Some Remarks on Conformal Symmetries and Bartnik’s Splitting Conjecture

Abstract

Inspired by the results in a recent paper by G. Galloway and C. Vega [11], we investigate a number of geometric consequences of the existence of a timelike conformal Killing vector field on a globally hyperbolic spacetime with compact Cauchy hypersurfaces, especially in connection with the so-called Bartnik's splitting conjecture. In particular we give a complementary result to the main theorem in [11].

Remarks on the existence of CMC Cauchy surfaces

“…As a final remark in this section, we note that the conjecture holds for those spacetimes, that, in addition, admit a future complete (in the sense of completeness of integral curves) timelike conformal Killing vector field. Indeed, as follows from results in [7], in this case, the future conformal boundary consists of a single point. Some further comments pertinent to this remark, and to the conjecture, are made in Section 3.…”

Existence of CMC Cauchy surfaces and spacetime splitting

“…In fact, the proof of Theorem 1.3 in [10] can be easily adapted to show that if a spacetime (M, g) with compact Cauchy surfaces admits a future complete timelike conformal Killing vector field then the future causal boundary consists of a single point. Theorem 12 then follows from Theorem 7 and Proposition 9.…”