Remarks on homogeneous solitons of the $\protect \mathrm{G}_{2}$-Laplacian flow

Anna Fino; Alberto Raffero

doi:10.5802/crmath.39

Géométrie différentielle

Remarks on homogeneous solitons of the

G_{2}

-Laplacian flow

Anna Fino ¹ ; Alberto Raffero ¹

¹ Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 401-406.

Résumé

We show the existence of expanding solitons of the $G_{2}$ -Laplacian flow on non-solvable Lie groups, and we give the first example of a steady soliton that is not an extremally Ricci pinched $G_{2}$ -structure.

Reçu le : 2019-09-16
Révisé le : 2020-03-10
Accepté le : 2020-03-16
Publié le : 2020-07-28

DOI : 10.5802/crmath.39

Affiliations des auteurs :

Anna Fino ¹ ; Alberto Raffero ¹

¹ Dipartimento di Matematica “G. Peano”, Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Anna Fino and Alberto Raffero},
     title = {Remarks on homogeneous solitons of the $\protect \mathrm{G}_{2}${-Laplacian} flow},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {401--406},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {4},
     year = {2020},
     doi = {10.5802/crmath.39},
     language = {en},
}

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Anna Fino; Alberto Raffero. Remarks on homogeneous solitons of the $\protect \mathrm{G}_{2}$-Laplacian flow. Comptes Rendus. Mathématique, Volume 358 (2020) no. 4, pp. 401-406. doi : 10.5802/crmath.39. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.39/

Bibliographie
Cité par

[1] Gavin Ball Seven Dimensional Geometries with Special Torsion, Ph. D. Thesis, Duke University (2019)

[2] Robert L. Bryant Some remarks on $G_{2}$ -structures, Proceedings of Gökova Geometry-Topology Conference 2005, International Press, 2006, pp. 75-109 | Zbl

[3] Marisa Fernández; Anna Fino; Alberto Raffero Exact $G_{2}$ -structures on unimodular Lie algebras (2020) (https://arxiv.org/abs/1904.11066v3, to appear in Monatsh. Math.) | DOI

[4] Anna Fino; Alberto Raffero Closed $G_{2}$ -structures on non-solvable Lie groups, Rev. Mat. Complut., Volume 32 (2019) no. 3, pp. 837-851 | DOI | MR | Zbl

[5] Anna Fino; Alberto Raffero A class of eternal solutions to the $G_{2}$ -Laplacian flow (2020) (https://arxiv.org/abs/1807.01128v2, to appear in J. Geom. Anal.) | DOI

[6] Anna Fino; Alberto Raffero Closed warped $G_{2}$ -structures evolving under the Laplacian flow, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 20 (2020), pp. 315-348 | DOI

[7] Michael Jablonski Homogeneous Ricci solitons are algebraic, Geom. Topol., Volume 18 (2014) no. 4, pp. 2477-2486 | DOI | MR | Zbl

[8] Ramiro Lafuente; Jorge Lauret Structure of homogeneous Ricci solitons and the Alekseevskii conjecture, J. Differ. Geom., Volume 98 (2014) no. 2, pp. 315-347 | DOI | MR | Zbl

[9] Jorge Lauret Laplacian flow of homogeneous $G_{2}$ -structures and its solitons, Proc. Lond. Math. Soc., Volume 114 (2017) no. 3, pp. 527-560 | DOI | MR | Zbl

[10] Jorge Lauret Laplacian solitons: questions and homogeneous examples, Differ. Geom. Appl., Volume 54 (2017), pp. 345-360 | DOI | MR | Zbl

[11] Jorge Lauret; Marina Nicolini Ricci pinched $G_{2}$ -structures on Lie groups (2020) (https://arxiv.org/abs/1902.06375v3, to appear in Commun. Anal. Geom.)

[12] Christopher Lin Laplacian solitons and symmetry in $G_{2}$ -geometry, J. Geom. Phys., Volume 64 (2013), pp. 111-119 | MR | Zbl

[13] Jason D. Lotay Geometric Flows of $G_{2}$ Structures (To appear in a forthcoming volume of the Fields Institute Communications, entitled “Lectures and Surveys on G $_{2}$ manifolds and related topics”)

[14] Jason D. Lotay; Yong Wei Laplacian flow for closed $G_{2}$ structures: Shi-type estimates, uniqueness and compactness, Geom. Funct. Anal., Volume 27 (2017) no. 1, pp. 165-233 | DOI | MR | Zbl

[15] Marina Nicolini Laplacian solitons on nilpotent Lie groups, Bull. Belg. Math. Soc. Simon Stevin, Volume 25 (2018) no. 2, pp. 183-196 | DOI | MR | Zbl

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