The fixed points of the generalized Ricci flow are the Bismut Ricci flat metrics, i.e., a generalized metric $(g,H)$ on a manifold $M$, where $g$ is a Riemannian metric and $H$ a closed $3$-form, such that $H$ is $g$-harmonic and $\operatorname{Rc}(g)=\tfrac{1}{4} H_g^2$. Given two standard Einstein homogeneous spaces $G_i/K$, where each $G_i$ is a compact simple Lie group and $K$ is a closed subgroup of them holding some extra assumption, we consider $M = G_1 \times G_2 / ΔK$. Recently, Lauret and Will proved the existence of a Bismut Ricci flat metric on any of these spaces. We proved that this metric is always asymptotically stable for the generalized Ricci flow on $M$ among a subset of $G$-invariant metrics and, if $G_1 = G_2$, then it is globally stable.
We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.
We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.
We consider four-dimensional homogeneous pseudo-Riemannian manifolds with non-trivial isotropy and completely classify the cases giving rise to non-trivial homogeneous Ricci solitons. In particular, we show the existence of non-compact homogeneous (and also invariant) pseudo-Riemannian Ricci solitons which are not isometric to solvmanifolds, and of conformally flat homogeneous pseudo-Riemannian Ricci solitons which are not symmetric.
We show that for an immortal homogeneous Ricci flow solution any sequence of parabolic blow-downs subconverges to a homogeneous expanding Ricci soliton. This is established by constructing a new Lyapunov function based on curvature estimates which come from real geometric invariant theory.
We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat homogeneous Ricci solitons.
We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of the identity plus a derivation) are precisely the fixed points of the system, and that a homogeneous Ricci soliton is isometric to an algebraic soliton if and only if the corresponding bracket flow solution is not chaotic, in the sense that its omega-limit set consists of a sing…
For Fano homogeneous toric bundles, we obtain a formula of the greatest lower bound on Ricci curvature. We also give a criteria for the ampleness of a kind of line bundles over general homogeneous toric bundles.
In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be normal. Using our previous work on warped product Einstein metrics, we show that every normal semi-algebraic Ricci soliton also admits a $k$-dimensional Einstein extension for any $k\geq 2$. We also prove converse theorems for these constructions and some geometric and topological structure results for homogeneous warped product Einstein metrics. In the appe…
In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow. We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.
A Finsler space is called Ricci-quadratic if its Ricci curvature $Ric(x,y)$ is quadratic in $y$. It is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold $M$. In this article, we prove that a homogeneous Randers space is Ricci-quadratic if and only if it is of Berwald type.
It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or a product $\mathbb{R}\times N(c)$, where $N(c)$ is a space of constant curvature.
In this short note, we show that homogeneous Ricci solitons are algebraic. As an application, we see that the generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture.
We consider the Ricci flow equation for invariant metrics on compact and connected homogeneous spaces whose isotropy representation decomposes into two irreducible inequivalent summands. By studying the corresponding dynamical system, we completely describe the behaviour of the homogeneous Ricci flow on this kind of spaces. Moreover, we investigate the existence of ancient solutions and relate this to the existence and non-existence of invariant Einstein metrics.
This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general, Ricci Flow is a nonlinear PDE whose solutions are rather difficult to calculate; however, in a homogeneous manifold, the Ricci Flow reduces to an ODE. The behavior of Ricci Flow in two, three, and four dimensional homogenous manifolds has been calculated and is well understood. The work presented here will describe efforts to better understand the behavior of R…
We discuss the geometry of homogeneous Ricci solitons. After showing the nonexistence of compact homogeneous and noncompact steady homogeneous solitons, we concentrate on the study of left invariant Ricci solitons. We show that, in the unimodular case, the Ricci soliton equation does not admit solutions in the set of left invariant vector fields. We prove that a left invariant soliton of gradient type must be a Riemannian product with a nontrivial Euclidean de Rham factor. As an application of our results we prove that any generalized metric Heisenberg Lie group is a nongradient left invariant…
It is proved that an homogeneous toric bundles over a flag manifold G^\C/P admits a Kaehler-Ricci solitonic metric if and only if it is Fano. In particular, an homogeneous toric bundle of this kind is Kaehler-Einstein if and only if it is Fano and its Futaki invariant vanishes identically.
A Lie hypersurface in the complex hyperbolic space is an orbit of a cohomogeneity one action without singular orbit. In this paper, we classify Ricci soliton Lie hypersurfaces in the complex hyperbolic spaces.
In this paper we study backward Ricci flow of locally homogeneous geometries of $4$-manifolds which admit compact quotients. We describe the long-term behavior of each class and show that many of the classes exhibit the same behavior near the singular time. In most cases, these manifolds converge to a sub-Riemannian geometry after suitable rescaling.
We discuss the Ricci flow on homogeneous 4-manifolds. After classifying these manifolds, we note that there are families of initial metrics such that we can diagonalize them and the Ricci flow preserves the diagonalization. We analyze the long time behavior of these families. We find that if a solution exists for all time, then the flow exhibits a type III singularity in the sense of Hamilton.
In this article we classify expanding homogeneous Ricci solitons up to dimension 5, according to their presentation as homogeneous spaces. We obtain that they are all isometric to solvsolitons, and this in particular implies that the generalized Alekseevskii conjecture holds in these dimensions. In addition, we prove that the conjecture holds in dimension 6 provided the transitive group is not semisimple.
The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In this paper, we consider homogeneous Finsler spaces with square metric and Randers change of square metric. First, we derive the explicit formulae for Ricci curvature of homogeneous Finsler spaces with these metrics. Next, we find a necessary and sufficient condition under which a homogeneous Finsler space with either of these metrics is of vanishing $S$-curvature. The formulae for Ricci curvature of homogeneous Finsler spaces with square metric an…
We study the quasi-convergence equivalence of some families of metrics on locally homogeneous closed 4-manifolds with trivial isotropy group, and identify the dimension of each equivalence class under certain conditions.
In this paper, we study the backward Ricci flow on locally homogeneous 3-manifolds. We describe the long time behavior and show that, typically and after a proper re-scaling, there is convergence to a sub-Riemannian geometry. A similar behavior was observed by the authors in the case of the cross curvature flow.
In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to $\mathbb R^n$. In the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of …
The Ricci iteration is a discrete analogue of the Ricci flow. We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not Kähler. The Ricci iteration in the non-Kähler setting exhibits new phenomena. Among them is the existence of so-called ancient Ricci iterations. As we show, these are closely related to ancient Ricci flows and provide the first nontrivial examples of Riemannian metrics to which the Ricci operator can be applied infinitely many times. In some of the cases we study, these ancient Ricci iterations emerge (in the Gromov--Hausdorff topology) fr…
In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.
Let $λ(t)$ be the first eigenvalue of $-Δ+aR\, (a>0)$ under the backward Ricci flow on locally homogeneous 3-manifolds, where $R$ is the scalar curvature. In the Bianchi case, we get the upper and lower bounds of $λ(t)$. In particular, we show that when the the backward Ricci flow converges to a sub-Riemannian geometry after a proper re-scaling, $λ^{+}(t)$ approaches zero, where $λ^{+}(t)=\max\{λ(t),0\}$.
We study the behavior of the normalized Ricci flow of invariant Riemannian homogeneous metrics at infinity for generalized Wallach spaces, generalized flag manifolds with four isotropy summands and second Betti number equal to one, and the Stiefel manifolds $V_2\mathbb{R}^n$ and $V_{1+k_2}\mathbb{R}^n$, with $n = 1+k_2+k_3$. We use techniques from the theory of differential equations, in particular the Poincaré compactification.
The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis.
We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov-Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.
The prescribed Ricci curvature problem in the context of G-invariant metrics on a homogeneous space M=G/K is studied. We focus on the metrics at which the Ricci curvature map is, locally, as injective and surjective as it can be. Our main result is that such property is generic in the compact case. Our main tool is a formula for the Lichnerowicz Laplacian we prove in terms of the moment map for the variety of algebras.
We construct new homogeneous Einstein spaces with negative Ricci curvature in two ways: First, we give a method for classifying and constructing a class of rank one Einstein solvmanifolds whose derived algebras are two-step nilpotent. As an application, we describe an explicit continuous family of ten-dimensional Einstein manifolds with a two-dimensional parameter space, including a continuous subfamily of manifolds with negative sectional curvature. Secondly, we obtain new examples of non-symmetric Einstein solvmanifolds by modifying the algebraic structure of non-compact irreducible symmetri…
Consider a compact Lie group $G$ and a closed Lie subgroup $H<G$. Let $\mathcal M$ be the set of $G$-invariant Riemannian metrics on the homogeneous space $M=G/H$. By studying variational properties of the scalar curvature functional on $\mathcal M$, we obtain an existence theorem for solutions to the prescribed Ricci curvature problem on $M$. To illustrate the applicability of this result, we explore cases where $M$ is a generalised Wallach space and a generalised flag manifold.
We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension $n$. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space. The a…
We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci curvature, i.e. $\mathrm{Ric}_2>0$, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and Hsiang-Kleiner theorem for positively curved manifolds do not hold in the $\mathrm{Ric}_2>0$ setting. These observations indicate that the class of manifolds with $\mathrm{Ric}_2>0$ is vastly different from the class of positively curved manifolds.
In this paper we investigate the behavior of three-dimensional homogeneous solutions of the cross curvature flow using Riemannian groupoids. The Riemannian groupoid technique, introduced by John Lott, allows us to investigate the long term behavior of collapsing solutions of the flow, producing soliton solutions in the limit. We also review Lott's results on the long term behavior of three-dimensional homogeneous solutions of Ricci flow, demonstrating the coordinates we choose and reviewing the groupoid technique. We find cross curvature soliton metrics on Sol and Nil, and show that the cross …
We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all simply connected homogeneous spaces of dimension n with a q-dimensional isotropy, which is proved to be equivalent in a precise sense to the Ricci flow. The approach is useful to better visualize the possible (nonflat) pointed limits of Ricci flow solutions, under diverse rescalings, as well as to determine the type of the possible singularities. Ancient solu…
This is the first of a sequence of two papers. Here, a simple algebraic characterization of the Fano manifolds in the class of homogeneous toric bundles over a flag manifold G^C/P is provided in terms of symplectic data. The result of this paper is used in the second paper, where it is proved that an homogeneous toric bundle over a flag manifold admits a Kaehler-Ricci solitonic metric if and only if it is Fano.
We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons and nilsolitons. We obtain that any homogeneous expanding Ricci soliton M=G/K is diffeomorphic to a product U/K x N, where U is a maximal reductive Lie subgroup of G and N is the maximal nilpotent normal subgroup of G, such tha…
We study the Ricci flow of the four-parameter family of Sp(n+1)-invariant metrics on spheres. We determine their forward behaviour and also classify ancient solutions. In doing so, we exhibit a new one-parameter family of ancient solutions on spheres. These (non-isometric) ancient solutions all have a larger isometry group, namely Sp(n+1)Sp(1), Sp(n+1)U(1), or U(2n+2). Two ancient solutions are non-collapsed and converge, under the backwards flow, to Jensen's second Einstein metric. One solution parametrizes the well known Berger metrics. The rest are new and collapse, under a rescaling of the…
Dipartimento di Matematica “Giuseppe Peano”, Università di Torino（都灵大学）；School of Mathematics and Physics, The University of Queensland（昆士兰大学）；School of Mathematics and Physics, University of Wollongong（伍伦贡大学）
We introduce a combinatorial method to construct indefinite Ricci-flat metrics on nice nilpotent Lie groups. We prove that every nilpotent Lie group of dimension $\leq6$, every nice nilpotent Lie group of dimension $\leq7$ and every two-step nilpotent Lie group attached to a graph admits such a metric. We construct infinite families of Ricci-flat nilmanifolds associated to parabolic nilradicals in the simple Lie groups ${\rm SL}(n)$, ${\rm SO}(p,q)$, ${\rm Sp}(n,\mathbb R)$. Most of these metrics are shown not to be flat.
This article grew out of the urge to realize explicit examples of solutions for the Ricci flow as families of isometrically embedded submanifolds, together with its Gromov-Hausdorff collapses. To this aim, we consider the Ricci flow of invariant metrics in a class of flag manifolds. On the one hand, we contrast with a previous result in literature by presenting entire flow lines of invariant metrics realized as orbits of a fixed representation. Indeed, we prove that the subset of realizable metrics has a global attractor with open interior, containing an expressive family of complete flow line…
