Current seminars

In last week's talk, Walter Simon presented singularity theorems for manifolds with spacelike Cauchy surfaces satifying certain convexity conditions.

The main theorem read:
If a globally hyperbolic spacetime admitting a closed, spacelike, 2-convex Cauchy surface is geodesically complete, then the Cauchy surface is either a spherical space or finitely covered by a surface bundle over the circle with totally geodesic fibres.

We will recall these results and focus on the proof of the main theorem.

The proof is divided into three parts. First, we identify a suitably embedded minimal surface. Second, assuming geodesic completeness, we construct a neighborhood foliated by minimal surfaces. Finally, we apply compactness theorems to extend this foliation to the entire manifold.

Our approach relies on several recent developments, including the positive resolution of the virtually Haken conjecture. In particular, we will observe that in certain special cases - such as when the Cauchy surface is Haken - our result admits a natural strengthening.

Inspired by the classical singularity theorems in General Relativity, Galloway and Ling (Commun. Math. Phys. 2017, 2024) have shown the following: If a globally hyperbolic spacetime satisfying the null energy condition contains a closed, spacelike Cauchy surface N which is strictly 2-convex (meaning that the sum of the lowest two eigenvalues of the future extrinsic curvature is positive), then N is either a spherical space or past null geodesically incomplete.

In recent work (with Eric Ling, Carl Rossdeutscher and Roland Steinbauer) we have relaxed the above convexity condition in essentially two respects. Firstly, we admit 2-convex extrinsic curvatures (for which the sum of the lowest two eigenvalues is non-negative). Secondly, if N admits a U(1) isometry group, we can impose even less restricive convexity conditions. In addition to spherical spaces and past null geodesically incompleteness, this allows for certain classes of surface bundles for N or a finite cover therof, which we classify. While our results do not use Einstein's equations, we provide several classes of vacuum solutions (with our without cosmological constant) of these equations as examples.

In my talk I will give a brief overview of singularity theorems and then focus on explaining the above results via examples. The proofs will only be outlined.

Preview: In his talk on 21.5. on the same subject Carl Rossdeutscher will focus on selected aspects of the proofs.

The character and status of very simple sounding questions in manifold theory depends crucially on the dimension of the manifold. An illustrative example is the famous Jordan curve theorem which roughly speaking states that a topological embedding of S^1 into R^2 separates R^2 into a bounded component (which is homeomorphic to a disk) and an unbounded one. Naively speaking this theorem seems obvious, if annoying to prove. In reality, it is a miracle it is true in the first place as corresponding statements in dimension 3 and higher are simply wrong. In this talk I will try to give an overview over questions in low dimensional topology (such as existence and uniques of smooth structures, classification of manifolds) and how these depend on the dimension.

The Standard Model (SM) is one of the greatest successes of modern theoretical physics. Despite this, mathematical references studying the full SM (rather than just its sectors in isolation) on curved spacetimes are somewhat scarce, even though it seems important to understand the theory at a classical level in a more geometric setting. In my talk, I will first briefly review the mathematical structure of the SM Lagrangian, the corresponding Euler-Lagrange equations, and some of their basic properties, particularly related to conformality. Then, I will present a global existence result for the SM equations on four-dimensional spacetimes of expanding type. The talk is focused on a geometrically intrinsic approach to the theory, and the main ingredient for the proof is a gauge-invariant energy estimate. This is joint work with Volker Branding.

The first half concerns Gregory—Laflamme (GL) instabilities which occur at the level of linearised gravity. Heuristic and numerical evidence suggests that GL instabilities plague black holes in dimensions greater than 4 which have an event horizon that has one direction that is large compared to all others. Sam will discuss a direct mathematical proof of the Gregory—Laflamme instability for the 5D Schwarzschild black string. The proof relies upon reducing the linearised vacuum Einstein equation to a Schrödinger equation to which direct variational methods can be applied.   The second half of the talk concerns the blue-shift instability in the interior of rotating Kerr black holes at the level of linearized gravity. This instability is intimately connected to Penrose’s strong cosmic censorship conjecture. In contrast to the GL instability, this instability is weak in the sense that the C^0 norms of metric perturbations do not grow. Jan will discuss a mathematical result on the blue-shift instability, illustrate the mechanism by a toy example, and, if time permits, discuss some elements of the proof.

Recently it was proposed that GME experiments may probe linearized gravity only as a quantum controlled field and not as a quantum field theory. I will argue that the concept of a relativistic quantum controlled field is fundamentally flawed because of causality violation. Quantum control is also implicit in a recent path-integral description of those experiments, which casts doubt on its validity beyond the Newtonian approximation.