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On algebraic solitons for geometric evolution equations on three-dimensional Lie groups. (English) Zbl 1350.53087

Summary: The relationship between algebraic soliton metrics and self-similar solutions of geometric evolution equations on Lie groups is investigated. After discussing the general relationship between algebraic soliton metrics and self-similar solutions to geometric evolution equations, we investigate the cross curvature ow and the second order renormalization group ow on simply-connected, three-dimensional, unimodular Lie groups, providing a complete classification of left invariant algebraic solitons that give rise to self-similar solutions of the corresponding ows on such spaces.

MSC:

53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010)
53C30 Differential geometry of homogeneous manifolds
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[1] [1] John A. Buckland, Short-time existence of solutions to the cross curvature ow on 3-manifolds, Proc. Amer. Math. Soc. 134 (2006), no. 6, 1803-1807 (electronic) MR2207496 (2007a:35061) · Zbl 1091.53043
[2] Xiaodong Cao, John Guckenheimer, and Laurent Saloff-Coste, The backward behavior of the Ricci and cross-curvature ows on SL(2;R), Comm. Anal. Geom. 17 (2009), no. 4, 777-796 MR2601352 (2012b:53138) · Zbl 1198.53068
[3] Xiaodong Cao, Yilong Ni, and Laurent Saloff-Coste, Cross curvature ow on locally homogenous three-manifolds. I, Pacific J. Math. 236 (2008), no. 2, 263-281 MR2407107 (2009a:53109) · Zbl 1152.53053
[4] Xiaodong Cao and Laurent Saloff-Coste, Cross curvature ow on locally homoge- neous three-manifolds (II), Asian J. Math. 13 (2009), no. 4, 421-458 MR2653711 (2011m:53116) · Zbl 1193.53142
[5] Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci ow: techniques and applications. Part I, Mathematical Surveys and Monographs, vol. 135, American Mathematical Society, Providence, RI, 2007 Geometric aspects MR2302600 (2008f:53088) · Zbl 1157.53034
[6] Bennett Chow and Richard S. Hamilton, The cross curvature ow of 3-manifolds with negative sectional curvature, Turkish J. Math. 28 (2004), no. 1, 1-10 MR2055396 (2005a:53107) · Zbl 1064.53045
[7] Bennett Chow and Dan Knopf, The Ricci ow: an introduction, Mathematical Surveys and Monographs, vol. 110, American Mathematical Society, Providence, RI, 2004 MR2061425 (2005e:53101)
[8] Bennett Chow, Peng Lu, and Lei Ni, Hamilton’s Ricci ow, Graduate Studies in Mathematics, vol. 77, American Mathematical Society, Providence, RI; Science Press, New York, 2006 MR2274812 (2008a:53068)
[9] Jason DeBlois, Dan Knopf, and Andrea Young, Cross curvature ow on a negatively curved solid torus, Algebr. Geom. Topol. 10 (2010), no. 1, 343-372 MR2602839 (2011h:53086) · Zbl 1211.53083
[10] Daniel Harry Friedan, Nonlinear models in 2 + ” dimensions, Ann. Physics 163 (1985), no. 2, 318-419 MR811072 (87f:81130) · Zbl 0583.58010
[11] Krzysztof Gawędzki, Lectures on conformal field theory, Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), 1999, pp. 727-805 MR1701610 (2001f:81175)
[12] Karsten Gimre, Christine Guenther, and James Isenberg, A geometric introduction to the two-loop renormalization group ow, J. Fixed Point Theory Appl. 14 (2013), no. 1, 3-20 MR3202021 · Zbl 1304.35002
[13] , Second-order renormalization group ow of three-dimensional homogeneous ge- ometries, Comm. Anal. Geom. 21 (2013), no. 2, 435-467 MR3043753
[14] , Short-time existence for the second order renormalization group ow in general dimensions, Proc. Amer. Math. Soc. 143 (2015), no. 10, 4397-4401 MR3373938 · Zbl 1323.53074
[15] David Glickenstein, Riemannian groupoids and solitons for three-dimensional homoge- neous Ricci and cross-curvature ows, Int. Math. Res. Not. IMRN 12 (2008), Art. ID rnn034, 49 MR2426751 (2009f:53100)
[16] David Glickenstein and Tracy L. Payne, Ricci ow on three-dimensional, unimodular metric Lie algebras, Comm. Anal. Geom. 18 (2010), no. 5, 927-961 MR2805148 · Zbl 1226.53066
[17] David Glickenstein and Liang Wu, Soliton metrics for two-loop renormalization group ow on 3d unimodular lie groups (2015), available at arXiv:1510.06136v2[math.DG]
[18] Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), no. 2, 255-306 MR664497 (84a:53050) · Zbl 0504.53034
[19] , The formation of singularities in the Ricci ow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993), 1995, pp. 7-136 MR1375255 (97e:53075)
[20] James Isenberg and Martin Jackson, Ricci ow of locally homogeneous geometries on closed manifolds, J. Differential Geom. 35 (1992), no. 3, 723-741 MR1163457 (93c:58049) · Zbl 0808.53044
[21] Michael Jablonski, Concerning the existence of Einstein and Ricci soliton metrics on solvable Lie groups, Geom. Topol. 15 (2011), no. 2, 735-764 MR2800365 (2012h:53100) · Zbl 1217.22005
[22] , Homogeneous Ricci solitons are algebraic, Geom. Topol. 18 (2014), no. 4, 2477- 2486 MR3268781 · Zbl 1301.53044
[23] , Homogeneous Ricci solitons, J. Reine Angew. Math. 699 (2015), 159-182 MR3305924 · Zbl 1315.53046
[24] Ramiro Lafuente and Jorge Lauret, On homogeneous Ricci solitons, Q. J. Math. 65 (2014), no. 2, 399-419 MR3230368 · Zbl 1301.53045
[25] Jorge Lauret, Ricci soliton homogeneous nilmanifolds, Math. Ann. 319 (2001), no. 4, 715-733 MR1825405 (2002k:53083)
[26] Jorge Lauret and Cynthia Will, On the diagonalization of the Ricci ow on Lie groups, Proc. Amer. Math. Soc. 141 (2013), no. 10, 3651-3663 MR3080187 · Zbl 1279.53065
[27] Li Ma and Dezhong Chen, Examples for cross curvature ow on 3-manifolds, Calc. Var. Partial Differential Equations 26 (2006), no. 2, 227-243 MR2222245 (2006m:53100) · Zbl 1098.53058
[28] John Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293-329 MR0425012 (54 #12970) · Zbl 0341.53030
[29] K. Onda, Examples of algebraic Ricci solitons in the pseudo-Riemannian case, Acta Math. Hungar. 144 (2014), no. 1, 247-265 MR3267185 · Zbl 1324.53063
[30] Tracy L. Payne, The Ricci ow for nilmanifolds, J. Mod. Dyn. 4 (2010), no. 1, 65-90 MR2643888 (2011d:53163) · Zbl 1206.53074
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