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49
W.Wylie, On the classification of gradient Ricci solitons
"... Abstract. We show that the only shrinking gradient solitons with vanishing Weyl tensor are quotients of the standard ones S n, S n−1 × R, and R n. This gives a new proof of the HamiltonIveyPerel’man classification of 3dimensional shrinking gradient solitons. We also show that gradient solitons wi ..."
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Cited by 35 (2 self)
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Abstract. We show that the only shrinking gradient solitons with vanishing Weyl tensor are quotients of the standard ones S n, S n−1 × R, and R n. This gives a new proof of the HamiltonIveyPerel’man classification of 3dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of H n, H n−1 × R, R n, S n−1 × R, or S n. 1.
Ricci solitons: the equation point of view
 Manuscripta Math
"... ABSTRACT. We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic prope ..."
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Cited by 33 (4 self)
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ABSTRACT. We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. New simpler proofs of some known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.
On gradient Ricci solitons with symmetry
"... We study gradient Ricci solitons with maximal symmetry. First we show that there are no nontrivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. ..."
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Cited by 33 (5 self)
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We study gradient Ricci solitons with maximal symmetry. First we show that there are no nontrivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of cohomogeneity one gradient solitons have been constructed. However, we apply the main result in [12] to show that there are no noncompact cohomogeneity one shrinking gradient solitons with nonnegative curvature.
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.
Linear stability of homogeneous Ricci solitons
 Int. Math. Res. Not. (2006), Art. ID 96253
"... Abstract. As a step toward understanding the analytic behavior of TypeIII Ricci ‡ow singularities, i.e. immortal solutions that exhibit j Rm j C=t curvature decay, we examine the linearization of an equivalent ‡ow at …xed points discovered recently by Baird–Danielo and Lott: nongradient homogeneous ..."
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Cited by 17 (1 self)
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Abstract. As a step toward understanding the analytic behavior of TypeIII Ricci ‡ow singularities, i.e. immortal solutions that exhibit j Rm j C=t curvature decay, we examine the linearization of an equivalent ‡ow at …xed points discovered recently by Baird–Danielo and Lott: nongradient homogeneous expanding Ricci solitons on nilpotent or solvable Lie groups. For all explicitly known nonproduct examples, we demonstrate linear stability of the ‡ow at these …xed points and prove that the linearizations generate C0 semigroups. 1.
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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Cited by 13 (2 self)
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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
CANONICAL MEASURES AND KÄHLERRICCI FLOW
"... We show that the KählerRicci flow on an algebraic manifold of positive Kodaira dimension and semiample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manif ..."
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Cited by 11 (1 self)
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We show that the KählerRicci flow on an algebraic manifold of positive Kodaira dimension and semiample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption of