Cosmological rigidity via $p$-harmonic temporal functions

Abstract

We prove a splitting theorem for cosmological spacetimes which admit a globally $p$-harmonic Cauchy temporal function for some $p < 1$, without making any use of timelike lines. In Bartnik's context of cosmological and timelike geodesically complete spacetimes, we establish a Liouville-type unboundedness result which shows that globally $p$-harmonic temporal functions, $p < 1$, are necessarily Cauchy temporal. Moreover, using the theory of elliptic PDEs, we show that locally Lipschitz $p$-harmonic temporal functions with $p < 1$ are in fact smooth. Altogether, our work reduces the resolution of Bartnik's cosmological splitting conjecture to the global weak solvability of the $p$-d'Alembert equation for some $p < 1$ in the class of locally Lipschitz temporal functions.