  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.
