Rigidity of spacelike graphs in generalized Robertson-Walker spacetimes via the infinity-Laplacian
Abstract
In the search for new characterizations of spacelike slices in generalized Robertson-Walker spacetimes, we introduce a new approach. The main tool for addressing this question is the infty-Laplacian with respect to the Riemannian metric induced on the spacelike hypersurface, together with a Liouville-type property for the infinity-Laplacian on any complete Riemannian manifold by Araujo, Mari and Pessoa in 2021. Starting from the infinity-Laplacian of the height function of a spacelike hypersurface, we derive several differential equations involving the infinity-Laplacian of suitable compositions of the height function, namely, the warping function f(h), its primitive P=\int f, and the logarithmic composition log(f(h)). Under some natural assumptions, including timelike and null energy conditions, global rigidity results are established in the parametric setting. Some of these results specialize naturally to complete spacelike hypersurfaces in the Steady State spacetime and in Einstein-de Sitter spacetime. Finally, we consider a second-order PDE involving the infty-Laplacian and study the uniqueness of its entire solutions on the fiber of the generalized Robertson-Walker spacetime. This leads to new Calabi-Bernstein-type uniqueness results in the non-parametric setting.
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BibTeX
@misc{jr2026rigidity,
title = {{Rigidity of spacelike graphs in generalized Robertson-Walker spacetimes via the infinity-Laplacian}},
author = {Eraldo Lima Jr. and Guilherme Nascimento and Alfonso Romero},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.09985},
eprint = {2611.09985},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.09985},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Rigidity of spacelike graphs in generalized Robertson-Walker spacetimes via the infinity-Laplacian
AU - Eraldo Lima Jr.
AU - Guilherme Nascimento
AU - Alfonso Romero
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.09985
UR - https://arxiv.org/abs/2611.09985
AB - In the search for new characterizations of spacelike slices in generalized Robertson-Walker spacetimes, we introduce a new approach. The main tool for addressing this question is the infty-Laplacian with respect to the Riemannian metric induced on the spacelike hypersurface, together with a Liouville-type property for the infinity-Laplacian on any complete Riemannian manifold by Araujo, Mari and Pessoa in 2021. Starting from the infinity-Laplacian of the height function of a spacelike hypersurface, we derive several differential equations involving the infinity-Laplacian of suitable compositions of the height function, namely, the warping function f(h), its primitive P=\int f, and the logarithmic composition log(f(h)). Under some natural assumptions, including timelike and null energy conditions, global rigidity results are established in the parametric setting. Some of these results specialize naturally to complete spacelike hypersurfaces in the Steady State spacetime and in Einstein-de Sitter spacetime. Finally, we consider a second-order PDE involving the infty-Laplacian and study the uniqueness of its entire solutions on the fiber of the generalized Robertson-Walker spacetime. This leads to new Calabi-Bernstein-type uniqueness results in the non-parametric setting.
ER -