On spacelike surfaces in 4-dimensional Lorentz-Minkowski spacetime through a lightcone

TL;DR

This paper investigates the geometry of spacelike surfaces within lightcones in four-dimensional Lorentz-Minkowski space, introducing a smooth function that encodes their intrinsic and extrinsic properties, and explores conditions for total umbilicity and eigenvalue estimates.

Contribution

It introduces a distinguished smooth function to analyze spacelike surfaces in lightcones, deriving new geometric characterizations and explicit examples, along with eigenvalue estimates.

Findings

A smooth function encodes intrinsic and extrinsic geometry.

Conditions for total umbilicity are established.

Explicit examples of spacelike surfaces are constructed.

Abstract

On any spacelike surface in a lightcone of four dimensional Lorentz-Minkowski space a distinguished smooth function is considered. It is shown how both extrinsic and intrinsic geometry of such a surface is codified by this function. The existence of a local maximum is assumed to decide when the spacelike surface must be totally umbilical, deriving a Liebmann type result. Two remarkable families of examples of spacelike surfaces in a lightcone are explicitly constructed. Finally, several results which involve the first eigenvalue of the Laplace operator of a compact spacelike surface in a lightcone are obtained.