A Hilbert-type theorem for spacelike surfaces with constant Gaussian curvature in $\mathbb{H}^2\times\mathbb{R}_1$

TL;DR

This paper proves a non-existence theorem for complete spacelike surfaces with constant Gaussian curvature greater than -1 in the Lorentzian product space , extending classical geometric results to Lorentzian settings.

Contribution

The paper establishes a Hilbert-type theorem showing the non-existence of complete spacelike surfaces with constant Gaussian curvature greater than -1 in .

Findings

No complete spacelike surfaces with K > -1 exist in .

Examples of complete spacelike surfaces with K  exist.

The result extends classical curvature theorems to Lorentzian product spaces.

Abstract

There are examples of complete spacelike surfaces in the Lorentzian product $\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K\leq -1$. In this paper, we show that there exists no complete spacelike surface in $\mathbb{H}^2\times\mathbb{R}_1$ with constant Gaussian curvature $K>-1$.