@article{nokey,

title = {The Weitzenbock formula for the Fueter\textendashDirac operator},

author = {Andr\'{e}s J. Moreno and Henrique N. S\'{a} Earp},

url = {https://www.intlpress.com/site/pub/pages/journals/items/cag/content/vols/0030/0001/a003/},

doi = {http://dx.doi.org/10.4310/CAG.2022.v30.n1.a3},

year  = {2022},

date = {2022-07-22},

journal = {Communications in Analysis and Geometry},

volume = {30},

issue = {1},

pages = {153 \textendash 205},

abstract = {We find a Weitzenb\"{o}ck formula for the Fueter\textendashDirac operator which controls infinitesimal deformations of an associative submanifold in a \textendashmanifold with a -structure. We establish a vanishing theorem to conclude rigidity under some positivity assumptions on curvature, which are particularly mild in the nearly parallel case. As applications, we find a different proof of rigidity for one of Lotay’s associatives in the round -sphere from those given by Kawai [14, 15]. We also provide simpler proofs of previous results by Gayet for the Bryant-Salamon metric [11]. Finally, we obtain an original example of a rigid associative in a compact manifold with locally conformal calibrated -structure obtained by Fern\'{a}ndez\textendashFino\textendashRaffero [9].},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@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 = {},

pubstate = {published},

tppubtype = {article}

}

@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 = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {Irreducible components of the moduli space of rank 2 sheaves of odd determinant on projective space},

author = {Charles Almeida and Marcos Jardim and Alexander S.Tikhomirov},

url = {https://www.sciencedirect.com/science/article/abs/pii/S0001870822001797?via%3Dihub},

doi = {https://doi.org/10.1016/j.aim.2022.108363},

year  = {2022},

date = {2022-03-29},

journal = {Advances in Mathematics},

volume = {402},

abstract = {We describe new irreducible components of the moduli space of rank 2 semistable torsion free sheaves on the three-dimensional projective space whose generic point corresponds to non-locally free sheaves whose singular locus is either 0-dimensional or consists of a line plus disjoint points. In particular, we prove that the moduli spaces of semistable sheaves with Chern classes  and  always contain at least one rational irreducible component. As an application, we prove that the number of such components grows as the second Chern class grows, and compute the exact number of irreducible components of the moduli spaces of rank 2 semistable torsion free sheaves with Chern classes  for non negative values for m; all components turn out to be rational. Furthermore, we also prove that these moduli spaces are connected, showing that some of sheaves here considered are smoothable.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{nokey,

title = {Invariant Einstein metrics on real flag manifolds with two or three isotropy summands},

author = {Brian Grajales and Lino Grama},

url = {https://www.sciencedirect.com/science/article/pii/S0393044022000444?via%3Dihub},

doi = {https://doi.org/10.1016/j.geomphys.2022.104494},

year  = {2022},

date = {2022-03-22},

journal = {Journal of Geometry and Physics},

volume = {176},

abstract = {We study the existence of invariant Einstein metrics on real flag manifolds associated to simple and non-compact split real forms of complex classical Lie algebras whose isotropy representation decomposes into two or three irreducible subrepresentations. In this situation, one can have equivalent submodules, leading to the existence of non-diagonal homogeneous Riemannian metrics. In particular, we prove the existence of non-diagonal Einstein metrics on real flag manifolds.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}