On Lorentzian Ricci solitons on nilpotent Lie groups

The aim of this article is to exhibit the variety of different Ricci soliton structures that a nilpotent Lie group can support when one allows for the metric tensor to be Lorentzian. In stark contrast to the Riemannian case, we show that a nilpotent Lie group can support a number of non‐isometric Lorentzian Ricci soliton structures with decidedly different qualitative behaviors and that Lorentzian Ricci solitons need not be algebraic Ricci solitons. The analysis is carried out by classifying all left invariant Lorentzian metrics on the connected, simply‐connected five‐dimensional Lie group having a Lie algebra with basis vectors and and non‐trivial bracket relations and , investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.     

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.     

Expanding solitons to the Hermitian curvature flow on complex Lie groups

We investigate the algebraic structure of complex Lie groups equipped with left-invariant metrics which are expanding semi-algebraic solitons to the Hermitian curvature flow (HCF). We show that the Lie algebras of such Lie groups decompose in the semidirect product of a reductive Lie subalgebra with their nilradicals. Furthermore, we give a structural result concerning expanding semi-algebraic solitons on complex Lie groups. It turns out that the restriction of the soliton metric to the nilradical is also an expanding algebraic soliton and we explain how to construct expanding solitons on complex Lie groups starting from expanding solitons on their nilradicals.     

Solvable extensions of negative Ricci curvature of filiform Lie groups

We give necessary and sufficient conditions of the existence of a left‐invariant metric of strictly negative Ricci curvature on a solvable Lie group the nilradical of whose Lie algebra is a filiform Lie algebra . It turns out that such a metric always exists, except for in the two cases, when is one of the algebras of rank two, or , and is a one‐dimensional extension of , in which cases the conditions are given in terms of certain linear inequalities for the eigenvalues of the extension derivation.     

Four-dimensional Lorentzian Lie groups

We describe four-dimensional Lie groups equipped with a left-invariant Lorentzian metric, obtaining a complete classification of the Einstein and Ricci-parallel examples.     

Ricci solitons on compact Kähler surfaces

We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming that the curvature is slightly more positive than that of the single known example of a soliton in this dimension.

     

Canonical Connections and Algebraic Ricci Solitons of Three-dimensional Lorentzian Lie Groups

In this paper, the author computes canonical connections and KobayashiNomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.     

On the Ricci curvature of steady gradient Ricci solitons

Assume (Mn,g) is a complete steady gradient Ricci soliton with positive Ricci curvature. If the scalar curvature approaches 0 towards infinity, we prove that , where O is the point where R obtains its maximum and γ(s) is a minimal normal geodesic emanating from O. Some other results on the Ricci curvature are also obtained.     

On Ricci negative solvmanifolds and their nilradicals

In the homogeneous case, the only curvature behavior which is still far from being understood is Ricci negative. In this paper, we study which nilpotent Lie algebras admit a Ricci negative solvable extension. Different unexpected behaviors were found. On the other hand, given a nilpotent Lie algebra, we consider the space of all the derivations such that the corresponding solvable extension has a metric with negative Ricci curvature. Using the nice convexity properties of the moment map for the variety of nilpotent Lie algebras, we obtain a useful characterization of such derivations and some applications.     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

Almost Kähler 4-dimensional Lie groups with J-invariant Ricci tensor

The J-invariance of the Ricci tensor is a natural weakening of the Einstein condition in almost Hermitian geometry. The aim of this paper is to determine left-invariant strictly almost Kähler structures (g,J,Ω) on real 4-dimensional Lie groups such that the Ricci tensor is J-invariant. We prove that all these Lie groups are isometric (up to homothety) to the (unique) 4-dimensional proper 3-symmetric space.     

Hilbert-Schmidt groups as infinite-dimensional Lie groups and their Riemannian geometry

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvature is -∞.     

The Ricci flow in a class of solvmanifolds

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 ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

Group-invariant solutions for the Ricci curvature equation and the Einstein equation

We consider the pseudo-Euclidean space $({\mathbb{R}}^n,g)$, $n\ge 3$, with coordinates $x=(x_1,\dots ,x_n)$ and metric $g_{ij}={\delta }_{ij}{?}_i$, ${?}_i=\pm 1$, where at least one ${?}_i$ is positive, and also tensors of the form $A={\sum }_{i,j}{A}_{ij}d{x}_{i}d{x}_j$, such that $A_{ij}$ are differentiable functions of x. For such tensors, we use Lie point symmetries to find metrics $\stackrel{￣}{g}=\frac{1}{u^2}g$ that solve the Ricci curvature and the Einstein equations. We provide a large class of group-invariant solutions and examples of complete metrics $\stackrel{￣}{g}$ defined globally in ${\mathbb{R}}^n$. As consequences, for certain functions $\stackrel{￣}{K}$, we show complete metrics $\stackrel{￣}{g}$, conformal to the pseudo-Euclidean metric g, whose scalar curvature is $\stackrel{￣}{K}$.     

Pseudo anti‐commuting Ricci tensor and Ricci soliton real hypersurfaces in complex hyperbolic two‐plane Grassmannians

In this paper, first we introduce a new notion of pseudo anti commuting Ricci tensor for real hypersurfaces in complex hyperbolic two‐plane Grassmannians and prove a complete classification theorem that such a hypersurface must be a tube over a totally real totally geodesic , , a horosphere whose center at the infinity is singular or an exceptional case.     

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota’s argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.     

Singular Ricci Solitons and Their Stability under the Ricci Flow

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3, and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sharp contrast to the evolution of their smooth counterparts. In particular, the family of diffeomorphisms associated with the Ricci flow “pushes away” from the singularity causing the evolving soliton to open up immediately becoming an incomplete (but non-singular) metric. The main objective of this paper is to study the local-in time stability of this dynamical evolution, under spherically symmetric perturbations of the singular initial metric. We prove a local well-posedness result for the Ricci flow in suitably weighted Sobolev spaces, which in particular implies that the “opening up” of the singularity persists for the perturbations as well.     

A compactness result for Kähler Ricci solitons

In this paper we prove a compactness result for compact Kähler Ricci gradient shrinking solitons. If (Mi,gi) is a sequence of Kähler Ricci solitons of real dimension n?4, whose curvatures have uniformly bounded Ln/2 norms, whose Ricci curvatures are uniformly bounded from below and μ(gi,1/2)?A (where μ is Perelman's functional), there is a subsequence (Mi,gi) converging to a compact orbifold (M_∞,g_∞) with finitely many isolated singularities, where g_∞ is a Kähler Ricci soliton metric in an orbifold sense (satisfies a soliton equation away from singular points and smoothly extends in some gauge to a metric satisfying Kähler Ricci soliton equation in a lifting around singular points).     

Translating solitons in Riemannian products

In this paper we study solitons invariant with respect to the flow generated by a complete parallel vector field in a ambient Riemannian manifold. A special case occurs when the ambient manifold is the Riemannian product $(\mathbb{R}\times P,\mathrm{d}{t}^2+g_0)$ and the parallel field is $X={?}_t$. Similarly to what happens in the Euclidean setting, we call them translating solitons. We see that a translating soliton in $\mathbb{R}\times P$ can be seen as a minimal submanifold for a weighted volume functional. Moreover we show that this kind of solitons appear in a natural way in the context of a monotonicity formula for the mean curvature flow in $\mathbb{R}\times P$. When $g_0$ is rotationally invariant and its sectional curvature is non-positive, we are able to characterize all the rotationally invariant translating solitons. Furthermore, we use these families of new examples as barriers to deduce several non-existence results.     

A note on the geometry of Lie groups

The degenerate semi-Riemannian geometry of a Lie group is studied. Then a naturally reductive homogeneous semi-Riemannian space is obtained from the Lie group in a natural way.