The grand challenges of contemporary fundamental physics-dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem-all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on 'Black holes, Gravitational waves and Fundamental Physics'. © 2019 IOP Publishing Ltd.

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Black holes, gravitational waves and fundamental physics: a roadmap

Three-dimensional geometry and topology. Volume 1, by William P. Thurston, (edited by Sylvio Levy). Princeton Mathematical Series, 35. £29.95. 1997. ISBN 0-691-08304-5 (Princeton University Press).

TOTAL MEAN CURVATURE AND SUBMANIFOLDS OF FINITE TYPE (World Scientific Series in Pure Mathematics—Volume 1) By Bang-Yen Chen: pp. 352. £15.70. (Published by World Scientific Publishing, distributed by John Wiley & Sons Ltd, 1984.)

Sur la dynamique de l’électron

https://www.math.uci.edu/~zlu/talks/2008-Chen-birthday/talk.pdf

Normal Scalar Curvature Conjecture and its applications

In this article we study surfaces in $ {\Bbb S}^2\times {\Bbb R} $ for which the unit normal makes a constant angle with the $ {\Bbb R} $-direction. We give a complete classification for surfaces satisfying this simple geometric condition.

Constant angle surfaces in $ {\Bbb S}^2\times {\Bbb R} $

In general relativity, a spatial surface in a 4-dimensional Lorentzian manifold is called marginally trapped if its mean curvature vector is lightlike at each point. In this paper, we completely classify marginally trapped spatial surfaces with positive relative nullity in Lorentzian space forms.

https://iopscience.iop.org/article/10.1088/0264-9381/24/3/003/pdf

Marginally trapped surfaces in Lorentzian space forms with positive relative nullity

In this paper we completely classify parallel non-degenerate surfaces in 4-dimensional Lorentzian space forms. In addition, we also completely classify non-degenerate surfaces with parallel mean curvature vector in 4-dimensional Lorentzian space forms.

/pdf/complete-classification-of-parallel-surfaces-in-4-2nsw9yatv9.pdf

Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms

In this article we extend the notion of constant angle surfaces in $$
\mathbb{S}^2 
$$
 × ℝ and ℍ2 × ℝ to general Bianchi-Cartan-Vranceanu spaces. We show that these surfaces have constant Gaussian curvature and we give a complete local classification in the Heisenberg group.

https://www.math.uaic.ro/~munteanu/papers/ConstHeisen_KUL.pdf

Constant angle surfaces in the Heisenberg group

Let L14(f,c)=(I×fS,gfc) be a Robertson-Walker space time which does not contain any open subset of constant curvature. In this paper, we provide a general study of nondegenerate surfaces in L14(f,c). First, we prove the nonexistence of marginally trapped surfaces with positive relative nullity. Then, we classify totally geodesic submanifolds. Finally, we classify the family of surfaces with parallel second fundamental form and the family of totally umbilical surfaces with parallel mean curvature vector.

Joeri Van der Veken

Papers

Constant angle surfaces in \( {\Bbb S}^2\times {\Bbb R} \)

Marginally trapped surfaces in Lorentzian space forms with positive relative nullity

Complete classification of parallel surfaces in 4-dimensional Lorentzian space forms

Constant angle surfaces in the Heisenberg group

Spatial and Lorentzian surfaces in Robertson-Walker space times