On the diagonalization of the Ricci flow on Lie groups
HTML articles powered by AMS MathViewer
- by Jorge Lauret and Cynthia Will PDF
- Proc. Amer. Math. Soc. 141 (2013), 3651-3663 Request permission
Abstract:
The main purpose of this note is to prove that any basis of a nilpotent Lie algebra for which all diagonal left-invariant metrics have diagonal Ricci tensor necessarily produce quite a simple set of structural constants; namely, the bracket of any pair of elements of the basis must be a multiple of one of them, and only the bracket of disjoint pairs can be a nonzero multiple of the same element. Some applications to the Ricci flow of left-invariant metrics on Lie groups concerning diagonalization are also given.References
- Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
- Xiaodong Cao and Laurent Saloff-Coste, Backward Ricci flow on locally homogeneous 3-manifolds, Comm. Anal. Geom. 17 (2009), no. 2, 305–325. MR 2520911, DOI 10.4310/CAG.2009.v17.n2.a6
- R. Carles, Weight systems for complex nilpotent Lie algebras and application to the varieties of Lie algebras. Publ. Univ. Poitiers, 96 (1996)
- Bing-Long Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363–382. MR 2520796
- Bing-Long Chen and Xi-Ping Zhu, Uniqueness of the Ricci flow on complete noncompact manifolds, J. Differential Geom. 74 (2006), no. 1, 119–154. MR 2260930
- E. A. Fernández-Culma, Classification of 7-dimensional Einstein nilradicals, Transform. Groups 17 (2012), no. 3, 639–656. MR 2956161, DOI 10.1007/s00031-012-9186-5
- Willem A. de Graaf, Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2, J. Algebra 309 (2007), no. 2, 640–653. MR 2303198, DOI 10.1016/j.jalgebra.2006.08.006
- David Glickenstein, Riemannian groupoids and solitons for three-dimensional homogeneous Ricci and cross-curvature flows, Int. Math. Res. Not. IMRN 12 (2008), Art. ID rnn034, 49. MR 2426751, DOI 10.1093/imrn/rnn034
- David Glickenstein and Tracy L. Payne, Ricci flow on three-dimensional, unimodular metric Lie algebras, Comm. Anal. Geom. 18 (2010), no. 5, 927–961. MR 2805148, DOI 10.4310/CAG.2010.v18.n5.a3
- James Isenberg and Martin Jackson, Ricci flow of locally homogeneous geometries on closed manifolds, J. Differential Geom. 35 (1992), no. 3, 723–741. MR 1163457
- James Isenberg, Martin Jackson, and Peng Lu, Ricci flow on locally homogeneous closed 4-manifolds, Comm. Anal. Geom. 14 (2006), no. 2, 345–386. MR 2255014, DOI 10.4310/CAG.2006.v14.n2.a5
- Dan Knopf and Kevin McLeod, Quasi-convergence of model geometries under the Ricci flow, Comm. Anal. Geom. 9 (2001), no. 4, 879–919. MR 1868923, DOI 10.4310/CAG.2001.v9.n4.a7
- Brett L. Kotschwar, Backwards uniqueness for the Ricci flow, Int. Math. Res. Not. IMRN 21 (2010), 4064–4097. MR 2738351, DOI 10.1093/imrn/rnq022
- Jorge Lauret, A canonical compatible metric for geometric structures on nilmanifolds, Ann. Global Anal. Geom. 30 (2006), no. 2, 107–138. MR 2234091, DOI 10.1007/s10455-006-9015-y
- Jorge Lauret, Einstein solvmanifolds and nilsolitons, New developments in Lie theory and geometry, Contemp. Math., vol. 491, Amer. Math. Soc., Providence, RI, 2009, pp. 1–35. MR 2537049, DOI 10.1090/conm/491/09607
- Jorge Lauret, The Ricci flow for simply connected nilmanifolds, Comm. Anal. Geom. 19 (2011), no. 5, 831–854. MR 2886709, DOI 10.4310/CAG.2011.v19.n5.a1
- Jorge Lauret and Cynthia Will, Einstein solvmanifolds: existence and non-existence questions, Math. Ann. 350 (2011), no. 1, 199–225. MR 2785768, DOI 10.1007/s00208-010-0552-0
- John Lott, On the long-time behavior of type-III Ricci flow solutions, Math. Ann. 339 (2007), no. 3, 627–666. MR 2336062, DOI 10.1007/s00208-007-0127-x
- L. Magnin, Adjoint and trivial cohomology tables for indecomposable nilpotent Lie algebras of dimension $\leq 7$ over $\mathbb {C}$, e-Book, 2nd Corrected Edition, 2007.
- John Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293–329. MR 425012, DOI 10.1016/S0001-8708(76)80002-3
- Yuri Nikolayevsky, Einstein solvmanifolds with free nilradical, Ann. Global Anal. Geom. 33 (2008), no. 1, 71–87. MR 2369187, DOI 10.1007/s10455-007-9077-5
- Yuri Nikolayevsky, Einstein solvmanifolds with a simple Einstein derivation, Geom. Dedicata 135 (2008), 87–102. MR 2413331, DOI 10.1007/s10711-008-9264-y
- Y. Nikolayevsky, Einstein solvmanifolds and the pre-Einstein derivation, Trans. Amer. Math. Soc. 363 (2011), no. 8, 3935–3958. MR 2792974, DOI 10.1090/S0002-9947-2011-05045-2
- Tracy L. Payne, The Ricci flow for nilmanifolds, J. Mod. Dyn. 4 (2010), no. 1, 65–90. MR 2643888, DOI 10.3934/jmd.2010.4.65
- S. M. Salamon, Complex structures on nilpotent Lie algebras, J. Pure Appl. Algebra 157 (2001), no. 2-3, 311–333. MR 1812058, DOI 10.1016/S0022-4049(00)00033-5
- Wan-Xiong Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301. MR 1001277
- Cynthia Will, Rank-one Einstein solvmanifolds of dimension 7, Differential Geom. Appl. 19 (2003), no. 3, 307–318. MR 2013098, DOI 10.1016/S0926-2245(03)00037-8
- Cynthia Will, The space of solvsolitons in low dimensions, Ann. Global Anal. Geom. 40 (2011), no. 3, 291–309. MR 2831460, DOI 10.1007/s10455-011-9258-0
- Michael Bradford Williams, Explicit Ricci solitons on nilpotent Lie groups, J. Geom. Anal. 23 (2013), no. 1, 47–72. MR 3010272, DOI 10.1007/s12220-011-9237-5
- Edward N. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata 12 (1982), no. 3, 337–346. MR 661539, DOI 10.1007/BF00147318
Additional Information
- Jorge Lauret
- Affiliation: FaMAF and CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina
- MR Author ID: 626241
- Email: lauret@famaf.unc.edu.ar
- Cynthia Will
- Affiliation: FaMAF and CIEM, Universidad Nacional de Córdoba, Córdoba, Argentina
- MR Author ID: 649211
- Email: cwill@famaf.unc.edu.ar
- Received by editor(s): November 1, 2011
- Received by editor(s) in revised form: December 30, 2011
- Published electronically: June 25, 2013
- Additional Notes: This research was partially supported by grants from CONICET (Argentina) and SeCyT (Universidad Nacional de Córdoba)
- Communicated by: Lei Ni
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 3651-3663
- MSC (2010): Primary 53C30, 53C44
- DOI: https://doi.org/10.1090/S0002-9939-2013-11813-7
- MathSciNet review: 3080187