Positive resolution of Bartnik's cosmological splitting conjecture

Abstract

We give a proof of the cosmological splitting conjecture of Robert Bartnik from 1988, which expresses the rigidity of the cosmological Hawking--Penrose singularity theorem. It states that a timelike geodesically complete, globally hyperbolic spacetime which has compact Cauchy surfaces and satisfies the strong energy condition must split isometrically as a Lorentzian product. Our methods combine the construction of global viscosity solutions to the Lorentzian eikonal equation by Zhu--Wu--Cui with our recently developed elliptic approach to the proof of Lorentzian splitting theorems in joint work with Braun, Gigli and Sämann, where we make use of the $p$-d'Alembertian operator for $p<1$.

arXiv commentThe starting point of our withdrawn resolution to Bartnik's conjecture (our Proposition 2.2 and the reference on which it is based) is shown to be false by a counterexample of Ehrlich and Galloway (1990). We are grateful to Greg Galloway for drawing this error to our attention. See our forthcoming joint work for further examples and discussion