2022
Loubeau, Eric; Moreno, Andrés J.; Earp, Henrique N. Sá; Saavedra, Julieth
Harmonic Sp(2)-Invariant G2-Structures on the 7-Sphere Journal Article
Em: The Journal of Geometric Analysis, vol. 32, iss. 240, 2022.
Resumo | Links | BibTeX | Tags: 7-Sphere, G2-Structures
@article{nokey,
title = {Harmonic Sp(2)-Invariant G2-Structures on the 7-Sphere},
author = {Eric Loubeau and Andr\'{e}s J. Moreno and Henrique N. S\'{a} Earp and Julieth Saavedra },
url = {http://dx.doi.org/10.1007/s12220-022-00953-9},
doi = {http://dx.doi.org/10.1007/s12220-022-00953-9},
year = {2022},
date = {2022-07-18},
journal = {The Journal of Geometric Analysis},
volume = {32},
issue = {240},
abstract = {We describe the 10-dimensional space of Sp(2)-invariant G2-structures on the homogeneous 7-sphere S7=Sp(2)/Sp(1) as Ω3+(S7)Sp(2)≃R+×Gl+(3,R). In those terms, we formulate a general Ansatz for G2-structures, which realises representatives in each of the 7 possible isometric classes of homogeneous G2-structures. Moreover, the well-known nearly parallel round and squashed metrics occur naturally as opposite poles in an S3-family, the equator of which is a new S2-family of coclosed G2-structures satisfying the harmonicity condition divT=0. We show general existence of harmonic representatives of G2-structures in each isometric class through explicit solutions of the associated flow and describe the qualitative behaviour of the flow. We study the stability of the Dirichlet gradient flow near these critical points, showing explicit examples of degenerate and nondegenerate local maxima and minima, at various regimes of the general Ansatz. Finally, for metrics outside of the Ansatz, we identify families of harmonic G2-structures, prove long-time existence of the flow and study the stability properties of some well-chosen examples.},
keywords = {7-Sphere, G2-Structures},
pubstate = {published},
tppubtype = {article}
}
We describe the 10-dimensional space of Sp(2)-invariant G2-structures on the homogeneous 7-sphere S7=Sp(2)/Sp(1) as Ω3+(S7)Sp(2)≃R+×Gl+(3,R). In those terms, we formulate a general Ansatz for G2-structures, which realises representatives in each of the 7 possible isometric classes of homogeneous G2-structures. Moreover, the well-known nearly parallel round and squashed metrics occur naturally as opposite poles in an S3-family, the equator of which is a new S2-family of coclosed G2-structures satisfying the harmonicity condition divT=0. We show general existence of harmonic representatives of G2-structures in each isometric class through explicit solutions of the associated flow and describe the qualitative behaviour of the flow. We study the stability of the Dirichlet gradient flow near these critical points, showing explicit examples of degenerate and nondegenerate local maxima and minima, at various regimes of the general Ansatz. Finally, for metrics outside of the Ansatz, we identify families of harmonic G2-structures, prove long-time existence of the flow and study the stability properties of some well-chosen examples.