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On the geometry of silent and anisotropic big bang singularities
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Analysis, Dynamics, Geometry, Number Theory and PDE.ORCID iD: 0000-0001-9383-0748
2026 (English)In: Journal of differential geometry, ISSN 0022-040X, E-ISSN 1945-743X, Vol. 132, no 3, p. 461-545Article in journal (Refereed) Published
Abstract [en]

This article is the second of two in which we develop a geometric framework for analysing silent and anisotropic big bang singularities. In the present article, we record geometric conclusions obtained by combining the geometric framework with Einstein’s equations. The main features of the results are the following: The assumptions do not involve any symmetry requirements and are weak enough to be consistent with most big bang singularities for which the asymptotic geometry is understood. The framework gives a clear picture of the asymptotic geometry. It also reproduces the Kasner map, conjectured in the physics literature to constitute the essence of the asymptotic dynamics for vacuum solutions to Einstein’s equations. When combined with Einstein’s equations, the framework yields partial improvements of the assumptions concerning, e.g., the expansion normalised Weingarten map K (one of the central objects of the framework, defined as the Weingarten map of the leaves of the foliation divided by the mean curvature). For example, the expansion normalised normal derivative of K can, under suitable assumptions concerning the eigenvalues of K , be demonstrated to decay exponentially and K can be demonstrated to converge exponentially, even though we initially only impose weighted bounds on these quantities. Finally, the framework gives a unified perspective on the existing results. Moreover, in 3+1 -dimensions, the only parameters necessary to interpret the results are the eigenvalues of K and an additional scalar function determined by the geometry induced on the leaves of the foliation.In the companion article, we obtain conclusionsconcerning the asymptotic behaviour of solutions to linear sys-tems of wave equations on the backgrounds consistent with the framework.

Place, publisher, year, edition, pages
International Press of Boston , 2026. Vol. 132, no 3, p. 461-545
National Category
Mathematical Analysis Geometry
Identifiers
URN: urn:nbn:se:kth:diva-377859DOI: 10.4310/jdg/1770826357ISI: 001717893000002Scopus ID: 2-s2.0-105030885181OAI: oai:DiVA.org:kth-377859DiVA, id: diva2:2044603
Note

QC 20260310

Available from: 2026-03-10 Created: 2026-03-10 Last updated: 2026-05-20Bibliographically approved

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Ringström, Hans

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