Boundedness of the total energy of relativistic membranes evolving in a curved spacetime

TL;DR

This paper proves the global boundedness of the total energy for relativistic membranes evolving in curved spacetimes, extending previous flat spacetime results using the Hyperboloidal Foliation Method.

Contribution

It provides a simplified proof of global existence and energy boundedness for relativistic membranes in curved Lorentzian manifolds, generalizing Lindblad's flat spacetime theorem.

Findings

Total energy of membranes remains globally bounded over time.

The Hyperboloidal Foliation Method effectively handles curved spacetime scenarios.

The proof simplifies previous approaches and extends results to curved backgrounds.

Abstract

We establish a global existence theory for the equation governing the evolution of a relativistic membrane in a (possibly curved) Lorentzian manifold, when the spacetime metric is a perturbation of the Minkowski metric. Relying on the Hyperboloidal Foliation Method introduced by LeFloch and Ma in 2014, we revisit a theorem established earlier by Lindblad (who treated membranes in the flat Minkowski spacetime) and we provide a simpler proof of existence, which is also valid in a curved spacetime and, most importantly, leads to the important property that the total energy of the membrane is globally bounded in time.