The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics

Understand

We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity.

• We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, and of $C^0$-trapped submanifolds.
• By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising.
• This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest.