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The Laplacian Flow of Locally Conformal Calibrated G2-Structures

1
Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
2
Departamento de Matemáticas—IUMA, Facultad de Ciencias Humanas y de la Educación, Universidad de Zaragoza, 22003 Huesca, Spain
*
Author to whom correspondence should be addressed.
Received: 8 November 2018 / Revised: 31 December 2018 / Accepted: 3 January 2019 / Published: 11 January 2019
(This article belongs to the Special Issue Applications of Differential Geometry)
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Abstract

We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( , T ) , where T > 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to , and they blow-up at a finite-time singularity. View Full-Text
Keywords: locally conformal calibrated G2-structures; Laplacian flow; solvable Lie algebras locally conformal calibrated G2-structures; Laplacian flow; solvable Lie algebras
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Fernández, M.; Manero, V.; Sánchez, J. The Laplacian Flow of Locally Conformal Calibrated G2-Structures. Axioms 2019, 8, 7.

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