arXiv:1807.08628gr-qc
Mathematical General Relativity
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Abstract
We present a number of open problems within general relativity. After a brief introduction to some technical mathematical issues and the famous singularity theorems, we discuss the cosmic censorship hypothesis and the Penrose inequality, the uniqueness of black hole solutions and the stability of Kerr spacetime and the final state conjecture, critical phenomena and the Einstein-Yang--Mills equations, and a number of other problems in classical general relativity. We then broaden the scope and discuss some mathematical problems motivated by quantum gravity, including AdS/CFT correspondence and problems in higher dimensions and, in particular, the instability of anti-de Sitter spacetime, and in cosmology, including the cosmological constant problem and dark energy, the stability of de Sitter spacetime and cosmological singularities and spikes. Finally, we briefly discuss some problems in numerical relativity and relativistic astrophysics.
> relatively far away from cosmology The preprint is in math.AP (Analysis of PDEs) alone. Most of the preprints by the three authors are found in math.AP, give or take some in math.CA (classical analysis & ODEs) and math.DG (diffgeo), with significant crossposting into math-ph…
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