Isometric embedding of negatively curved complete surfaces in Lorentz-Minkowski space

TL;DR

This paper proves that complete negatively curved surfaces can be smoothly embedded into Lorentz-Minkowski space, extending classical embedding results from Euclidean space to a Lorentzian setting.

Contribution

It establishes the existence of smooth isometric embeddings for negatively curved complete surfaces into Lorentz-Minkowski space, generalizing the Hilbert-Efimov theorem.

Findings

Complete negatively curved surfaces admit smooth embeddings into Lorentz-Minkowski space.

Extension of classical embedding theorems to Lorentzian geometry.

Provides new insights into the geometry of negatively curved surfaces.

Abstract

Hilbert-Efimov theorem states that any complete surface with curvature bounded above by a negative constant can not be isometrically imbedded in $\mathbb{R}^3.$ We demonstrate that any simply-connected smooth complete surface with curvature bounded above by a negative constant admits a smooth isometric embedding into the Lorentz-Minkowski space $\mathbb{R}^{2,1}$.