Initial Data Sets with Ends of Cylindrical Type: I. The Lichnerowicz Equation

Abstract

We construct large classes of vacuum general relativistic initial data sets, possibly with a cosmological constant Λ ∈ R, containing ends of cylindrical type.

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

“…This provides a satisfactory complement to the results of [5] in the CMC setting, and the two sets of results together establish the existence of solutions to the full constraint equations under reasonably general hypotheses.…”

“…In a companion paper to this one [5] we have constructed large families of solutions of the Lichnerowicz equation on manifolds with cylindrical ends. This paper addresses the complementary problem of constructing solutions of the vacuum vector constraint equation.…”

“…As explained in [5], there are compelling physical reasons for studying this problem when the initial metric (M, g) is complete with a finite number of cylindrically bounded ends, possibly accompanied by a finite number of asymptotically hyperbolic or asymptotically Euclidean or conic ends. That paper initiated the general study of the constraint equations on manifolds with ends of cylindrical type, but focused exclusively on the Lichnerowicz equation.…”

“…We refer to [5] for much of the terminology, definitions and notation used below. In the current paper, we are primarily interested in initial metrics (M, g), the ends of which are either asymptotically cylindrical or asymptotically periodic.…”

Bifurcating Solutions of the Lichnerowicz Equation

“…For this note that, since ∆φ integrates to zero, there exists a point p on S 1 × S 2 such that ∆φ(p) = 0. At this point we have 6) which implies that the constant function φ ≡ φ(p) solves (2.4). A first corollary of this, keeping in mind (2.6), is thatR ≤ 0 is incompatible with the existence of positive solutions of (2.4) with β 2 ≥ 0, therefore S 2 cannot be replaced by another two-dimensional surface in our model.…”