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

TL;DR

This paper proves that under certain low-regularity bounds on initial data, trapped surfaces cannot form initially in solutions to Einstein's equations, highlighting the importance of regularity in gravitational collapse.

Contribution

It establishes the non-existence of trapped surfaces at the initial level under low-regularity bounds in Besov and Sobolev norms, extending understanding of initial data constraints.

Findings

No trapped surfaces exist initially near Minkowski data in Besov $B^{3/2}_{2,1}$ norm.

Results suggest regularity bounds are crucial for trapped surface formation.

Discussion on extending results to $H^{3/2}$ smallness.

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.