The Hawking Singularity Theorem for H\"older Continuous Metrics with $L^p$-Bounded Curvature

TL;DR

This paper extends Hawking's singularity theorem to low-regularity Lorentzian metrics with $W^{1,p}$ regularity and $L^p$-bounded curvature, establishing new global hyperbolicity and incompleteness results.

Contribution

It introduces a low-regularity version of Hawking's theorem for metrics with $W^{1,p}$ regularity, using novel regularisation and mean curvature notions.

Findings

Proves a low-regularity Hawking singularity theorem for $W^{1,p}$ metrics.

Establishes global hyperbolicity bounds and timelike geodesic incompleteness under low regularity.

Shows $W^{1,p}$-metrics with $L^p$-curvature are causally plain and extends Myers's theorem.

Abstract

We prove a low-regularity version of Hawking's singularity theorem for Lorentzian metrics in $W^{1,p}$ with Riemann curvature in $L^p$, where $p>2n$ and $n$ the dimension of spacetime. This extends previous results beyond the Lipschitz regime. Under suitable lower Ricci bounds and upper mean curvature assumptions, expressed in terms of temporal functions, we establish both the globally hyperbolic version of Hawking's theorem, in the form of an upper bound on the time separation from a spacelike Cauchy hypersurface, and the version with a compact achronal spacelike hypersurface, yielding timelike RT-geodesic incompleteness. The proof combines regularisations, based on the elliptic RT-equations, to raise the regularity of the metric by one derivative, with a refinement of the previously used manifold convolution. We introduce a new smeared-out notion of mean curvature adapted to the low metric regularity before, and the $W^{2,p}$-hypersurfaces arising after regularisation. As further consequences, we show that $W^{1,p}$-Lorentzian metrics with $L^p$-bounded curvature are causally plain, and we prove a corresponding low-regularity version of Myers's theorem in the Riemannian setting.