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The existence of soliton metrics for nilpotent Lie groups. (English) Zbl 1189.53048
The primary aim of this work is to investigate nilsoliton metrics using algebraic and combinatorial methods. Each nonabelian metric algebra (𝔫 μ ,Q) is associated to a vector [α 2 ] listing the squares of the non trivial structure constants. It is shown that the nil Ricci endomorphism satisfies the nilsoliton condition if and only if the vector [α 2 ] is a solution to the matrix equation Uv=[1] being [1] a vector with every entry a one. A generalized Cartan matrix A is defined in terms of U. So the authors characterizes the solution spaces of (cA+d[1])v=-2β[1], c,d positive, by the properties of A. Several examples of nilsoliton metrics are presented, and also a metric nilpotent Lie algebra admitting a semisimple derivation with rational eigenvalues but no nilsoliton metric.
53C30Homogeneous manifolds (differential geometry)
53C25Special Riemannian manifolds (Einstein, Sasakian, etc.)
22E25Nilpotent and solvable Lie groups
22F30Homogeneous spaces
[1]Besse A.L.: Volume 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Einstein manifolds. Springer–Verlag, Berlin (1987)
[2]Chow B., Knopf D.: The Ricci flow: an introduction. Mathematical surveys and monographs, vol 110, pp. R1. American Mathematical Society, Providence (2004)
[3]Deloff, E.: Naturally reductive metrics and metrics with volume preserving geodesic symmetries. Thesis, Rutgers (1979)
[4]Goze M., Hakimjanov Y.: Sur les algèbres de Lie nilpotentes admettant un tore de dérivations. Manuscripta. Math. 84(2), 115–124 (1994) · Zbl 0823.17009 · doi:10.1007/BF02567448
[5]Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Differ. Geom. 17(2), 255–306 (1982)
[6]Heber J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133(2), 279–352 (1998) · Zbl 0906.53032 · doi:10.1007/s002220050247
[7]Jablonski, M.: Detecting orbits along subvarieties via the moment map. arXiv:math.DG/0810.5697.
[8]Jablonski, M.: Moduli of Einstein and non-Einstein nilradicals. arXiv:math.DG/0902.1698.
[9]Jensen G.R.: Homogeneous Einstein spaces of dimension four. J. Differ. Geom. 3, 309–349 (1969)
[10]Kac V.G.: Infinite-dimensional Lie algebras. 3rd edn. Cambridge University Press, Cambridge (1990)
[11]Karidi R.: Ricci structure and volume growth for left invariant Riemannian metrics on nilpotent and some solvable Lie groups. Geom. Dedicata. 46(3), 249–277 (1993) · Zbl 0781.53039 · doi:10.1007/BF01263618
[12]Khakimdjanov, Y.: Characteristically nilpotent, filiform and affine Lie algebras. In: Bajo, I. (ed.) Recent Advances in Lie Theory (Vigo, 2000) Res. Exp. Math., vol. 25, pp. 271–287. Heldermann, Lemgo (2002)
[13]Khakimdzhanov, Y.B.: Characteristically nilpotent Lie algebras. Algebra i Logika 28(6),722–737, 744 (1989)
[14]Lauret, J.: Einstein solvmanifolds and nilsolitons. New Dev. Lie Theory Geom. Contemp. Math. 491, (2009)
[15]Lauret, J.: Einstein solvmanifolds are standard. arXiv:math.DG/0703472v1.
[16]Lauret J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319(4), 715–733 (2001) · Zbl 0987.53019 · doi:10.1007/PL00004456
[17]Lauret J.: Standard Einstein solvmanifolds as critical points. Q. J. Math. 52(4), 463–470 (2001) · Zbl 1015.53025 · doi:10.1093/qjmath/52.4.463
[18]Lauret J.: Finding Einstein solvmanifolds by a variational method. Math. Z. 241(1), 83–99 (2002) · Zbl 1015.53028 · doi:10.1007/s002090100407
[19]Lauret, J., Will, C.: Einstein solvmanifolds: existence and nonexistence questions. arXiv:math-/0602502v3.
[20]Miatello I.D.: Ricci curvature of left invariant metrics on solvable unimodular Lie groups. Math. Z. 180(2), 257–263 (1982) · Zbl 0471.53033 · doi:10.1007/BF01318909
[21]Nikolayevsky, Y.: Einstein solvmanifolds and the pre-Einstein derivation. Trans. Amer. Math. Soc. (to appear)
[22]Nikolayevsky Y.: Einstein solvmanifolds with a simple Einstein derivation. Geom. Dedicata. 135, 87–102 (2008) · Zbl 1145.53040 · doi:10.1007/s10711-008-9264-y
[23]Nikolayevsky Y.: Einstein solvmanifolds with free nilradical. Ann. Global Anal. Geom. 33(1), 71–87 (2008) · Zbl 1156.53032 · doi:10.1007/s10455-007-9077-5
[24]Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:mathDG/0211159.
[25]Perelman G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:mathDG/0307245.
[26]Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:mathDG/0303109.
[27]Will, C.: A curve of nilpotent Lie algebras which are not Einstein nilradicals. arXiv:math.DG/0802.2544.
[28]Will C.: Rank-one Einstein solvmanifolds of dimension 7. Differ. Geom. Appl. 19(3), 307–318 (2003) · Zbl 1045.53032 · doi:10.1016/S0926-2245(03)00037-8
[29]Wolter T.H.: Einstein metrics on solvable groups. Math. Z. 206(3), 457–471 (1991) · Zbl 0707.53038 · doi:10.1007/BF02571355