Pure and Applied Mathematics Quarterly

Volume 20 (2024)

Number 4

On the non-existence of trapped surfaces under low-regularity bounds

Pages: 1463 – 1504

DOI: https://dx.doi.org/10.4310/PAMQ.2024.v20.n4.a1

Authors

Jonathan Luk (Department of Mathematics, Stanford University, Stanford, CA, USA)

Georgios Moschidis (Department of Mathematics, Princeton University, Princeton, NJ, USA)

Abstract

The emergence of trapped surfaces in solutions to the Einstein field equations is intimately tied to the well-posedness properties of the corresponding Cauchy problem in the low regularity regime. In this paper, we study the question of existence of trapped surfaces already at the level of the initial hypersurface when the scale invariant size of the Cauchy data is assumed to be bounded. Our main theorem states that no trapped surfaces can exist initially when the Cauchy data are close to the data induced on a spacelike hypersurface of Minkowski spacetime (not necessarily a flat hyperplane) in the Besov $B^{3/2}{2,1}$ norm. We also discuss the question of extending the above result to the case when merely smallness in $H^{3/2}$ is assumed.

Received 20 April 2022

Received revised 11 April 2023

Accepted 31 May 2023

Published 18 July 2024