Jorge Lauret
The purpose of the present expository paper is to give an account of the recent progress and present status of the classification of solvable Lie groups admitting an Einstein left invariant Riemannian metric, the only known examples so far of noncompact Einstein homogeneous manifolds. The problem turns to be equivalent to the classification of Ricci soliton left invariant metrics on nilpotent Lie groups.
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A nilsoliton is a nilpotent Lie algebra $\mathfrak{g}$ with a metric such that $\operatorname{Ric}=λ\operatorname{Id}+D$, with $D$ a derivation. For indefinite metrics, this determines four different geometries, according to whether $λ$ and $D$ are zero or not. We illustrate with examples the greater flexibility of the indefinite case compared to the Riemannian setting. We determine the algebraic properties that $D$ must satisfy when it is nonzero. For each of the four geometries, we show that under suitable assumptions it is possible to extend the nilsoliton metric to an Einstein solvmanifold…
A Riemannian manifold (M,g) is said to be Einstein if its Ricci tensor satisfies ric(g) = cg, for some real number c. In the homogeneous case, a problem that is still open is the so called Alekseevskii Conjecture. This conjecture says that any homogeneous Einstein space with negative scalar curvature (i.e. c < 0) is a solvmanifold: a simply connected solvable Lie group endowed with a left invariant Riemannian metric. The aim of this paper is to classify Einstein solvmanifolds of dimension 9 whose nilradicals are 7-step nilpotent Lie algebras of dimension 8.
All known examples of nontrivial homogeneous Ricci solitons are left-invariant metrics on simply connected solvable Lie groups whose Ricci operator is a multiple of the identity modulo derivations (called solsolitons, and nilsolitons in the nilpotent case). The tools from geometric invariant theory used to study Einstein solvmanifolds, turned out to be useful in the study of solsolitons as well. We prove that, up to isometry, any solsoliton can be obtained via a very simple construction from a nilsoliton together with any abelian Lie algebra of symmetric derivations of its metric Lie algebra. …
A construction of Einstein solvmanifolds not based on nilsolitons
本文构造了标准但非 pseudo-Iwasawa 类型的不定 Einstein solvmanifold，基于 nice diagram 上非对角三元组的概念提出分类算法，并证明所得维数至多为 5 的流形不与 nilsoliton 的标准扩张等距。
Einstein metrics and Killing spinors on pseudo-Riemannian solvmanifolds
本文通过nilsoliton构造了一类任意维数和符号的伪Iwasawa可解流形，证明了所有允许Killing旋量的伪Iwasawa可解流形都属于该族，并指出当度量爱因斯坦时唯一可能是双曲半空间。
In this paper, we study the solvmanifolds constructed from any parabolic subalgebras of any semisimple Lie algebras. These solvmanifolds are naturally homogeneous submanifolds of symmetric spaces of noncompact type. We show that the Ricci curvatures of our solvmanifolds coincide with the restrictions of the Ricci curvatures of the ambient symmetric spaces. Consequently, all of our solvmanifolds are Einstein, which provide a large number of new examples of noncompact homogeneous Einstein manifolds. We also show that our solvmanifolds are minimal, but not totally geodesic submanifolds of symmetr…
We obtain new examples of non-symmetric Einstein solvmanifolds by combining two techniques. In \cite{T2}, H. Tamaru constructs new {\em attached} solvmanifolds, which are submanifolds of the solvmanifolds corresponding to noncompact symmetric spaces, endowed with a natural metric. Extending this construction, we apply it to {\em associated} solvmanifolds, described in \cite{GK}, obtained by modifying the algebraic structure of the solvable Lie algebras corresponding to noncompact symmetric spaces. Our new examples are Einstein solvmanifolds with nilradicals of high nilpotency, which are geomet…
We show that homogeneous Einstein metrics on Euclidean spaces are Einstein solvmanifolds, using that they admit periodic, integrally minimal foliations by homogeneous hypersurfaces. For the geometric flow induced by the orbit-Einstein condition, we construct a Lyapunov function based on curvature estimates which come from real GIT.
We study Einstein manifolds admitting a transitive solvable Lie group of isometries (solvmanifolds). It is conjectured that these exhaust the class of noncompact homogeneous Einstein manifolds. J. Heber has showed that under certain simple algebraic condition called standard (i.e. the orthogonal complement of the derived algebra is abelian), Einstein solvmanifolds have many remarkable structural and uniqueness properties. In this paper, we prove that any Einstein solvmanifold is standard, by applying a stratification procedure from geometric invariant theory due to F. Kirwan.
A non-Standard Indefinite Einstein Solvmanifold
本研究构造了一个非标准的可解流形不定不变爱因斯坦度量示例，其在幂零根基上的限制具有非退化性，为微分几何领域相关研究提供了新的案例。
A Riemannian Einstein solvmanifold is called standard, if the orthogonal complement to the nilradical of its Lie algebra is abelian. No examples of nonstandard solvmanifolds are known. We show that the standardness of an Einstein metric solvable Lie algebra is completely detected by its nilradical and prove that many classes of nilpotent Lie algebras (Einstein nilradicals, algebras with less than four generators, free Lie algebras, some classes of two-step nilpotent ones) contain no nilradicals of nonstandard Einstein metric solvable Lie algebras. We also prove that there are no nonstandard Ei…
We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-Kähler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $φ$ such that the induced metric $g_φ$ is Einstein, unless $g_φ$ is flat. We give an example of 7-dimensional solvmanifold admitting a left-invariant calibrated $G_2$-structure $φ$ such that $g_φ$ is Ricci-soliton. Moreover, we show that a 7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ cannot admit any left-invariant cocalibrated $G_2$-structure $φ$ such that the i…
The structure of a solvable Lie groups admitting an Einstein left-invariant metric is, in a sense, completely determined by the nilradical of its Lie algebra. We give an easy-to-check necessary and sufficient condition for a nilpotent algebra to be an Einstein nilradical whose Einstein derivation has simple eigenvalues. As an application, we classify filiform Einstein nilradicals (modulo known classification results on filiform graded Lie algebras).
The general aim of this paper is to study which are the solvable Lie groups admitting an Einstein left invariant metric. The space N of all nilpotent Lie brackets on R^n parametrizes a set of (n+1)-dimensional rank-one solvmanifolds, containing the set of all those which are Einstein in that dimension. The moment map for the natural GL(n)-action on N evaluated in a point of N encodes geometric information on the corresponding solvmanifold, allowing us to use strong and well-known results from geometric invariant theory. For instance, the functional on N whose critical points are precisely the …
An Einstein nilradical is a nilpotent Lie algebra, which can be the nilradical of a metric Einstein solvable Lie algebra. The classification of Riemannian Einstein solvmanifolds (possibly, of all noncompact homogeneous Einstein spaces) can be reduced to determining, which nilpotent Lie algebras are Einstein nilradicals and to finding, for every Einstein nilradical, its Einstein metric solvable extension. For every nilpotent Lie algebra, we construct an (essentially unique) derivation, the pre-Einstein derivation, the solvable extension by which may carry an Einstein inner product. Using the pr…
Inhomogeneous deformations of Einstein solvmanifolds
本文针对非平坦单模Ricci孤子可解流形，构造了单参数族的完备扩张梯度Ricci孤子，其轨道为位似超曲面，且度量一端渐近爱因斯坦可解流形，族中恰有一个爱因斯坦度量。
Einstein solvmanifolds as submanifolds of symmetric spaces
本文结合 Ado 定理与 Nash 嵌入定理，为包含 Einstein 和 Ricci 孤立子可解流形的一类可解李群构造同时为等距嵌入的忠实表示，进而证明所有 Einstein 可解流形均可实现为对称空间的子流形。
All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics on a Lie group, we prove that any left-invariant Einstein metric of negative Ricci curvature on a solvable Lie group is maximally symmetric. This theorem is motivated both by the Alekseevskii Conjecture and by the question of stability of Einstein metrics under the Ricci flow. We also address questions of existence of maximally symmetric left-invariant Riema…
A Riemannian Einstein solvmanifold (possibly, any noncompact homogeneous Einstein space) is almost completely determined by the nilradical of its Lie algebra. A nilpotent Lie algebra, which can serve as the nilradical of an Einstein metric solvable Lie algebra, is called an Einstein nilradical. Despite a substantial progress towards the understanding of Einstein nilradicals, there is still a lack of classification results even for some well-studied classes of nilpotent Lie algebras, such as the two-step ones. In this paper, we give a classification of two-step nilpotent Einstein nilradicals in…
In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the nilradical is of two-step. New examples of Einstein solvmanifolds with three-step and four-step nilradicals are also given.
Weyl-Einstein structures on conformal solvmanifolds
该文研究共形可解李群及其紧商上的Weyl-Einstein结构，证明紧共形可解流形带该结构则为Einstein流形，非交换幂零共形李群无左不变该结构，还分类了几乎交换情形及3维单连通可解李群的左不变度量。
We classify solvable Lie groups with a free nilradical admitting an Einstein left-invariant metric. Any such group is essentially determined by the nilradical of its Lie algebra, which is then called an Einstein nilradical. We show that among the free Lie algebras, there are very few Einstein nilradicals. Except for the one-step (abelian) and the two-step ones, there are only six others: f(2,3), f(2,4), f(2,5), f(3,3), f(4,3), f(5,3) (where f(m,p) is a free p-step Lie algebra on m generators). The reason for that is the inequality-type restrictions on the eigenvalue type of an Einstein nilradi…
The only known examples of noncompact Einstein homogeneous spaces are standard solvmanifolds (special solvable Lie groups endowed with a left invariant metric), and according to a long standing conjecture, they might be all. The classification of Einstein solvmanifolds is equivalent to the one of Einstein nilradicals, i.e. nilpotent Lie algebras which are nilradicals of the Lie algebras of Einstein solvmanifolds. Up to now, there have been found very few examples of graded nilpotent Lie algebras that can not be Einstein nilradicals. In particular, in each dimension, there are only finitely man…
We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear stability, we turn our attention to left-invariant soliton metrics on (non-compact) simply connected solvable Lie groups and prove linear stability of many such metrics. These include an open set of two-step solvsolitons, all two-step nilsolitons, two infinite families of three-step solvable Einstein metrics, all nilsolitons of dimensions six or less, and all…
We show that for any solvable Lie group of real type, any homogeneous Ricci flow solution converges in Cheeger-Gromov topology to a unique non-flat solvsoliton, which is independent of the initial left-invariant metric. As an application, we obtain results on the isometry groups of non-flat solvsoliton metrics and Einstein solvmanifolds.
Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily non-unimodular. In this work we consider unimodular solvable Lie groups and prove that there is always some metric with maximal symmetry. Further, if the group at hand admits a Ricci soliton, then it is the isometry group of the Ricci soliton which is maximal.
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…
Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds
研究可解李群上的左不变 Einstein 伪 Riemann Sasaki 度量，通过与带相容导子的伪 Kähler 幂零李代数对应来刻画并分类相关结构。
The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL(n)-orbits in this variety have a critical point of the squared norm of the moment map. In dimension 7, there are 148 complex nilpotent Lie algebras and 6 curves of pairwise non-isomorphic nilpotent Lie algebras, and we give in this paper a complete classification of the aforementioned distinguished orbits.
New Special Einstein Pseudo-Riemannian Metrics on Solvable Lie Algebras
本研究提出可解李代数上爱因斯坦伪凯勒与仿凯勒度量的构造方法，完成低维(Nil4)型幂零孤立子的秩一伪伊瓦萨瓦扩张分类，证明该度量在广义海森堡李代数对应扩张中存在，并给出后续研究思路。
A left invariant metric on a nilpotent Lie group is called minimal, if it minimizes the norm of the Ricci tensor among all left invariant metrics with the same scalar curvature. Such metrics are unique up to isometry and scaling and the groups admitting a minimal metric are precisely the nilradicals of (standard) Einstein solvmanifolds. If $N$ is endowed with an invariant symplectic, complex or hypercomplex structure, then minimal compatible metrics are also unique up to isometry and scaling. The aim of this paper is to give more evidence of the existence of minimal metrics, by presenting seve…
This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons exhibit such maximal symmetry? Question (i) is addressed both for semisimple and for solvable Lie groups. Building on previous work of the authors on Einstein metrics, a complete answer is given to (ii): expanding homogeneous Ricci solitons have maximal isometry algebras although not always maximal isometry groups. As a consequence of the tools developed to addre…
Given a Riemannian space $N$ of dimension $n$ and a field $D$ of symmetric endomorphisms on $N$, we define the extension $M$ of $N$ by $D$ to be the Riemannian manifold of dimension $n+1$ obtained from $N$ by a construction similar to extending a Lie group by a derivation of its Lie algebra. We find the conditions on $N$ and $D$ which imply that the extension $M$ is Einstein. In particular, we show that in this case, $D$ has constant eigenvalues; moreover, they are all integer (up to scaling) if $\det D \ne 0$. They must satisfy certain arithmetic relations which imply that there are only fini…
In this paper, we consider half-flat $SU(3)$-structures and the subclasses of coupled and double structures. In the general case we show that the intrinsic torsion form $w_1^-$ is constant in each of the two subclasses. We then consider the problem of finding half-flat structures inducing Einstein metrics on homogeneous spaces. We give an example of a left invariant half-flat (non coupled and non double) structure inducing an Einstein metric on $S^3\times S^3$ and we show there does not exist any left invariant coupled structure inducing an ${\rm Ad}(S^1)$-invariant Einstein metric on it. Fina…
We construct explicit proper p-harmonic functions on rank-one Lie groups of Iwasawa type. This class of Lie groups includes many classical Riemannian manifolds such as the rank-one symmetric spaces of non-compact type, Damek-Ricci spaces, rank-one Einstein solvmanifolds and Carnot spaces.
$η$-Einstein Sasakian Lie algebras
该研究针对李代数上的$η$-Einstein Sasakian结构，分中心非平凡与无中心两类情形展开分析，给出相关刻画与构造方法，得到存在性限制条件并构建新实例。
Uniform Lie algebras are combinatorially defined two-step nilpotent Lie algebras which can be used to define Einstein solvmanifolds. These Einstein spaces often have nontrivial isotropy groups. We derive basic properties of uniform Lie algebras and we classify uniform Lie algebras with five or fewer generators. We define a type of directed colored graph called a uniformly colored graph and establish a correspondence between uniform Lie algebras and uniformly colored graphs. We present several methods of constructing infinite families of uniformly colored graphs and corresponding uniform Lie al…
We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group $H_{2n + 1}$. Furthermore, we classify Sasakian Lie algebras of dimension 5 and determine which of them carry a Sasakian $α$-Einstein structure. We show that a 5-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either $H_5$ or a semidirect product $\R \ltimes (H_3 \times \R)$. In p…
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