Splitting theorems play a vital role in both Riemannian and Lorentzian geometry. Under the strong energy condition from general relativity, Yau conjectured in 1982 that a timelike geodesically complete spacetime ought to be exceptional: if even one of its complete geodesics is timelike and maximizing, then the space is a stationary, static, geometric product.
Although Yau's conjecture was proved by Newman (1990) following works by Eschenberg (1988) and Galloway (1999), the proof is complicated relative to the Riemannian case by the fact that the Lorentzian Laplacian is not elliptic. We describe a new proof of the Lorentizan splitting theorems, in which simplicity is gained by
sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity $p$-d'Alembert operator for this purpose. This allows us to bring the Eschenburg, Galloway, and Newman theorems into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.
Our proof relies on a $p$-d'Alembert comparison result obtained with Beran, Braun, Calisto, Gigli, Ohanyan, Rott, Saemann. We anticipate that work in progress will confirm that our method can be used to lower the regularity requirements on the Lorentzian metric tensor for the splitting to occur.
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