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.
Lotay, Jason D.; Earp, Henrique N. Sá; Saavedra, Julieth
Flows of G2-structures on contact Calabi–Yau 7-manifolds Journal Article
Em: Annals of Global Analysis and Geometry, vol. 62, pp. 367–389, 2022.
Resumo | Links | BibTeX | Tags: Calabi–Yau, G2-Structures
@article{nokey,
title = {Flows of G2-structures on contact Calabi\textendashYau 7-manifolds},
author = {Jason D. Lotay and Henrique N. S\'{a} Earp and Julieth Saavedra },
url = {https://link.springer.com/article/10.1007/s10455-022-09854-0},
doi = {http://dx.doi.org/10.1007/s10455-022-09854-0},
year = {2022},
date = {2022-06-21},
journal = {Annals of Global Analysis and Geometry},
volume = {62},
pages = {367\textendash389},
abstract = {We study the Laplacian flow and coflow on contact Calabi\textendashYau 7-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas it produces an immortal (though neither eternal nor self-similar) solution of the Laplacian coflow which has an infinite time singularity of Type IIb, unless the transverse Calabi\textendashYau geometry is flat. The flows in each case collapse (under normalised volume) to a lower-dimensional limit, which is either R, for the Laplacian flow, or standard C3, for the Laplacian coflow. We also study the Hitchin flow in this setting, which we show coincides with the Laplacian coflow, up to reparametrisation of time, and defines an (incomplete) Calabi\textendashYau structure on the spacetime track of the flow.},
keywords = {Calabi\textendashYau, G2-Structures},
pubstate = {published},
tppubtype = {article}
}
We study the Laplacian flow and coflow on contact Calabi–Yau 7-manifolds. We show that the natural initial condition leads to an ancient solution of the Laplacian flow with a finite time Type I singularity which is not a soliton, whereas it produces an immortal (though neither eternal nor self-similar) solution of the Laplacian coflow which has an infinite time singularity of Type IIb, unless the transverse Calabi–Yau geometry is flat. The flows in each case collapse (under normalised volume) to a lower-dimensional limit, which is either R, for the Laplacian flow, or standard C3, for the Laplacian coflow. We also study the Hitchin flow in this setting, which we show coincides with the Laplacian coflow, up to reparametrisation of time, and defines an (incomplete) Calabi–Yau structure on the spacetime track of the flow.
2018
del Barco, Viviana; Grama, Lino
On generalized G2-structures and T-duality Journal Article
Em: Journal of Geometry and Physics, vol. 132, pp. 109-113, 2018.
Resumo | Links | BibTeX | Tags: G2-Structures, T-duality
@article{nokey,
title = {On generalized G2-structures and T-duality},
author = {Viviana del Barco and Lino Grama},
url = {https://www.sciencedirect.com/science/article/pii/S0393044018303413?via%3Dihub},
doi = {https://doi.org/10.1016/j.geomphys.2018.05.021},
year = {2018},
date = {2018-06-18},
journal = {Journal of Geometry and Physics},
volume = {132},
pages = {109-113},
abstract = {This is a short note on generalized -structures obtained as a consequence of a -dual construction given in del Barco et al. (2017). Given classical -structure on certain seven dimensional manifolds, either closed or co-closed, we obtain integrable generalized -structures which are no longer a usual one, and with non-zero three form in general. In particular we obtain manifolds admitting closed generalized -structures not admitting closed (usual) -structures.},
keywords = {G2-Structures, T-duality},
pubstate = {published},
tppubtype = {article}
}
This is a short note on generalized -structures obtained as a consequence of a -dual construction given in del Barco et al. (2017). Given classical -structure on certain seven dimensional manifolds, either closed or co-closed, we obtain integrable generalized -structures which are no longer a usual one, and with non-zero three form in general. In particular we obtain manifolds admitting closed generalized -structures not admitting closed (usual) -structures.