Low regularity singularity theorems: C^1,1, C¹ (and beyond)

One of my main areas of interest going back to the beginning of my PhD studies has been extending the singularity theorems of General Relativity to lower regularity metrics. In a recent paper (arxiv:1910.13915, published in Commun. Math. Phys.) I give a complete proof of both the Hawking and the Penrose singularity theorem for C¹-Lorentzian metrics. This is a significant step down in regularity from previous work on singularity theorems for C^1,1-metrics because one looses uniqueness for solutions of the geodesic equation (and therefore the exponential map is no longer well-defined). Hopefully this an be a stepping stone towards pushing this analytic approach further to eventually cover the physically very relevant classes of Lipschitz metrics and of continuous metrics with locally square integrable first derivatives. We also recently proved a singularity/incompleteness theorem for generalized cones in the very abstract setting of Lorentzian length spaces (see below).

The Riemannian side of things: Low regularity versions of Myers’s theorem

Due to similarities to the singularity theorems, especially the Hawking theorem, many of the low regularity methods developed for the singularity theorems can also be applied in the Riemannian case to generalize Myers’s theorem. A C¹-version of Myers’s theorem is included as an appendix in the C¹ singularity theorems preprint.

Lorentzian length spaces: Generalized cones (with S. B. Alexander, M. Kunzinger and C. Sämann)

In 2017, M. Kunzinger and C. Sämann introduced so called Lorentzian length spaces in the hopes of establishing an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. For these spaces one defines synthetic timelike curvature bounds, akin to the theory of Alexandrov and CAT(k)-spaces, based on triangle comparison.
In a recent preprint (arxiv:1909.09575, to appear in Commun. Anal. Geom.) we study generalizations of Lorentzian warped products with one-dimensional base of the form I ×_f X, where I is an interval, X is a length space and f is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces , displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, timelike sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber X. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.

Rigidity of AdS₂ x S² spacetimes (Collaboration with G. J. Galloway and E. Ling)

The spacetime AdS₂ x S² arises as the ‘near horizon’ geometry of the extremal Reissner-Nordstrom solution, and for that reason it has been studied in connection with the AdS/CFT correspondence.  We show that certain features  (namely two transversal foliations by totally geodesic null hypersurfaces which intersect in isometric, totally geodesic round 2-spheres) of AdS₂ x S² have to be present in any asymptotically AdS₂ x S² spacetime (defined via asymptotics of the metric in suitable coordinates) satisfying the null energy condition.  This line of research follows a conjectural viewpoint of Juan Maldacena stating that any such spacetime must have a very special structure and might already be isometric to AdS₂ x S². Recent examples of Paul Tod show that this is not the case. He further shows that, assuming enough differentiability of the foliations, such spacetimes must split as a product.
We also would like to develop a better understanding of ”asymptotically AdS₂ x S^q ” more in line with the usual definition of asymptotically flat (AdS₂ x S^q conformally embeds into the Einstein static universe R x S^q+1 but its boundary in R x S^q+1 consists of two timelike lines and is not a hypersurface).

Causal character of maximizers & an inextendibility result (with E. Ling)

In a recent paper together with E. Ling we showed that even for spacetime metrics that are merely Lipschitz maximizing causal curves must have fixed causal character (i.e., be either timelike or null). This then gives that timelike geodesically complete spacetimes cannot be extended with a Lipschitz continuous metric. This inextendibility result has since been improved by E. Minguzzi and S. Suhr.

Side project: Degenerate parabolic equations (with D. Mitrovic and M. Kunzinger)

I also work on a joint project with Darko Mitrovic, University of Montenegro, and  Michael Kunzinger, University of Vienna, studying degenerate parabolic equations on Riemannian manifolds.

My PhD project

The singularity theorems of General Relativity form the mathematical foundation for the prediction of the existence of black holes and, on the cosmic scale, the existence of an initial singularity (Big Bang). Initiated by R. Penrose in 1965 and continued by S. W. Hawking, R. Penrose, G. F. R. Ellis, R. Geroch and others, the investigation of singularity theorems to this day constitutes a central research field in mathematical relativity.

The general form of a singularity theorem, as described by J. Senovilla, is captured in the following ‘pattern singularity theorem’:

Pattern Singularity Theorem. If a spacetime with a C²-metric satisfies

• a condition on the curvature
• a causality condition
• an appropriate initial and/or boundary condition

then it contains endless but incomplete causal geodesics.

The goal of my dissertation project was to extend the range of validity of the classical singularity theorems and to develop new geometrical methods for this field. In particular we generalized the singularity theorem of Hawking and Penrose to spacetime metrics with locally Lipchitz continuous first derivatives. Another focus was to extend results from Riemannian comparison geometry to the setting of Lorentzian manifolds and to study rigidity properties resulting from the typical assumptions of the singularity theorems. This project was funded by a DOC-fellowship of the Austrian Academy of Sciences (ÖAW).