Laplacian coflow on the $7$-dimensional Heisenberg group

We study the Laplacian coflow and the modified Laplacian coflow of $G_2$-structures on the $7$-dimensional Heisenberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval $(-\infty, T)$, with $0 \lt T \lt +\infty$. However, for the modified Laplacian coflow, we prove that in some cases the solution is defined only on a finite interval while in other cases the solution is ancient or eternal, that is it is defined on $(-\infty, \infty)$.

Keywords

$G_2$-structure, Laplacian coflow

2010 Mathematics Subject Classification

Primary 53C15. Secondary 53C30, 53C44.

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The first and third authors are supported by the project FIRB “Geometria differenziale e teoria geometrica delle funzioni”, the project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” and by G.N.S.A.G.A. of I.N.d.A.M.

The second author is supported through Project MINECO (Spain) PGC2018-098409-B-100 and Basque Government Project IT1094-16.

Received 23 January 2018

Accepted 6 June 2019

Published 8 September 2020