturbation of Photon Surfaces

Simone Coli (Universität Tübingen)

It is well known that many static, spherically symmetric space-times admit a photon sphere, with the Schwarzschild solution serving as the canonical example. The notion of a photon sphere can be generalized to that of a photon surface, in the sense of Claudel–Virbhadra–Ellis. The uniqueness of static space-times containing an equipotential photon surface has been rigorously established, in both vacuum and electro-vacuum cases, in works by C. Cederbaum and G. Galloway and by C. Cederbaum, S. Jahns, and O. Vicanek Martinez. Furthermore, the uniqueness of space-times containing a photon sphere has been studied under non-spherical perturbations of the Schwarzschild metric, under the assumption of a constant lapse function and allowing for simultaneous deformations of the spatial geometry and of the photon sphere’s location, in work by H. Yoshino. Building upon these results, we investigate the stability and deformation of photon surfaces when the lapse function is no longer assumed to be constant, and we study the behaviour of photon surfaces in the Schwarzschild space-time under small non-spherical perturbations of the full static data.

Donnerstag, 04.12.2025: Static and Stationary Vacuum Extensions Realizing Bartnik Boundary Data: Local Well-Posedness near Schwarzschild Spheres

Dr. Ahmed Ellithy (Uppsala University)

We consider the Bartnik extension problem, which originates from Bartink's definition of quasi-local mass. In this problem, we consider Bartnik boundary data on a topological 2-sphere (induced metric and mean curvature, plus suitable stationary data), and we find a unique asymptotically flat initial data set $(M,g,K)$ that arises as a slice in a stationary spacetime $\mathcal{M}$ solving Einstein's vacuum equations, with $\partial M = S^2$ realizing the boundary data. Furthermore, if the Bartnik data is time-symmetric, then $K = 0$ and $\mathcal{M}$ is static. In this talk, we will address the local theory near Schwarzschild spacetimes. We present a new analytic framework for this extension problem. In this approach, we write the putative stationary spacetime in a double-geodesic gauge in which Einstein's equations reduce to a coupled elliptic system on the lapse function and the twist 1-form, together with transport equations for the second fundamental form of the geodesic leaves. In this gauge, the linearized static equations decouple and reduce to a non-local elliptic problem of Dirichlet-to-Neumann type on the boundary, while the genuinely stationary degrees of freedom reduce to an elliptic boundary value problem for the twist 1-form. To accommodate the mixed elliptic/transport structure, we work in Bochner-type spaces (specifically, continuous on the geodesic parameter with angular Sobolev regularity) instead of the classical Sobolev and Hölder spaces traditionally used for elliptic problems. These Bochner spaces we use are in fact traditionally used for hyperbolic and parabolic PDEs. Within this framework, we prove the local well-posedness of the Bartnik extension problem for both static and stationary data near Schwarzschild spheres: existence, uniqueness, and smooth dependence on the prescribed boundary data. Along the way, we develop an elliptic solvability theory for boundary value problems in Bochner spaces, which has not to our knowledge been previously used for elliptic problems and maybe of independent interest as they provide an appropriate setting for systems with both elliptic and transport/hyperbolic structure.

Donnerstag, 11.12.2025: Rigidity aspects of a singularity theorem

Carl Rossdeutscher (Universität Wien)

In 2018 Galloway and Ling established the following cosmological singularity theorem: If a (3+1)-dimensional spacetime satisfying the null energy condition contains a compact Cauchy surface with a positive definite second fundamental form (i.e., it’s expanding in all directions), then the spacetime is past null geodesically incomplete unless the Cauchy surface is a spherical space. We present some rigidity results for this singularity theorem. In particular if the second fundamental form is only positive semidefinite and the spacetime is past null geodetically complete, we show that the Cauchy surface (or at least a finite cover thereof) is a surface bundle over the circle with totally geodesic fibers or a spherical space. Under certain additional assumptions on the Cauchy surface, we show that passing to a cover is unnecessary. Our results make in particular use of the positive resolution of the virtual positive first Betti number conjecture by Agol. If a spacetime admits a U(1) isometry group, we can relax the assumption on the second fundamental form further.

Donnerstag, 11.12.2025: TBA

Xabier Oianguren-Asua (Tübingen)

Donnerstag, 18.12.2025: TBA

Dr. Shahnaz Farhat (Tübingen)

Montag, 12.01.2026: tba

Stefan Friedl

Donnerstag, 22.01.2026: TBA

Cameron Peters (Vancouver)

Freitag, 30.01.2026: Static Black Holes with a Negative Cosmological Constant

PhD Brian Harvie (Universität Kopenhagen)

The classification of stationary black hole solutions of the Einstein field equations, broadly referred to as the "no-hair conjecture", is a challenging and fundamental line of research in general relativity. The problem is more tractable for black hole spacetimes which are static, but even under this stronger assumption the existing results are mostly limited to static black holes with zero or positive cosmological constant. In this talk, I will present a geometric inequality for isolated static vacuum black holes with a negative cosmological constant which has far-reaching implications for their geometry and uniqueness. The inequality relates the surface gravity, area, and topology of a horizon in a static spacetime to its conformal infinity, and equality is achieved only by the Kottler black holes. From this, we deduce several new static uniqueness theorems for Kottler. Namely, we show: (1) the Kottler black hole over the sphere which minimizes surface gravity is unique, (2) the Kottler black hole over the torus is unique, assuming the horizons have non-spherical topology, and (3) uniqueness for the higher-genus Kottler black holes is equivalent to the Riemannian Penrose inequality. This is based on joint work with Ye-Kai Wang.

Freitag, 30.01.2026: TBA

Olivia Vicanek Martinez (Universität Tübingen)

TBA

Freitag, 30.01.2026: Quaternion Kähler manifolds of non-negative sectional curvature

Profl Dr. Uwe Semmelmann (Universität Stuttgart)

Quaternion Kähler manifolds, i.e., Riemannian manifolds with holonomy contained in Sp(m)Sp(1), are Einstein. In the case of positive scalar curvature, there is a longstanding conjecture by LeBrun and Salamon stating that all such manifolds should be symmetric. So far, the conjecture has been confirmed only up to dimension 12. In the first part of my talk I will give an introduction to the geometry of quaternion Kähler manifolds of positive scalar curvature In the second part I will make a few remarks on the proof of the conjecture under the additional assumption of non-negative sectional curvature. This extends earlier work by Berger, who proved that quaternion Kähler manifolds of positive sectional curvature are isometric to the quaternionic projective space. My talk is based on a joint article with Simon Brendle and on earlier work by Simon Brendle.