Flows of Co-closed $$G_{2}$$-Structures

Grigorian, Sergey

doi:10.1007/978-1-0716-0577-6_12

Cited by 5 publications

(10 citation statements)

References 20 publications

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“…In order to use such formulas, we introduce the following notation. We write the positive 3-form given in (12) as…”

Section: Formulas For the Torsion And The Laplaciansmentioning

confidence: 99%

“…where T is the (full) torsion tensor (that is, ι T (X) ψ = ∇ X ϕ). We refer to the survey [12] and the references therein for more information. A G 2 -structure ψ flows self-similarly according to the Laplacian coflow (respectively, modified Laplacian coflow) if and only if Δψ = λψ + L X ψ (respectively, Δψ + 2d((m − tr T )ϕ) = λψ + L X ψ) for some λ ∈ R and X ∈ X(M ), and in that case it is called a (modified) Laplacian coflow soliton.…”

Section: Laplacian Coflow Solitonsmentioning

confidence: 99%

“…Natural ways to evolve coclosed

G_{2}

‐structures on a given manifold

M

are the Laplacian coflow , defined by

\frac{\partial}{\partial t} ψ false(t false) = Δ ψ false(t false),

and the modified Laplacian coflow given by

\frac{\partial}{\partial t} ψ false(t false) = Δ ψ false(t false) + 2 d (false(m - prefixtr T false) φ false(t false)), m \in R,

where

T

is the (full) torsion tensor (that is,

ι_{T (X)} ψ = \nabla_{X} φ

). We refer to the survey [12] and the references therein for more information. A

G_{2}

‐structure

ψ

flows self‐similarly according to the Laplacian coflow (respectively, modified Laplacian coflow) if and only if

Δ ψ = λ ψ + {scriptL}_{X} ψ

(respectively,

Δ ψ + 2 d false((m - tr T) φ false) = λ ψ + {scriptL}_{X} ψ

) for some

λ \in R

and

X \in X (M)

, and in that case it is called a (modified) Laplacian coflow soliton .…”

Section: G2‐geometry On the Group Gjmentioning

confidence: 99%

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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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“…In order to use such formulas, we introduce the following notation. We write the positive 3-form given in (12) as…”

Section: Formulas For the Torsion And The Laplaciansmentioning

confidence: 99%

Section: Laplacian Coflow Solitonsmentioning

confidence: 99%

“…Natural ways to evolve coclosed

G_{2}

‐structures on a given manifold

M

are the Laplacian coflow , defined by

\frac{\partial}{\partial t} ψ false(t false) = Δ ψ false(t false),

and the modified Laplacian coflow given by

\frac{\partial}{\partial t} ψ false(t false) = Δ ψ false(t false) + 2 d (false(m - prefixtr T false) φ false(t false)), m \in R,

where

T

is the (full) torsion tensor (that is,

ι_{T (X)} ψ = \nabla_{X} φ

). We refer to the survey [12] and the references therein for more information. A

G_{2}

‐structure

ψ

flows self‐similarly according to the Laplacian coflow (respectively, modified Laplacian coflow) if and only if

Δ ψ = λ ψ + {scriptL}_{X} ψ

(respectively,

Δ ψ + 2 d false((m - tr T) φ false) = λ ψ + {scriptL}_{X} ψ

) for some

λ \in R

and

X \in X (M)

, and in that case it is called a (modified) Laplacian coflow soliton .…”

Section: G2‐geometry On the Group Gjmentioning

confidence: 99%

See 1 more Smart Citation

A new example of a compact ERP G2‐structure

Kath

Lauret

2021

Bulletin of London Math Soc

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“…The condition of the induced metric to be Ricci soliton is preserved along the coflow. For overviews on these topics, see [13] and [17].…”

Section: Introductionmentioning

confidence: 99%

“…‚ cp m 6 : f 17 ptq " f 36 ptq " f 45 ptq, f 27 ptq " f 35 ptq " f 46 ptq. ‚ cp m 7 : f 17 ptq " f 36 ptq, f 23 ptq " f 57 ptq, f 26 ptq " f 47 ptq.The final result is obtained by substituting the defining parameters of the Lie algebras cp m s listed in Table2in both the expressions(13) and(14).After solving ϕptq P X 4 , let us focus on the evolution equation dϕptq dt " ∆ t ϕptq. Next, we get a generic expression of the Laplacian ∆ t ϕptq suitable for any of the Lie algebras cp m s .…”

mentioning

confidence: 99%

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

2020

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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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Flows of G2-structures on 7-manifolds with symmetry

Paola Saavedra Ramirez

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ResumoA tese é composta de duas partes em relação as G 2 -estruturas. Na primeira parte, vamos descrever o espaço de G2-estruturas Sp(2)-invariantes de dimensão 10 sobre o espaço homogêneo S 7 = Sp(2)/Sp(1), em que S 7 é a esfera 7-dimensional. Nesta parte, foi formulado um Ansatz geral para as G 2 -estruturas que realizam os representantes em cada uma das 7 possíveis classes isométricas de G 2 -estruturas homogêneas. Mais ainda, as bem conhecidas G 2 -estruturas quase-paralelas na esfera redonda e esmagada acontecem em polos diferentes de uma S 3 -família, cujo equador é uma nova S 2 -família de G 2 -estrutura cofechadas satisfazendo a condição divT = 0. Na segunda parte, nos estudamos o cofluxo laplaciano de G2-estruturas proposto por Karigiannis, McKay and Tsui in [25] sobre variedades de contato Calabi-Yau usando como valor inicial a G 2 -estrutura dada por Habib e Vezoni em [22] encontrando uma singularidade. Nós demostramos que a métrica e o volume colapsam nesta singularidade. Também analisamos soluções, tipo soliton do cofluxo laplaciano de G2-estruturas sobre variedades de contato Calabi-Yau dadas por Sá Earp e Lotay em [32]. Palavras-chave: G 2 -estruturas, G 2 -fluxo, espaços homogêneos.

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Flows of Co-closed $$G_{2}$$-Structures

Cited by 5 publications

References 20 publications

A new example of a compact ERP G2‐structure

A new example of a compact ERP G2‐structure

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

Flows of G2-structures on 7-manifolds with symmetry

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