On the local isometric embedding of trapped surfaces into three-dimensional Riemannian manifolds

TL;DR

This paper investigates the local isometric embedding of trapped surfaces into three-dimensional Riemannian manifolds, providing new methods to solve the embedding problem in cosmological models with different spatial curvatures.

Contribution

It introduces a novel generalization of Darboux's technique, reducing the complex embedding problem to solving a single non-linear PDE in Friedmann cosmologies.

Findings

Embedding of trapped surfaces can be achieved by solving one non-linear PDE.

The method applies to Friedmann models with positive, zero, and negative curvature.

New technique simplifies the classical Darboux approach.

Abstract

We study trapped surfaces from the point of view of local isometric embedding into three-dimensional Riemannian manifolds. When a two-surface is embedded into three-dimensional Euclidean space, the problem of finding all surfaces applicable upon it gives rise to a non-linear partial differential equation of Monge-Ampere type, first discovered by Darboux, and later reformulated by Weingarten. Even today, this problem remains very difficult, despite some remarkable results. We find an original way of generalizing the Darboux technique, which leads to a coupled set of 6 non-linear partial differential equations. For the 3-manifolds occurring in Friedmann-(Lemaitre)-Robertson-Walker cosmologies, we show that the local isometric embedding of trapped surfaces into them can be proved by solving just one non-linear equation. Such an equation is here solved for the three kinds of Friedmann model associated with positive, zero, negative curvature of spatial sections, respectively.