Trading linearity for ellipticity: a low regularity Lorentzian splitting theorem

Abstract: While Einstein's theory of gravity is formulated in a smooth setting,   the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually break down.  In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of non-smooth limits under dimension and curvature bounds analogous to the energy conditions in relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

This lecture explores recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a non-smooth theory of gravity. In particular,  we establish a low regularity splitting theorem by sacrificing linearity of the d'Alembertian to recover ellipticity.   We exploit a negative homogeneity, $p$-d'Alembert operator for this purpose.   The same technique yields a simplified proof of Eschenberg (1988) Galloway (1989) and Newman's (1990) confirmation of Yau's (1982) conjecture,  bringing all three Lorentzian splitting results into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.