Initial data sets with ends of cylindrical type: II. The vector constraint equation.

Abstract

We construct solutions of the vacuum vector constraint equations on manifolds with cylindrical ends.

A Limit Equation Criterion for Solving the Einstein Constraint Equations on Manifolds with Ends of Cylindrical Type

“…Any vector field in the nullspace of L is called a conformal Killing field, and the operator divL is known as the conformal vector Laplacian. The mapping properties of this operator acting between weighted Sobolev spaces were studied extensively in [3]. The most important mapping property for us will be [3, Thm 6.1].…”

“…(2.1)-(2.2) which haveφ as an asymptote for the conformal factor is proved in precisely the same way as Theorem 2.1, for one can show that the central theorems used in Sect. 3 all apply in the asymptotically periodic case except for a minor change in the definition of Y (see [3], [15]). As in the conformally asymptotically cylindrical case, the arguments in Sect.…”

“…As in [3], we will denote the function space W k,2 δ by H k δ . Next, assume without loss of generality that the radius of injectivity for (M, g) is >1.…”

“…General results for the solvability of the LCBY equations on manifolds with asymptotically cylindrical or periodic geometries (which we collectively call geometries of cylindrical type) appeared in 2012 with the pair of papers [2] and [3] which analyzed the Lichnerowicz equation and vector equation separately. It was noted in the first of these that several known examples of black hole spacetimes admit CMC hypersurfaces with ends of cylindrical type, including the extreme Kerr metrics.…”

Hamiltonian mass of asymptotically Schwarzschild–de Sitter space-times

“…3. (See [3][4][5][6][7] for further families of vacuum initial data sets with various ends of cylindrical type.) This raises the question of existence of a natural notion of mass in this context.…”