Laplacian coflow on the 7-dimensional Heisenberg group

Abstract

We study the Laplacian coflow and the modified Laplacian coflow of G 2 -structures on the 7-dimensional Heinseberg group. For the Laplacian coflow we show that the solution is always ancient, that is it is defined in some interval (−∞, T ), with 0 < T < +∞. 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 defined for every positive time.

On $G_2$-structures, special metrics and related flows

“…It turns out that the behaviour of these flows on solvable Lie groups is slightly different from the behaviour of the Laplacian flow. We refer the reader to [5,6] for a detailed treatment.…”

Stability of geometric flows of closed forms

“…To overcome this technical difficulty, Grigorian modified in [9] the Laplacian coflow by introducing two extra terms, one of which depends on a parameter A. In the compact case, this modification is always well-posed for any choice of A ∈ R [9], but it has been shown that the behaviour of the flow may significantly depend on the choice of A (see [1]).…”

The Laplacian coflow on almost-abelian Lie groups

“…No general result is known about the short time existence of the coflow (1). In [2] the Laplacian coflow on the seven-dimensional Heiseberg group has been studied, showing that the solution is always ancient, that is it is defined in some interval (−∞, T ), with 0 < T < +∞. Other examples of flows of G 2 -structures are the modified Laplacian coflow [11,12] and Weiss and Witt's heat flow [24].…”

“…where b i (t) = e −σ i (t) for a function (t) satisfying ˙ (t) = f (t) 2 . In order to determine the function f (t), note that, for every t where it is defined, the 3-form χ t is negative, compatible with ω 0 and it defines a complex structure J t , given by…”

“…is the solution of (7). To see this, first observe that the choice of f (t) ensures that ε(χ t ) 2 = f (t) 2 . The only thing we have to prove is that the adjoint…”