Laplacian coflow for warped G2-structures

Manero, Víctor; Otal, Antonio; Villacampa, Raquel

doi:10.1016/j.difgeo.2020.101593

Cited by 6 publications

(7 citation statements)

References 23 publications

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“…We start by recalling that the torsion forms of a

G_{2}

‐structure

φ

on a manifold

M

are the components of the intrinsic torsion

\nabla φ

, where

\nabla

is the Levi–Civita connection of the metric

g

attached to

φ

. If we set

ψ : = * φ,

then they are defined as the unique differential forms

τ_{i} \in {normalΩ}^{i} M

i = 0, 1, 2, 3

, such that

d φ = τ_{0} ψ + 3 τ_{1} \land φ + * τ_{3}, d ψ = 4 τ_{1} \land ψ + τ_{2} \land φ,

and they are given by (see, for example, [23, (4)]),

\begin{matrix} τ_{0} = \frac{1}{7} * (d φ \land φ), τ_{1} = - \frac{1}{12} * (* d φ \land φ), \\ τ_{2} = - * d ψ + 4 * (τ_{1} \land ψ), τ_{3} = * d φ - τ_{0} φ - 3 * (τ_{1} \land φ) . \end{matrix}

…”

Section: G2‐geometry On the Group Gjmentioning

confidence: 99%

A new example of a compact ERP G2‐structure

Kath

Lauret

2021

Bulletin of London Math Soc

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We provide the second-known example of an extremally Ricci pinched closed G2-structure on a compact 7-manifold, by finding a lattice in the only unimodular solvable Lie group admitting a left-invariant G2-structure. Furthermore, the Laplacian coflow and its solitons are studied on a 6parameter family of left-invariant coclosed G2-structures on this Lie group. In this way, we obtain a 4-parameter subfamily of expanding solitons. The family is locally pairwise non-equivalent. Contents

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“…We start by recalling that the torsion forms of a

G_{2}

‐structure

φ

on a manifold

M

are the components of the intrinsic torsion

\nabla φ

, where

\nabla

is the Levi–Civita connection of the metric

g

attached to

φ

. If we set

ψ : = * φ,

then they are defined as the unique differential forms

τ_{i} \in {normalΩ}^{i} M

i = 0, 1, 2, 3

, such that

d φ = τ_{0} ψ + 3 τ_{1} \land φ + * τ_{3}, d ψ = 4 τ_{1} \land ψ + τ_{2} \land φ,

and they are given by (see, for example, [23, (4)]),

\begin{matrix} τ_{0} = \frac{1}{7} * (d φ \land φ), τ_{1} = - \frac{1}{12} * (* d φ \land φ), \\ τ_{2} = - * d ψ + 4 * (τ_{1} \land ψ), τ_{3} = * d φ - τ_{0} φ - 3 * (τ_{1} \land φ) . \end{matrix}

…”

Section: G2‐geometry On the Group Gjmentioning

confidence: 99%

A new example of a compact ERP G2‐structure

Kath

Lauret

2021

Bulletin of London Math Soc

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“…where τ 0 ∈ R, τ 1 ∈ Λ 1 g * , τ 2 ∈ Λ 2 14 g * and τ 3 ∈ Λ 3 27 g * are the torsion forms of ϕ. In [MOV,(3)], the authors gave the following useful formulas for the torsion forms: such that sp{e 1 , e 2 } is abelian, g 0 := sp{e 7 , e 1 , e 2 } is a subalgebra, g 1 := sp{e 3 , e 4 , e 5 , e 6 } is an abelian ideal and h := sp{e 1 , . .…”

Section: Note Thatmentioning

confidence: 99%

New examples of shrinking Laplacian solitons

Nicolini¹

2020

Preprint

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We give a one-parameter family of examples of shrinking Laplacian solitons, which are the second known solutions to the closed G2-Laplacian flow with a finite-time singularity. The torsion forms and the Laplacian and Ricci operators of a large family of G2-structures on different Lie groups are also studied. We apply these formulas to prove that, under a suitable extra condition, there is no closed eigenform for the Laplacian on such family.

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“…, 7q. With this notation, (20) implies (21) γ pδ l `δm `δn `δo q " ´α pβ i `βj `βk q, for all pl, m, n, oq P K and pl, m, n, oq " { pi, j, kq. Now we prove the two statements of the theorem.…”

Section: Long Time Solutions Of the Laplacian Coflow Of An Lcp G 2 -Structurementioning

confidence: 99%

“…Now we prove the two statements of the theorem. (i) Let us consider two pair of indexes pi 1 , j 1 , k 1 q, pi 2 , j 2 , k 2 q P A Y B such that they have a common index, let us say k 1 " k 2 " k. Under this hypothesis and making use of (21) the following identities hold: γpδ 1 `. .…”

Section: Long Time Solutions Of the Laplacian Coflow Of An Lcp G 2 -Structurementioning

confidence: 99%

“…Therefore, bearing in mind (21) the following sequence of identities hold: p r f s 7 ptqq 2 r ∆ lmno " pf s 7 ptqq 2 ∆ ijk " ´αmpβ i `βj `βk q " γmpδ l `δm `δn `δo q, for every pl, m, n, oq P K Y tp2, 4, 6, 7qu, that is, r ψptq is a solution of the coflow. The converse of the statement is basically the same and we omit the proof.…”

Section: Long Time Solutions Of the Laplacian Coflow Of An Lcp G 2 -Structurementioning

confidence: 99%

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Solutions of the Laplacian flow and coflow of a locally conformal parallel $$\mathrm {G}_2$$-structure

2020

Self Cite

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We study the Laplacian flow of a G 2 -structure where this latter structure is claimed to be Locally Conformal Parallel. The first examples of long time solutions of this flow with the Locally Conformal Parallel condition are given. All of the solutions are ancient and Laplacian soliton of shrinking type. These examples are one-parameter families of Locally Conformal Parallel G 2 -structures on rankone solvable extensions of six-dimensional nilpotent Lie groups. The found solutions are used to construct long time solutions to the Laplacian coflow starting from a Locally Conformal Parallel structure. We also study the behavior of the curvature of the solutions obtaining that for one of the examples the induced metric is Einstein along all the flow (resp. coflow).

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Laplacian coflow for warped G2-structures

Cited by 6 publications

References 23 publications

A new example of a compact ERP G2‐structure

A new example of a compact ERP G2‐structure

New examples of shrinking Laplacian solitons

Solutions of the Laplacian flow and coflow of a locally conformal parallel $$\mathrm {G}_2$$-structure

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