Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity

TL;DR

This paper investigates the existence of isometric immersions of Lorentzian hypersurfaces into Minkowski spacetime under minimal regularity assumptions, allowing for metrics with Sobolev regularity and distributional curvature.

Contribution

It extends classical results to metrics with low regularity, including Sobolev spaces and distributional curvature, covering hypersurfaces with arbitrary and changing signatures.

Findings

Established existence results under minimal regularity assumptions.

Applicable to hypersurfaces with arbitrary and variable signatures.

Framework accommodates metrics with Sobolev regularity and distributional curvature.

Abstract

Assuming minimal regularity assumptions on the data, we revisit the classical problem of finding isometric immersions into the Minkowski spacetime for hypersurfaces of a Lorentzian manifold. Our approach encompasses metrics having Sobolev regularity and Riemann curvature defined in the distributional sense, only. It applies to timelike, spacelike, or null hypersurfaces with arbitrary signature that possibly changes from point to point.