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

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.

21 pages, 3 figures

References in corpus (4)

• Curvature estimates for immersed hypersurfaces in Riemannian manifolds
• Geometrical properties of trapped surfaces and apparent horizons
• On the Local Isometric Embedding in $R^3$ of Surfaces with Zero Sets of Gaussian Curvature Forming Cusp Domains
• A concise introduction to trapped surface formation in general relativity

Cited by in corpus (1)

• Einstein, Planck and Vera Rubin: relevant encounters between the Cosmological and the Quantum Worlds