TY - JOUR

T1 - Curvature-dimension bounds for Lorentzian splitting theorems

AU - Woolgar, Eric

AU - Wylie, William

N1 - Funding Information:
The work of EW was supported by a Discovery Grant RGPIN 203614 from the Natural Sciences and Engineering Research Council (NSERC) . The work of WW was supported by a grant from the Simons Foundation ( #355608 , William Wylie) and a grant from the National Science Foundation ( DMS-1654034 ).
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/10

Y1 - 2018/10

N2 - We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry–Émery–Ricci tensor. We extend the Hawking–Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions N≤1, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when N=1. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry–Émery–Ricci tensor for all N∈(−∞,2]∪(n,∞) and for the N=∞ case as well, with reduced rigidity if N=2. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-Émery theory now cover all synthetic dimensions for which such theorems are possible. The splitting theorems are found always to exhibit reduced rigidity at the critical synthetic dimension.

AB - We analyze Lorentzian spacetimes subject to curvature-dimension bounds using the Bakry–Émery–Ricci tensor. We extend the Hawking–Penrose type singularity theorem and the Lorentzian timelike splitting theorem to synthetic dimensions N≤1, including all negative synthetic dimensions. The rigidity of the timelike splitting reduces to a warped product splitting when N=1. We also extend the null splitting theorem of Lorentzian geometry, showing that it holds under a null curvature-dimension bound on the Bakry–Émery–Ricci tensor for all N∈(−∞,2]∪(n,∞) and for the N=∞ case as well, with reduced rigidity if N=2. In consequence, the basic singularity and splitting theorems of Lorentzian Bakry-Émery theory now cover all synthetic dimensions for which such theorems are possible. The splitting theorems are found always to exhibit reduced rigidity at the critical synthetic dimension.

KW - Bakry–Emery–Ricci curvature

KW - Lorentzian geometry

KW - Singularity theorems

KW - Splitting theorems

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U2 - 10.1016/j.geomphys.2018.06.001

DO - 10.1016/j.geomphys.2018.06.001

M3 - Article

AN - SCOPUS:85048752828

VL - 132

SP - 131

EP - 145

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -