The Alekseevskii conjecture in low dimensions

Arroyo, Romina M.; Lafuente, Ramiro A.

doi:10.1007/s00208-016-1386-1

Cited by 18 publications

(12 citation statements)

References 37 publications

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“…Nothing changes by allowing a derivation of the form

D = false[\begin{matrix} * & 0 \\ 0 & D_{p} \end{matrix} false] \in Der false(frakturg false)

in the definition of algebraic soliton since

D k = 0

must necessarily hold (see [, Remark 7]). It is proved in [, Section 4] (see also ) that algebraic solitons are precisely the fixed points, and hence the possible limits of any normalized bracket flow. Furthermore, given a starting point, one can obtain at most one non‐flat algebraic soliton as a limit by running all possible normalized bracket flow solutions (see Corollary ).…”

Section: Laplacian Flow On Homogeneous Spacesmentioning

confidence: 99%

“…Let p be a real vector space of dimension 7. A 3-form ϕ ∈ Λ 3 p * is called positive if it can be written as in (2) in terms of some basis, or equivalently, if it belongs to the open orbit GL(p) · φ ⊂ Λ 3 p * . It follows that the set of all positive 3-forms is parameterized by the 35-dimensional homogeneous space GL 7 (R)/G 2 .…”

Section: Positive 3-formsmentioning

confidence: 99%

“…A G 2 -structure on a seven-dimensional differentiable manifold M is a differential 3-form ϕ ∈ Ω 3 M such that ϕ p is positive on T p M for any p ∈ M , or in other words, ϕ p can be written as in (2) with respect to some basis {e 1 , . .…”

Section: G 2 -Structuresmentioning

confidence: 99%

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Laplacian flow of homogeneous G2-structures and its solitons

Lauret

2017

Proc. London Math. Soc.

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We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a $G$-invariant $G_2$-structure on a homogeneous space $G/K$ that flows by pull-back of automorphisms of $G$ up to scaling). Algebraic solitons are geometrically characterized among Laplacian solitons as those with a `diagonal' evolution. Unlike the Ricci flow case, where any homogeneous Ricci soliton is isometric to an algebraic soliton, we have found, as an application of the above characterization, an example of a left-invariant closed semi-algebraic soliton on a nilpotent Lie group which is not equivalent to any algebraic soliton. The (normalized) bracket flow evolution of such a soliton is periodic. In the context of solvable Lie groups with a codimension-one abelian normal subgroup, we obtain long time existence for any closed Laplacian flow solution; furthermore, the norm of the torsion is strictly decreasing and converges to zero. We also classify algebraic solitons in this class and exhibit several explicit examples of closed expanding Laplacian solitons.Comment: 33 pages, final version accepted in Proc. London Math. So

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“…Nothing changes by allowing a derivation of the form

D = false[\begin{matrix} * & 0 \\ 0 & D_{p} \end{matrix} false] \in Der false(frakturg false)

in the definition of algebraic soliton since

D k = 0

Section: Laplacian Flow On Homogeneous Spacesmentioning

confidence: 99%

Section: Positive 3-formsmentioning

confidence: 99%

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Laplacian flow of homogeneous G2-structures and its solitons

Lauret

2017

Proc. London Math. Soc.

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“…As u is reductive, we have u = [u, u] + z(u) (vector space direct sum). Recently, Arroyo and Lafuente proved that this direct sum is orthogonal using Lemma 3.6 (i) below, see [4].…”

Section: Remark 31mentioning

confidence: 99%

A step towards the Alekseevskii conjecture

Jablonski

Petersen²

2016

Math. Ann.

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We refine existing structure results for non-compact, homogeneous, Einstein manifolds and provide a reduction in the classification problem of such spaces. Using this work, we verify the (Generalized) Alekseevskii Conjecture for a large class of homogeneous spaces.A longstanding open question in the study of Riemannian homogeneous spaces is the classification of non-compact, Einstein spaces. In the 1970s, it was conjectured by D. Alekseevskii that any (non-compact) homogeneous Einstein space of negative scalar curvature is diffeomorphic to R n . Equivalently, this conjecture can be phrased as follows:Classical Alekseevskii Conjecture: Given a homogeneous Einstein space G/K with negative scalar curvature, K must be a maximal compact subgroup of G.

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“…As smooth manifolds, the spaces a2) and a3) are diffeomorphic to S 3 × S 3 × S 2 , while the space a4) is diffeomorphic to S 5 × S 3 . b) The compact, simply connected, symmetric spaces admitting a Spin(7)-structure are exhausted by SU(3), S 3 × S 3 × S 2 , S 5 × S 3 , HP 2 and the exceptional Wolf space G 2 SO (4) .…”

Section: Introductionmentioning

confidence: 99%

Homogeneous 8-manifolds admitting invariant Spin(7)-structures

Alekseevsky,

Chrysikos,

Fino

et al. 2019

Preprint

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We study compact, simply connected, homogeneous 8-manifolds admitting invariant Spin(7)-structures, classifying all canonical presentations G/H of such spaces, with G simply connected. For each presentation, we exhibit explicit examples of invariant Spin( 7)-structures and we describe their type, according to Fernández classification. Finally, we analyse the associated Spin(7)-connection with torsion.

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The Alekseevskii conjecture in low dimensions

Cited by 18 publications

References 37 publications

Laplacian flow of homogeneous G2-structures and its solitons

Laplacian flow of homogeneous G2-structures and its solitons

A step towards the Alekseevskii conjecture

Homogeneous 8-manifolds admitting invariant Spin(7)-structures

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