Ricci soliton homogeneous nilmanifolds

Lauret, Jorge

doi:10.1007/pl00004456

Cited by 192 publications

(219 citation statements)

References 23 publications

(14 reference statements)

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“…This implies, in particular, that the nilradical n of an Einstein metric solvable Lie algebra admits an N-gradation defined by the eigenspaces of Φ. By [La1,Theorem 3.7], a necessary and sufficient condition for a metric nilpotent algebra (n, ·, · ) to be the nilradical of an Einstein metric solvable Lie algebra is (6) Ric n = c id n +Φ, for some Φ ∈ Der(n), where c dim g < 0 is the scalar curvature of (g, ·, · ). This equation, in fact, defines (g, ·, · ) in the following sense: given a metric nilpotent Lie algebra whose Ricci operator satisfies (6), with some constant c < 0 and some Φ ∈ Der(n), one can define g as a one-dimensional extension of n by Φ.…”

Section: Einstein Solvmanifoldsmentioning

confidence: 99%

“…For such an extension g = RH ⊕ n, ad H|n = Φ, and the inner product defined by H, n = 0, H 2 = Tr Φ (and coinciding with the existing one on n) is Einstein, with scalar curvature c dim g. A nilpotent Lie algebra n which admits an inner product ·, · and a derivation Φ satisfying (6) is called an Einstein nilradical, the corresponding derivation Φ is called an Einstein derivation, and the inner product ·, · the nilsoliton metric. As proved in [La1,Theorem 3.5], a nilpotent Lie algebra admits no more than one nilsoliton metric, up to conjugation by Aut(n) and scaling (and hence, an Einstein derivation, if it exists, is unique, up to conjugation and scaling). Equation (6), together with (4), implies that if n is an Einstein nilradical, with Φ the Einstein derivation, then for some c < 0 Throughout the paper, ⊕ means the direct sum of linear spaces (even when the summands are Lie algebras).…”

Section: Einstein Solvmanifoldsmentioning

confidence: 99%

“…From the assumption and assertion 1 we obtain that Tr ( (6) holds, with some c ∈ R and some ψ ∈ Der(g 0 .μ). Then ·, · is a nilsoliton metric for (R n , g 0 .μ) by [La1,Theorem 3.7]. The nilsoliton inner product for n = (R n , μ) can then be taken as X,…”

Section: ])mentioning

confidence: 99%

“…On the Lie algebra level, all the metric Einstein solvable Lie algebras can be obtained as the result of the following construction [Heb,La1,La5,LW]. One starts with the three pieces of data: a nilpotent Lie algebra n, a semisimple derivation Φ of n, and an inner product ·, · n on n, with respect to which Φ is symmetric.…”

Section: Introductionmentioning

confidence: 99%

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Einstein solvmanifolds and the pre-Einstein derivation

Nikolayevsky

2011

Trans. Amer. Math. Soc.

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Abstract. 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 pre-Einstein derivation, we then give a variational characterization of Einstein nilradicals. As an application, we prove an easy-to-check convex geometry condition for a nilpotent Lie algebra with a nice basis to be an Einstein nilradical and also show that a typical two-step nilpotent Lie algebra is an Einstein nilradical.

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Section: Einstein Solvmanifoldsmentioning

confidence: 99%

Section: Einstein Solvmanifoldsmentioning

confidence: 99%

Section: ])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Einstein solvmanifolds and the pre-Einstein derivation

Nikolayevsky

2011

Trans. Amer. Math. Soc.

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“…nilsolitons, solvsolitons), as a result of work by J. Lauret [La1], [La3], M. Jablonski [Ja], and many others (cf the survey [La2]). These expanders are however not of gradient type, i.e., they satisfy the more general equation…”

Section: Introductionmentioning

confidence: 99%

Non-Kähler expanding Ricci solitons, Einstein metrics, and exotic cone structures

et al. 2015

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Abstract. We produce new non-Kähler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form R 2 × M2 × · · · × Mr, where r ≥ 2 and Mi are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for complete expanding solitons on the vector bundles whose sphere bundles are the twistor or Sp(1) bundles over quaternionic projective space.Mathematics Subject Classification (2000): 53C25, 53C44 IntroductionIn [BDGW] we constructed complete steady gradient Ricci soliton structures (including Ricci-flat metrics) on manifolds of the form R 2 × M 2 × . . . × M r where M i , 2 ≤ i ≤ r, are arbitrary closed Einstein manifolds with positive scalar curvature. We also produced numerical solutions of the steady gradient Ricci soliton equation on certain non-trivial R 3 and R 4 bundles over quaternionic projective spaces. In the current paper we will present the analogous results for the case of expanding solitons on the same underlying manifolds.Recall that a gradient Ricci soliton is a manifold M together with a smooth Riemannian metric g and a smooth function u, called the soliton potential, which give a solution to the equation:for some constant ǫ. The soliton is then called expanding, steady, or shrinking according to whether ǫ is greater, equal, or less than zero. A gradient Ricci soliton is called complete if the metric g is complete. The completeness of the vector field ∇u follows from that of the metric (cf [Zh]). If the metric of a gradient Ricci soliton is Einstein, then either Hess u = 0 (i.e., ∇u is parallel) or we are in the case of the Gaussian soliton (cf [PW] or [PRS]).At present most examples of non-Kählerian expanding solitons arise from left-invariant metrics on nilpotent and solvable Lie groups (resp. nilsolitons, solvsolitons), as a result of work by J. Lauret [La1], [La3], M. Jablonski [Ja], and many others (cf the survey [La2]). These expanders are however not of gradient type, i.e., they satisfy the more general equationwhere X is a vector field on M and L denotes Lie differentiation.A large class of complete, non-Einstein, non-Kählerian expanders of gradient type (with dimension ≥ 3) consists of an r-parameter family of solutions to (0.1) on R k+1 × M 2 × . . . × M r where k > 1 and M i are positive Einstein manifolds. The special case r = 1 (i.e., no M i ) is due to Bryant [Bry] and the solitons have positive sectional curvature. The r = 2 case is due to Gastel and Kronz [GK], who adapted Böhm's construction of complete Einstein metrics with negative scalar curvature to the soliton case. The case of arbitrary r was treated in [DW3] via a generalization

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On Lorentzian Ricci solitons on nilpotent Lie groups

Wears

2016

Math. Nachr.

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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 boldE1,boldE2,boldE3,boldE4 and E5 and non‐trivial bracket relations E1,E5=boldE3 and E2,E5=boldE4, investigating the various curvature properties of the resulting families of metrics, and classifying all Lorentzian Ricci soliton structures.

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Ricci soliton homogeneous nilmanifolds

Cited by 192 publications

References 23 publications

Einstein solvmanifolds and the pre-Einstein derivation

Einstein solvmanifolds and the pre-Einstein derivation

Non-Kähler expanding Ricci solitons, Einstein metrics, and exotic cone structures

On Lorentzian Ricci solitons on nilpotent Lie groups

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