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Axioms 2019, 8(1), 7; doi:10.3390/axioms8010007
Article
The Laplacian Flow of Locally Conformal Calibrated G2-Structures
1
Departamento de Matemáticas, Facultad de Ciencia y Tecnología, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
2
Departamento de Matemáticas—IUMA, Facultad de Ciencias Humanas y de la Educación, Universidad de Zaragoza, 22003 Huesca, Spain
*
Author to whom correspondence should be addressed.
Received: 8 November 2018 / Accepted: 3 January 2019 / Published: 11 January 2019
Abstract
:We consider the Laplacian flow of locally conformal calibrated -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated manifolds and, in both cases, we obtain a flow of locally conformal calibrated -structures, which are ancient solutions, that is they are defined on a time interval of the form , where is a real number. Moreover, for each of these examples, we prove that the underlying metrics of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to , and they blow-up at a finite-time singularity.
Keywords:
locally conformal calibrated G2-structures; Laplacian flow; solvable Lie algebras1. Introduction
A -structure on a 7-manifold M can be characterized by the existence of a globally defined 3-form (the form) on M, which can be written at each point as
with respect to some local coframe on M. Here, stands for , and so on. A -structure induces a Riemannian metric and a volume form on M given by
for any pair of vector fields on M, where denotes the contraction by X.
The classes of -structures can be described in terms of the exterior derivatives of the 3-form and the 4-form [1,2], where is the Hodge operator defined from and . If the 3-form is closed and coclosed, then the holonomy group of is a subgroup of the exceptional Lie group [2], and the metric is Ricci-flat [3]. When this happens, the -structure is said to be torsion-free [4]. This condition has a variational formulation, due to Hitchin [5,6]. The first compact examples of Riemannian manifolds with holonomy were constructed first by Joyce [7,8], and then by Kovalev [9]. Recently, other examples of compact manifolds with holonomy were obtained in [10,11]. Explicit examples on solvable Lie groups were also constructed in [12]. A -structure is called locally conformal parallel if satisfies the two following conditions
for some closed non-vanishing 1-form , which is known as the Lee form of the -structure. Such a -structure is locally conformal to one which is torsion-free. Ivanov, Parton and Piccinni in [13] prove that a compact locally conformal parallel manifold is a mapping torus bundle over the circle with fibre a simply connected nearly Kähler manifold of dimension six and finite structure group.
We remind that a -structure is called closed (or calibrated according to [14]) if . In this paper we will focus our attention on the class of locally conformal calibrated -structures, which are characterized by the condition
where is a closed non-vanishing 1-form, which is also known as the Lee form of the -structure. We will refer to a manifold equipped with such a structure as a locally conformal calibrated manifold. Each point of such a manifold has an open neighborhood U where , for some with being the algebra of the real differentiable functions on U, and the 3-form defines a calibrated -structure on U. Hence, locally conformal calibrated -structures are locally conformal equivalent to calibrated -structures, and they can be considered analogous in dimension 7 to the locally conformal symplectic manifolds, which have been studied in [15,16,17,18,19,20,21] and the references therein. Some results of locally conformal calibrated manifolds were given in [22,23,24,25]. In fact, in [24] the first author and Ugarte introduced a differential complex for locally conformal calibrated manifolds, and such manifolds were characterized as the ones endowed with a -structure for which the space of differential forms annihilated by becomes a differential subcomplex of the de Rham’s complex. Moreover, in [23] it is proved that a similar result to that of Ivanov, Parton and Piccinni holds for compact 7-manifolds with a suitable locally conformal calibrated -structure. More recently, a structure result for Lie algebras with an exact locally conformal calibrated -structure was proved by Bazzoni and Raffero in [22], where it is also shown that none of the non-Abelian nilpotent Lie algebras with closed -structures admits locally conformal calibrated -structures.
Compact -calibrated manifolds have interesting curvature properties. As we mentioned before, a holonomy manifold is Ricci-flat or, equivalently, both Einstein and scalar-flat. But on a compact calibrated manifold, both the Einstein condition [26] and scalar-flatness [27] are equivalent to the holonomy being contained in . In fact, Bryant in [27] shows that the scalar curvature is always non-positive.
Locally conformal calibrated -structures whose underlying Riemannian metric is Einstein have been studied in [25], where it was shown that in the compact case the scalar curvature of can not be positive. Then, Fino and Raffero in [25] show that a compact homogeneous 7-manifold cannot admit an invariant Einstein locally conformal calibrated -structure unless the underlying metric is flat. However, in contrast to the compact homogeneous case, a non-compact example of homogeneous manifold S endowed with a locally conformal calibrated -structure whose associated Riemannian metric is Einstein and non Ricci-flat was given in [25]. The manifold S is a simply connected solvable Lie group which is not unimodular (see Section 4.2 for details).
On the other hand, in [23] it is given an example of a compact manifold N with a locally conformal calibrated -structure. The manifold N is a compact solvmanifold, that is N is a compact quotient of a simply connected solvable Lie group K by a lattice, endowed with an invariant locally conformal calibrated -structure.
Since Hamilton introduced the Ricci flow in 1982 [28], geometric flows have been an important tool in studying geometric structures on manifolds. In geometry, geometric flows for different -structures have been proposed. Let M be a 7-manifold endowed with a calibrated -structure . The Laplacian flow starting from is the initial value problem
where is a closed form on M, and is the Hodge Laplacian operator associated with the metric induced by the 3-form . This flow was introduced by Bryant in [27] as a tool to find torsion-free -structures on compact manifolds. Short-time existence and uniqueness of the solution when M is compact were proved in [29]. The analytic and geometric properties of the Laplacian flow have been deeply investigated in the series of papers [30,31,32]. Non-compact examples where the flow converges to a flat -structure have been given in [33].
In [34], a flow evolving the 4-form in the direction of minus its Hodge Laplacian was introduced, and it is called Laplacian coflow of . This flow preserves the condition of the -structure being coclosed, that is is closed for any t, and it was studied in [34] for two explicit examples of coclosed -structures. But no general result is known about the short time existence of the coflow. A modified Laplacian coflow was introduced by Grigorian in [35] (see also [36]). There it was proved that for compact manifolds, the modified Laplacian coflow has a unique solution for the short time period , for some . Geometric properties of both coflows on the 7-dimensional Heisenberg group and on 7-dimensional almost-abelian Lie groups were proved in [37,38], respectively.
Some work has also been done on other related flows of -structures—such as the Laplacian flow and the Laplacian coflow, for locally conformal parallel -structures. These flows has been originally proposed by the second author with Otal and Villacampa in [39], and the first examples of long time solutions of the flows are given in [39].
In this note, for any locally conformal calibrated -structure on a manifold M, we consider the Laplacian flow of given by
We do not known any general result on the short time existence of solution for this flow. Nevertheless, in Section 4 (Theorems 1 and 2), for each of the aforementioned examples of solvable Lie groups K and S with a locally conformal calibrated -structure, we show that the solution of the before Laplacian flow is ancient, that is it is defined on a time interval of the form , where is a real number. Moreover, for each of the two examples K and S, we show that the underlying metrics of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to , and they blow-up in finite-time. As we mentioned before, the Lie group S has a locally conformal calibrated -structure inducing an Einstein metric. We prove that the solution of the flow on S induces an Einstein metric for all time t where is defined.
2. -Structures
Let M be a 7-dimensional manifold with a -structure defined by a 3-form . Denote by the 4-form , where is the Hodge star operator of the metric induced by . Let be the de Rham complex of differential forms on M. Then, Bryant in [27] proved that the forms and are such that
where and . Here and are the spaces
The differential forms () that appear in (3), are called the intrinsic torsion forms of .
In terms of the torsion forms, some classes of -structures with the defining conditions are recalled in the Table 1.
3. The Laplacian Flow of Locally Conformal Calibrated G2-Structures
In this section, we introduce the Laplacian flow of a locally conformal calibrated -structure on a manifold M and, for its equations, we show some properties that help us solve the flow when M is a Lie group.
Definition 1.
Let M be a 7-manifold with a locally conformal calibrated -structure φ. We say that a time-dependent -structure on M, defined for t in some real open interval, satisfies the Laplacian flow system of φ if, for all times t for which is defined, we have
where is the Lee form of , and is the Hodge Laplacian operator associated with the metric induced by the 3-form .
In order to solve the first equation of the flow (5) for our examples, we follow the approach of [39].
Let G be a simply connected solvable Lie group of dimension 7 with Lie algebra g. Let be a basis of the dual space of g, and let be some differentiable real functions depending on a parameter such that and , for any , where I is a real open interval. For each , we consider the basis of defined by
We consider the one-parameter family of left invariant -structures on G given by
where stands for the product .
Now, we introduce the function on ordered indices as follows:
Therefore,
Moreover, we have
where is the coefficient in of if (i.e., if ), and is the coefficient in of if and . Consequently, the first equation of the flow (5) is equivalent to the system of differential equations
that is,
We will also use the following properties of .
Lemma 1.
Proof.
The first statement of this Lemma was proved in [39]. Nevertheless, we point out how to prove it. Since , the system (8) implies that . Hence, , for some constant C. Now, using that , for , we have that . So, , which proves i).
Now, let us suppose that , for some with and . From (7), we get
Integrating this equation, we obtain , for some constant C. Since , for all , we have , and so . This proves ii).
Finally, let us suppose that , for some with and . Then, using (7), we get . Integrating this equation, we obtain , for some constant C. But since , for all . Thus, , which completes the proof. □
4. Solutions of the Laplacian Flow on Locally Conformal Calibrated Solvmanifolds
Lie groups admitting left invariant locally conformal calibrated -structures constitute a convenient setting where it is possible to investigate the behaviour of the Laplacian flow (5) in the non-compact case.
In this section, we consider two examples of solvable Lie groups K and S, each of them with a left invariant locally conformal calibrated -structure, and we show that in both cases the solution is ancient (i.e., it is defined in some interval , with ) and the induced metrics blow-up at a finite-time singularity.
4.1. The Laplacian Flow on K
Let K be the simply connected and solvable Lie group of dimension 7 whose Lie algebra k is defined by
Here, stands for , and so on; and means that there is a basis of the dual space of k, satisfying
where d denotes the Chevalley-Eilenberg differential on .
The 3-form on K given by
defines a left invariant locally conformal calibrated -structure on the Lie group K, with Lee form , and so with torsion form . In fact,
In [23] it is proved that there exists a lattice in K, so that the quotient space of right cosets is a compact solvmanifold endowed with an invariant locally conformal calibrated -structure , with Lee form .
However, we should note that in the following Theorem, we will show a solution of the Laplacian flow (5) of the form (defined by (10)) on the Lie group K. Such a solution does not solve the Laplacian flow of on the compact quotient since we will consider the Hodge Laplacian operator on the Lie algebra k of K and we cannot check the Hodge Laplacian operator on the compact space .
Theorem 1.
The family of locally conformal calibrated -structures on K given by
is the solution for the Laplacian flow (5) of the form φ given by (10), where . The Lee form of is . Moreover, the underlying metrics of this solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in K, as t goes to , and they blow-up as t goes to .
Proof.
As in Section 2, let be some differentiable real functions depending on a parameter such that and , for any , where I is a real open interval. For each , we consider the basis of left invariant 1-forms on K defined by
From now on, we write , , and so forth. Then, for any , we consider the -structure on K given by
Note that the 3-form defined by (13) is such that and, for any t, determines the metric on K such that the basis of left invariant vector fields on K dual to is orthonormal. So, , and hence .
To solve the flow (5) of we determine firstly the functions and the interval I so that , for . We know that
We calculate separately each of the terms and of . Taking into account (12) and the fact that the basis is orthonormal, we have
and, on the other hand, we obtain
Each of these equalities implies that , and so
since .
The equalities (19) imply that the functions and defined in (18) are such that . Hence, . So, from (17), we have . Now, Lemma 1– ii) and (19) imply
Thus,
From this equality and (21), we obtain
Now, we can suppose that (see below Lemma 2). Then, the previous conditions reduce to
This implies that the unique non-zero components of the Laplacian of are
Then, the system of differential Equations (7) reduces to
Integrating this equation, we obtain
But implies . Hence,
Therefore, the one-parameter family of 3-forms given by (11) is the solution of the Laplacian flow of on K, and it exists for every .
A simple computation shows that
and so the Lee form of is .
Now we study the behavior of the underlying metric of such a solution in the limit for . If we think of the Laplacian flow as a one parameter family of manifolds with a locally conformal calibrated -structure, it can be checked that, in the limit, the resulting manifold has vanishing curvature. For , let us consider the metric on K induced by the form given by (11). Then,
Then, taking into account the symmetry properties of the Riemannian curvature we obtain
where . Therefore, .
Furthermore, the curvatures of blow-up as t goes to , and the finite-time singularity is of Type I since as ; in fact,
□
To complete the proof of Theorem 1, we show that under the conditions (19)–(22) the assumption , that we made in its proof, is correct.
Lemma 2.
Proof.
Take and . We know that if the 3-form defined in (13) is the solution for the Laplacian flow (5) of the form , then the equalities (19)–(22) are satisfied. Now, taking into account (17), the equalities (19)–(22) imply that the Hodge Laplacian of has the following expression
Thus, for , the equation of the system (7) becomes in both cases
while for with , the equation is expressed as
Therefore, the system (7) becomes
Thus,
To solve this differential equation, we consider the change of variable . Then, (26) can be expressed as follows:
We solve this differential equation by applying separation of variables, and we get the following solution
for some constant C. This equation is equivalent to
for some constant . Thus, since . Therefore, since for all t, for the functions u and v we have three possibilities: , or . But , hence the only possibility is , that is, . (Here, we would like to note that since , the second differential equation of the system (25) reduces to , that is the differential Equation (24), which we have solved before.) □
Remark 1.
Note that proceeding in a similar way as Lauret did in [40] for the Ricci flow, we can evolve the Lie brackets instead of the 3-form defining the -structure, and we can show that the corresponding bracket flow has a solution for every t. In fact, if we fix on the 3-form , the basis defines, for every real number , a solvable Lie algebra with bracket such that is the Lie bracket of the Lie algebra k of K. Moreover, the solution of the bracket flow converges to the null bracket corresponding to the abelian Lie algebra as t goes to , and it blows-up as t goes to .
4.2. The Laplacian Flow on S
Now we consider the simply connected and solvable Lie group S whose Lie algebra s is defined as follows:
Then, the 3-form given by
defines a left invariant locally conformal calibrated -structure on the Lie group S, with Lee form , and so with torsion form . In fact,
Since S is a nonunimodular Lie group, S cannot admit a lattice such that the quotient space is a compact solvmanifold. In fact, the linear map , is such that is non-zero, where is the basis of s dual to the basis of .
Theorem 2.
The family of locally conformal calibrated -structures on S given by
is the solution for the Laplacian flow (5) of the form φ given by (28), where . The Lee form of is . Moreover, the underlying metrics of this solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric, uniformly on compact sets in S, as t goes to , and they blow-up as t goes to .
Proof.
To study the flow (5) of the form defined in (28), we should proceed as in Theorem 1. However, in order to short the proof, we will show directly that the one-parameter family of -structures given by (29) is the solution for the flow (5). For this, we consider the differentiable real functions given by
These functions are defined for all ; moreover, , for .
Now, for each , we consider the basis of left invariant 1-forms on S defined by
For any , we consider the 3-form on S given by
Then, this 3-form defines a -structure on S, and it is equal to the 3-form defined in (29). Note that the 3-form is such that and, for any t, determines the metric on S such that the basis of left invariant vector fields on S dual to is orthonormal. So, .
Moreover, for every , defines a locally conformal calibrated -structure on S. In fact,
since on the right-hand side of (29) the terms and are both closed and . So, the Lee form of is .
On the other hand, we have
and
To complete the proof, we study the behavior of the underlying metrics of such a solution in the limit for . If we think of the Laplacian flow as a one parameter family of manifolds with a locally conformal calibrated -structure, it can be checked that, in the limit, the resulting manifold has vanishing curvature. Denote by , , the metric on S induced by the form given by (29). Then, has the following expression
Now, one can check that every non-vanishing coefficient appearing in the expression of the Riemannian curvature of is proportional to . Therefore, .
Furthermore, the curvatures of blow-up as t goes to , and the finite-time singularity is of Type I since as ; in fact
□
Remark 3.
As we have noticed in Remark 1, we can also evolve the Lie brackets instead of the 3-form defining the left invariant -structure on S, and we can show that the corresponding bracket flow has a solution for every . In fact, if we fix on the 3-form , the basis defines, for every real number , a solvable Lie algebra with bracket such that is the Lie bracket of the Lie algebra s of S. As for the Lie group K (see Remark 1), the solution of the bracket flow converges to the null bracket corresponding to the abelian Lie algebra as t goes to , and it blows-up as t goes to .
Remark 4.
Thus, . However, the solution does not converge to a locally conformal parallel -structure as t goes to since, by (29), the forms blow-up when , and degenerate as t goes to . Note that the metrics behaves differently for S than for K. Indeed, the induced metrics by the solution of the Laplacian flow on S blow-up at infinity and at the finite time, while the induced metrics by the solution of the Laplacian flow on K only blow-up as t goes to .
Remark 5.
Note that, for every , the metric is an Einstein metric with negative scalar curvature on the Lie group S. In fact, with respect to the orthonormal basis , we have
Author Contributions
The three authors have contributed equally to the realization and writing of this article.
Funding
The first and third authors were partially supported by MINECO-FEDER Grant MTM2014-54804-P and Gobierno Vasco Grant IT1094-16, Spain. The second author was partially supported by the project MTM2017-85649-P (AEI/Feder, UE) and Gobierno de Aragón/Fondo Social Europeo—Grupo Consolidado E22-17R Algebra y Geometría.
Acknowledgments
We are grateful to the anonymous referees for useful comments and improvements. Moreover, we would like to thank Guest Editor.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Bryant, R.L. Metrics with exceptional holonomy. Ann. Math. 1987, 126, 525–576. [Google Scholar] [CrossRef]
- Fernández, M.; Gray, A. Riemannian manifolds with structure group G2. Ann. Mat. Pura Appl. 1982, 132, 19–45. [Google Scholar] [CrossRef]
- Bonan, E. Sur des variétés riemanniennes a groupe d’holonomie G2 ou Spin(7). C. R. Acad. Sci. Paris 1966, 262, 127–129. [Google Scholar]
- Salamon, S. Riemannian Geometry and Holonomy Groups; Longman Scientific and Technical: Harlow Essex, UK, 1989. [Google Scholar]
- Hitchin, N.J. The geometry of three-forms in six dimensions. J. Diff. Geom. 2000, 55, 547–576. [Google Scholar] [CrossRef]
- Hitchin, N.J. Stable forms and special metrics. In Global Differential Geometry: The Mathematical Legacy of Alfred Gray, Proceedings of the International Congress on Differential Geometry, Bilbao, Spain, 18–23 September 2000; Fernández, M., Wolf, J.A., Eds.; Contemporary Mathematics; American Mathematical Society: Providence, RI, USA, 2001; Volume 288, pp. 70–89. [Google Scholar]
- Joyce, D.D. Compact Riemannian 7-manifolds with holonomy G2. I. J. Differ. Geom. 1996, 43, 291–328. [Google Scholar] [CrossRef]
- Joyce, D.D. Compact Riemannian 7-manifolds with holonomy G2. II. J. Differ. Geom. 1996, 43, 329–375. [Google Scholar] [CrossRef]
- Kovalev, A. Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 2003, 565, 125–160. [Google Scholar] [CrossRef]
- Corti, A.; Haskins, M.; Nordström, J.; Pacini, T. G2-manifolds and associative submanifolds via semi-Fano 3-folds. Duke Math. J. 2015, 164, 1971–2092. [Google Scholar] [CrossRef]
- Joyce, D.D.; Karigiannis, S. A new construction of compact torsion-free G2-manifolds by gluing families of Eguchi-Hanson spaces. arXiv, 2017; arXiv:1707.09325. [Google Scholar]
- Chiossi, S.; Fino, A. Conformally parallel G2 structures on a class of solvmanifolds. Math. Z. 2006, 252, 825–848. [Google Scholar] [CrossRef]
- Ivanov, S.; Parton, M.; Piccinni, P. Locally conformal parallel G2 and Spin(7) manifolds. Math. Res. Lett. 2006, 13, 167–177. [Google Scholar] [CrossRef]
- Harvey, R.; Lawson, H.B., Jr. Calibrated geometries. Acta Math. 1982, 148, 47–157. [Google Scholar] [CrossRef]
- Banyaga, A. On the geometry of locally conformal symplectic manifolds. In Infinite Dimensional Lie Groups in Geometry and Representation Theory, Proceedings of the 2000 Howard Conference, Washington, DC, USA, 17–21 August 2000; World Scientific Publishing: River Edge, NJ, USA, 2002; pp. 79–91. [Google Scholar]
- Bazzoni, G. Locally conformally symplectic and Kähler geometry. arXiv, 2017; arXiv:1711.02440. [Google Scholar]
- Bazzoni, G.; Marrero, J.C. On locally conformal symplectic manifolds of the first kind. Bull. Sci. Math. 2018, 143, 1–57. [Google Scholar] [CrossRef][Green Version]
- Dragomir, S.; Ornea, L. Locally Conformal Kähler Geometry; Progress in Mathematics; Birkhäuser: Boston, MA, USA, 1998; Volume 155, p. xiv+327. [Google Scholar]
- Eliashberg, Y.; Murphy, E. Making cobordisms symplectic. arXiv, 2015; arXiv:1504.06312. [Google Scholar]
- Ornea, L.; Verbitsky, M. A report on locally conformally Kähler manifolds. Contemp. Math. 2011, 542, 135–150. [Google Scholar]
- Vaisman, I. Locally conformal symplectic manifolds. Int. J. Math. Math. Sci. 1985, 8, 521–536. [Google Scholar] [CrossRef]
- Bazzoni, G.; Raffero, A. Special types of locally conformal closed G2-structures. Axioms 2018, 7, 90. [Google Scholar] [CrossRef]
- Fernández, M.; Fino, A.; Raffero, A. Locally conformal calibrated G2-manifolds. Annali Matematica Pura Applicata 2016, 195, 1721–1736. [Google Scholar] [CrossRef]
- Fernández, M.; Ugarte, L. A differential complex for locally conformal calibrated G2 manifolds. Ill. J. Math. 2000, 44, 363–390. [Google Scholar]
- Fino, A.; Raffero, A. Einstein locally conformal calibrated G2-structures. Math. Z. 2015, 280, 1093–1106. [Google Scholar] [CrossRef]
- Cleyton, R.; Ivanov, S. On the geometry of closed G2-structures. Commun. Math. Phys. 2007, 270, 53–67. [Google Scholar] [CrossRef]
- Bryant, R.L. Some remarks on G2-structures. In Proceedings of the Gökova Geometry-Topology Conference, Gökova, Turkey, 30 May–3 June 2005; International Press of Boston: Somerville, MA, USA, 2006; pp. 75–109. [Google Scholar]
- Hamilton, R.S. Three-manifolds with positive Ricci curvature. J. Diff. Geom. 1982, 17, 255–306. [Google Scholar] [CrossRef]
- Bryant, R.L.; Xu, F. Laplacian flow for closed G2-structures: Short time behavior. arXiv, 2011; arXiv:1101.2004v1. [Google Scholar]
- Lotay, J.D.; Wei, Y. Laplacian flow for closed G2-structures: Shi-type estimates, uniqueness and compactness. Geom. Funct. Anal. 2017, 27, 165–233. [Google Scholar] [CrossRef]
- Lotay, J.D.; Wei, Y. Stability of torsion free G2-structures along the Laplacian flow. arXiv, 2015; arXiv:1504.07771. [Google Scholar]
- Lotay, J.D.; Wei, Y. Laplacian flow for closed G2 structures: Real analyticity. arXiv, 2015; arXiv:1601.04258. [Google Scholar]
- Fernández, M.; Fino, A.; Manero, V. Laplacian flow of closed G2-structures inducing nilsolitons. J. Geom. Anal. 2016, 26, 1808–1837. [Google Scholar] [CrossRef]
- Karigiannis, S.; McKay, B.; Tsui, M.P. Soliton solutions for the Laplacian coflow of some G2-structures with symmetry. Diff. Geom. Appl. 2012, 30, 318–333. [Google Scholar] [CrossRef]
- Grigorian, S. Short-time behavior of a modified Laplacian coflow of G2-structures. Adv. Math. 2013, 248, 378–415. [Google Scholar] [CrossRef]
- Grigorian, S. Flows of co-closed G2-structures. arXiv, 2011; arXiv:1811.10505. [Google Scholar]
- Bagaglini, L.; Fernández, M.; Fino, A. Laplacian coflow on the 7-dimensional Heisenberg group. arXiv, 2017; arXiv:1704.00295. [Google Scholar]
- Bagaglini, L.; Fino, A. The laplacian coflow on almost-abelian Lie groups. Ann. Mat. Pura Appl. 2018, 197, 1855–1873. [Google Scholar] [CrossRef]
- Manero, V.; Otal, A.; Villacampa, R. Solutions of the Laplacian flow and coflow of a locally conformal parallel G2-structure. arXiv, 2017; arXiv:1711.08644v1. [Google Scholar]
- Lauret, J. The Ricci flow for simply connected nilmanifolds. Commun. Anal. Geom. 2011, 19, 831–854. [Google Scholar] [CrossRef][Green Version]
Table 1.
Some classes of -structures.
Class | Type | Conditions |
---|---|---|
parallel | ||
closed, calibrated | ||
locally conformal parallel | ||
locally conformal calibrated |
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