Positive resolution of Bartnik's cosmological splitting conjecture

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 $p<1$.