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In a 2004 paper, Lindblad demonstrated that the minimal surface equation on ℝ1,1 describing graphical timelike minimal surfaces embedded in ℝ1,2 enjoy small data global existence for compactly supported initial data, using Christodoulou's conformal method. Here we give a different, geometric proof of the same fact, which exposes more clearly the inherent null structure of the equations, and which allows us to also close the argument using relatively few derivatives and mild decay assumptions at infinity. [ABSTRACT FROM AUTHOR]

Many classical results in relativity theory concerning spherically symmetric space-times have easy generalizations to warped product space-times, with a two-dimensional Lorentzian base and arbitrary dimensional Riemannian fibers. We first give a systematic presentation of the main geometric constructions, with emphasis on the Kodama vector field and the Hawking energy; the construction is signature independent. This leads to proofs of general Birkhoff-type theorems for warped product manifolds; our theorems in particular apply to situations where the warped product manifold is not necessarily Einstein, and thus can be applied to solutions with matter content in general relativity. Next we specialize to the Lorentzian case and study the propagation of null expansions under the assumption of the dominant energy condition. We prove several non-existence results relating to the Yamabe class of the fibers, in the spirit of the black-hole topology theorem of Hawking–Galloway–Schoen. Finally we discuss the effect of the warped product ansatz on matter models. In particular we construct several cosmological solutions to the Einstein–Euler equations whose spatial geometry is generally not isotropic. [ABSTRACT FROM AUTHOR]