We study the Anomaly flow on $2$-step nilmanifolds with respect to any Hermitian connection in the Gauduchon line. In the case of flat holomorphic bundle, the general solution to the Anomaly flow is given for any initial invariant Hermitian metric. The solutions depend on two constants $K_1$ and $K_2$, and we study the qualitative behaviour of the Anomaly flow in terms of their signs, as well as the convergence in Gromov-Hausdorff topology. The sign of $K_1$ is related to the conformal invariant introduced by Fu, Wang and Wu. In the non-flat case, we find the general evolution equations of the Anomaly flow under certain initial assumptions. This allows us to detect non-flat solutions to the Hull-Strominger-Ivanov system on a concrete nilmanifold, which appear as stationary points of the Anomaly flow with respect to the Bismut connection.
In this work we study the geodesic flow on nilmanifolds associated to graphs. We are interested in the construction of first integrals to show complete integrability on some compact quotients. Also examples of integrable geodesic flows and of non-integrable ones are shown.
Two Riemannian manifolds are said to have $C^k$-conjugate geodesic flows if there exist an $C^k$ diffeomorphism between their unit tangent bundles which intertwines the geodesic flows. We obtain a number of rigidity results for the geodesic flows on compact 2-step Riemannian nilmanifolds: For generic 2-step nilmanifolds the geodesic flow is $C^2$ rigid. For special classes of 2-step nilmanifolds, we show that the geodesic flow is $C^0$ or $C^2$ rigid. In particular, there exist continuous families of 2-step nilmanifolds whose Laplacians are isospectral but whose geodesic flows are not $C^0$ co…
The Bach flow is a fourth order geometric flow defined on four manifolds. For a compact manifold, it is a conformally modified gradient flow for the $L^2$-norm of the Weyl curvature. In this paper we study the Bach flow on four-dimensional simply connected nilmanifolds whose Lie algebra is indecomposable. We show that the Bach flow beginning at an arbitrary left invariant metric exists for all positive times and after rescaling converges in the pointed Cheeger-Gromov sense to an expanding Bach soliton which is non-gradient. Combining our results with previous results of Helliwell gives a compl…
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 as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.'s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow.
We consider magnetic flows on 2-step nilmanifolds $M = Γ\backslash G$, where the Riemannian metric $g$ and the magnetic field $σ$ are left-invariant. Our first result is that when $σ$ represents a rational cohomology class and its restriction to $\mathfrak{g} = T_eG$ vanishes on the derived algebra, then the associated magnetic flow has zero topological entropy. In particular, this is the case when $σ$ represents a rational cohomology class and is exact. Our second result is the construction of a magnetic field on a 2-step nilmanifold that has positive topological entropy for arbitrarily high …
We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving tamed symplectic forms on a complex man…
In this paper we study the geodesic flow on nilmanifolds equipped with a left-invariant metric. We write the underlying definitions and find general formulas for the Poisson involution. As an example we develop the Heisenberg Lie group equipped with its canonical metric. We prove that a family of first integrals giving the complete integrability can be read off at the Lie algebra of the isometry group. We also explain the complete integrability on compact quotients and for any invariant metric.
Let T be the nilpotent group of 4 x 4 real upper triangular matrices. In this note we show that the Euler equations of certain left-invariant riemannian metrics on T have a horseshoe. We also show, with the aid of a numerical computation of a Melnikov-type integral, that the Euler equations of the sub-riemannian Carnot metric on T has a horseshoe. This sharpens an earlier result of Montgomery, Shapiro and Stolin who had shown that the equations are algebraically non-integrable.
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 negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling |R(g(t))|g(t) converges smoothly to a Ricci soliton, uniformly on compact sets. The Ricci soliton limit is also invariant by some transitive nilpotent Lie group,…
The aim of this paper is to study self-similar solutions to the symplectic cuvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention in the family of symplectic Two- and Three-step nilpotent Lie algebras admitting a "minimal compatible metric" and we give a complete classification of these algebras together with their respective metric. Such classification is given by using our generalization of Nikolayevsky's nice basis criterium (arXiv:1301.4949), which will be repeated here in the context of canonical compatible metrics for geometric structures on nilmanifolds, f…
Magnetic Killing tensors and first integrals of the magnetic flow
Université Paris-Saclay（巴黎萨克雷大学）；Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西蒙·斯托伊洛数学研究所）；Universidad Nacional de Rosario（罗萨里奥国立大学）
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, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the `invariant Ricci' flow and are characterized as the critical points of a natural variational problem. The uniqueness…
In this paper, we study the dual Anomaly flow, which is a dual version of the Anomaly flow under T-duality. A family of monotone functionals is introduced and used to estimate the dilaton function along the flow. Many examples and reductions of the dual Anomaly flow are worked out in detail.
We study the Hull-Strominger system and the Anomaly flow on a special class of 2-step solvmanifolds, namely the class of almost-abelian Lie groups. In this setting, we characterize the existence of invariant solutions to the Hull-Strominger system with respect to the family of Gauduchon connections in the anomaly cancellation equation. Then, motivated by the results on the Anomaly flow, we investigate the flow of invariant metrics in our setting, proving that it always reduces to a flow of a special form. Finally, under an extra assumption on the initial metrics, we show that the flow is immor…
A new formulation of the Anomaly flow in the case of vanishing slope parameter is given, where the dependence on the global section of the canonical bundle appears only in the initial data. This allows a natural unification of the Anomaly flow with the Kähler-Ricci flow.
The Hull-Strominger system for supersymmetric vacua of the heterotic string allows general unitary Hermitian connections with torsion and not just the Chern unitary connection. Solutions on unimodular Lie groups exploiting this flexibility were found by T. Fei and S.T. Yau. The Anomaly flow is a flow whose stationary points are precisely the solutions of the Hull-Strominger system. Here we examine its long-time behavior on unimodular Lie groups with general unitary Hermitian connections. We find a diverse and intricate behavior, which depends very much on the Lie group and the initial data.
We initiate the study of a new nonlinear parabolic equation on a Riemann surface. The evolution equation arises as a reduction of the Anomaly flow on a fibration. We obtain a criterion for long-time existence for this flow, and give a range of initial data where a singularity forms in finite time, as well as a range of initial data where the solution exists for all time. A geometric interpretation of these results is given in terms of the Anomaly flow on a Calabi-Yau threefold.
The Anomaly flow is a flow which implements the Green-Schwarz anomaly cancellation mechanism originating from superstring theory, while preserving the conformally balanced condition of Hermitian metrics. There are several versions of the flow, depending on whether the gauge field also varies, or is assumed known. A distinctive feature of Anomaly flows is that, in $m$ dimensions, the flow of the Hermitian metric has to be inferred from the flow of its $(m-1)$-th power $ω^{m-1}$. We show how this can be done explicitly, and we work out the corresponding flows for the torsion and the curvature te…
The Anomaly flow is shown to converge on toric fibrations with the Fu-Yau ansatz, for both positive and negative values of the slope parameter $α'$. This implies both results of Fu and Yau on the existence of solutions for Hull-Strominger systems, which they proved using different methods depending on the sign of $α'$. It is also the first case where the Anomaly flow can even be shown to exist for all time. This is in itself remarkable from the point of view of the theory of fully nonlinear partial differential equations, as the elliptic terms in the flow are not concave.
We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety of 2n-dimensional Lie algebras, called the bracket flow, whose solutions differ from those to the original curvature flow by only pull-back by time-dependent diffeomorphisms. The approach, which has already been used to study the Ricci flow on homogeneous manifolds, is useful to …
We prove that the parabolic flow of conformally balanced metrics introduced by Phong, Picard and Zhang in "A flow of conformally balanced metrics with Kähler fixed points", is stable around Calabi-Yau metrics. The result shows that the flow can converge on a Kähler manifold even if the initial metric is not conformally Kähler.
