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

We establish a global existence theory for the equation governing the evolution of a relativistic membrane in a (curved) background spacetime, when the spacetime metric is a perturbation of the Minkowski metric. Relying on a hyperboloidal foliation and generalizing an observation made by LeFloch and Ma, we revisit a theorem established by Lindblad (who treated membranes in the flat Minkowski spacetime) and provide a simpler proof of existence. Our result is valid in a curved spacetime and, most importantly, establishes the important property that the total energy of the membrane is globally bounded in time.