Related papers: Laplacian flow of closed $G_2$-structures inducing…

We investigate the existence of closed $G_2$-structures which are solitons for the Laplacian flow on nilpotent Lie groups. We obtain that seven of the twelve Lie algebras admitting a closed $G_2$-structure do admit a Laplacian soliton.…

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the…

We use the bracket flow/algebraic soliton approach to study the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case. We prove that any homogeneous Laplacian soliton is equivalent to a semi-algebraic soliton (i.e.\ a…

Let $\varphi(t), t\in [0,T]$ be a smooth solution to the Laplacian flow for closed G_2 structures on a compact 7-manifold $M$. We show that for each fixed positive time $t\in (0,T]$, $(M,\varphi(t),g(t))$ is real analytic, where $g(t)$ is…

In this work, we approach the Laplacian coflow of a coclosed $G_2$-structure $\varphi$ using the formulae for the irreducible $G_2$-decomposition of the Hodge Laplacian and the Lie derivative of the Hodge dual $4$-form of $\varphi$. In…

We develop a general approach to study geometric flows on homogeneous spaces. Our main tool will be a dynamical system defined on the variety of Lie algebras called the bracket flow, which coincides with the original geometric flow after a…

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.

Let (N,g) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their `almost' versions). We define a left invariant Riemannian metric on N compatible with g to be minimal,…

In this paper, it is shown that (with no additional assumptions) on a compact 7-dimensional manifold which admits a $G_2$-structure soliton solutions to the Laplacian flow of R. Bryant can only be shrinking or steady. We also show that the…

We find explicit solutions of the Laplacian coflow of $G_2-$structures on seven-dimensional almost-abelian Lie groups. Moreover, we construct new examples of solitons for the Laplacian coflow which are not eigenforms of the Laplacian and we…

We provide the second known example of an extremally Ricci pinched closed G2-structure on a compact 7-manifold, by finding a lattice in the only unimodular solvable Lie group admitting a left-invariant G2-structure. Furthermore, the…

Let N be a nilpotent Lie group and let S be an invariant geometric structure on N (cf. symplectic, complex or hypercomplex). We define a left invariant Riemannian metric on N compatible with S to be "minimal", if it minimizes the norm of…

We review results about $G_2$-structures in relation to the existence of special metrics, such as Einstein metrics and Ricci solitons, and the evolution under the Laplacian flow on non-compact homogeneous spaces. We also discuss some…

We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well…

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The…

We give the first examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity. Extremally Ricci pinched G2-structures (introduced by Bryant) which are…

We prove short time existence and uniqueness of solutions to the Laplacian flow for closed $G_2$ structures on a compact manifold $M^7$. The result was claimed in \cite{BryantG2}, but its proof has never appeared.

We apply the general Ansatz in geometric flows on homogeneous spaces proposed by Jorge Lauret for the Laplacian co-flow of invariant $G_2$-structures on a Lie group, finding an explicit soliton on a particular almost Abelian $7$-manifold.

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the omega-limit of bracket flow solutions…

We consider the Laplacian "co-flow" of $G_2$-structures: $\frac{d}{dt} \psi = - \Delta_d \psi$ where $\psi$ is the dual 4-form of a $G_2$-structure $\phi$ and $\Delta_d$ is the Hodge Laplacian on forms. This flow preserves the condition of…