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Geometry of Weighted Finsler Spacetimes

Lu, Yufeng 大阪大学 DOI:10.18910/81994

2021.03.24

概要

In this thesis, we develop the theory of weighted Ricci curvature for the weighted Lorentz– Finsler framework. The aim of this thesis is two-fold. One is to extend the classical singularity theorems of general relativity to weighted Finsler spacetimes. The other is to establish some comparison theorems in the same setting.

In order to achieve the first goal, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. In favor of these equations of new versions, weighted Finsler Penrose’s theorem, weighted Finsler Hawking–Penrose’s theorem and weighted Finsler Hawking’s theorem are proved.

In order to achieve the second goal, we establish firstly a weighted Lorentz–Finsler version of the Bonnet–Myers theorem based on a generalized Bishop inequality. Using the Riccati equation techniques, we prove the Bishop–Gromov volume comparison for the so-called standard for comparisons of Lorentzian volumes (SCLVs). We also prove the Gu¨nther volume comparison for SCLVs when the flag curvature is bounded above.

This thesis is based on [LMO] and [Lu].

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