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17
RECENT PROGRESS ON RICCI SOLITONS
, 2009
"... In recent years, there has seen much interest and increased research activities in Ricci solitons. Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton’s Ricci flow and play important roles in the singularity study of the Ricci flow. In this pap ..."
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In recent years, there has seen much interest and increased research activities in Ricci solitons. Ricci solitons are natural generalizations of Einstein metrics. They are also special solutions to Hamilton’s Ricci flow and play important roles in the singularity study of the Ricci flow. In this paper, we survey some of the recent progress on Ricci solitons.
Einstein solvmanifolds and nilsolitons
, 2009
"... Abstract. 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 ..."
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Abstract. 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. Contents
Dimensional reduction and the longtime behavior of Ricci flow
 COMM. MATH. HELV
, 2007
"... If g(t) is a threedimensional Ricci flow solution, with sectional curvatures that are O(t−1) and diameter that is O(t 1 2), then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton. ..."
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Cited by 18 (4 self)
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If g(t) is a threedimensional Ricci flow solution, with sectional curvatures that are O(t−1) and diameter that is O(t 1 2), then the pullback Ricci flow solution on the universal cover approaches a homogeneous expanding soliton.
RICCI FLOW ON THREEDIMENSIONAL, UNIMODULAR METRIC LIE ALGEBRAS
, 2009
"... We give a global picture of the Ricci flow on the space of threedimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a twodimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respe ..."
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Cited by 14 (0 self)
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We give a global picture of the Ricci flow on the space of threedimensional, unimodular, nonabelian metric Lie algebras considered up to isometry and scaling. The Ricci flow is viewed as a twodimensional dynamical system for the evolution of structure constants of the metric Lie algebra with respect to an evolving orthonormal frame. This system is amenable to direct phase plane analysis, and we find that the fixed points and special trajectories in the phase plane correspond to special metric Lie algebras, including Ricci solitons and special Riemannian submersions. These results are one way to unify the study of Ricci flow on left invariant metrics on threedimensional, simplyconnected, unimodular Lie groups, which had previously been studied by a casebycase analysis of the different Bianchi classes. In an appendix, we prove a characterization of the space of threedimensional, unimodular, nonabelian metric Lie algebras modulo isometry and scaling.
Lorentz Ricci solitons on 3dimensional Lie groups
, 906
"... The threedimensional Heisenberg group H3 has three leftinvariant Lorentz metrics g1, g2 and g3 as in [R92]. They are not isometric each other. In this paper, we characterize the leftinvariant Lorentzian metric g1 as a Lorentz Ricci soliton. This Ricci soliton g1 is a shrinking nongradient Ricci ..."
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The threedimensional Heisenberg group H3 has three leftinvariant Lorentz metrics g1, g2 and g3 as in [R92]. They are not isometric each other. In this paper, we characterize the leftinvariant Lorentzian metric g1 as a Lorentz Ricci soliton. This Ricci soliton g1 is a shrinking nongradient Ricci soliton. Likewise we prove that the isometry group of flat Euclid plane E(2) has Lorentz Ricci solitons. 1
The Ricci Flow for Nilmanifolds
, 2008
"... We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product with respect to time and the evolution of structure constants with respect to time, as well as the evolution of ..."
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Cited by 10 (0 self)
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We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product with respect to time and the evolution of structure constants with respect to time, as well as the evolution of these quantities modulo rescaling. We set up systems of O.D.E.’s for some of these flows and describe their qualitative properties. We also present some explicit solutions for the evolution of soliton metrics under the Ricci flow. 1
Algebraic Ricci Solitons of threedimensional Lorentzian Lie groups
, 2012
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Future nonlinear stability for solutions of the EinsteinVlasov system of Bianchi types II and VI0
 J. Math. Phys
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