Global Regularity for the 2+1 Dimensional Equivariant Einstein-Wave Map System

Abstract

In this paper we consider the equivariant 2+1 dimensional Einsteinwave map system and show that if the target satisfies the so called Grillakis condition, then global existence holds. In view of the fact that the 3+1 vacuum Einstein equations with a spacelike translational Killing field reduce to a 2+1 dimensional Einstein-wave map system with target the hyperbolic plane, which in particular satisfies the Grillakis condition, this work proves global existence for the equivariant class of such spacetimes.

On Scattering for Small Data of $$\varvec{2 + 1}$$ 2 + 1 -Dimensional Equivariant Einstein-Wave Map System

“…Equivalently, scattering backwards in time can be proven using time reversal. It should be noted that the assumptions (28) are consistent with the results proven for the fully nonlinear system (1) in [6,1]. The proof of Theorem 1.4 is based on an argument that the linear part of the equation (25) dominates the nonlinear part in the large.…”

“…Then (M, g, u) is regular and causally geodesically complete. 2 where q 0 is the metric of Σ 0 and K 0 is a symmetric 2-tensor Actually, as a consequence of the Theorem 5.1 in [1] (also Theorem 1.3.1 in [6]), Theorem 1.8 in [1] also holds without the smallness restriction on the initial energy, with the following additional condition on the target manifold (N, h)…”

“…Theorem 1.8 in [1] carried forward the program initiated in [6] to understand global behavior of the 2+1 wave maps coupled to Einstein's equations. The motivation to study 2+1 Einstein-wave map system comes from the fact that the system arises naturally in 3+1 vacuum Einstein's equations with one isometry group (see [6] for a detailed discussion).…”

Mathematical general relativity

“…Even global existence for the wave map equation in n = 2 has not been proven. However, utilizing energy estimates, global existence can be proven in the case n = 1, [13,88]. It is perhaps of interest to study the U(1) symmetric vacuum case in 4D in the presence of a spacelike Killing vector field which is hypersurface orthogonal.…”

Open problems in mathematical physics

“…But global existence for large data is known only for symmetric solutions and, in particular, the global existence problem for the wave map equation is open for the case n = 2. For the case n = 1, global existence can be proved using energy estimates [61,132]. The U(1) symmetric vacuum 1+3 case in which the Einstein equations reduce to 1+2 gravity coupled to wave map matter in the presence of a hypersurface orthogonal spacelike Killing field, is of intermediate difficulty between the full 1+3 Einstein equations and the highly symmetric Gowdy equations [133].…”