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31
Bachflat gradient steady Ricci solitons
 Calc. Var. Partial Differential Equations
, 2014
"... Abstract. In this paper we prove that any ndimensional (n ≥ 4) complete Bachflat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a threedimensional gradient steady Ricci soliton with divergencefree Bach tensor is either flat or is ..."
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Cited by 16 (8 self)
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if there exists a smooth function f on Mn such that the Ricci tensor Rij of the metric gij satisfies the equation Rij +∇i∇jf = ρ gij for some constant ρ. For ρ = 0 the Ricci soliton is steady, for ρ> 0 it is shrinking and for ρ < 0 expanding. The function f is called a potential function of the gradient
M~iiB'9~.oMff~~~m1:EA~..~~if~ B'9~~r~~IJ0
"... {J3M1Nymffl~IillLF~~oIJ;!1:E ' eJ~7f~mJlJB'9~gJE1.J$Jf~U~5JT§~5t1f~~ ..."
Curvature tensor under the Ricci flow
"... Abstract. Consider the unnormalized Ricci flow (gij)t = −2Rij for t ∈ [0, T), where T < ∞. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times t ∈ [0, T), then the solution can be extended beyond T. We prove that if the Ricci curvature is unifo ..."
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Cited by 34 (2 self)
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Abstract. Consider the unnormalized Ricci flow (gij)t = −2Rij for t ∈ [0, T), where T < ∞. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times t ∈ [0, T), then the solution can be extended beyond T. We prove that if the Ricci curvature
A SIMPLE PROOF ON THE NONEXISTENCE OF SHRINKING BREATHERS FOR THE RICCI FLOW
, 2006
"... Abstract. Suppose M is a compact ndimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T). We will give a simple proof of a recent result of Perelman on the nonexistence of shrinking breather without using the logarithmic Sobolev inequality. ..."
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Cited by 9 (7 self)
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Abstract. Suppose M is a compact ndimensional manifold, n ≥ 2, with a metric gij(x, t) that evolves by the Ricci flow ∂tgij = −2Rij in M × (0, T). We will give a simple proof of a recent result of Perelman on the nonexistence of shrinking breather without using the logarithmic Sobolev inequality
j − ∇j (div v) i
"... 4 Short time existence and curvature estimates We give a very rough description of how DeTurck’s proof (a.k.a. DeTurck’s trick) of short time existence of the Ricci flow ∂ ∂t gij = −2Rij on closed manifolds works. Recall that given a variation ∂ ∂s gij = vij of a metric, the associated variation of ..."
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4 Short time existence and curvature estimates We give a very rough description of how DeTurck’s proof (a.k.a. DeTurck’s trick) of short time existence of the Ricci flow ∂ ∂t gij = −2Rij on closed manifolds works. Recall that given a variation ∂ ∂s gij = vij of a metric, the associated variation
Entropy and reduced distance for Ricci expanders
 J. Geom. Anal
"... ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij /∂t =−2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expan ..."
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Cited by 40 (6 self)
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ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij /∂t =−2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case
Convergence of the Ricci flow toward a soliton
 Comm. Anal. Geom
"... We will consider a τflow, given by the equation d dtgij = −2Rij + 1 τ gij on a closed manifold M, for all times t ∈ [0, ∞). We will prove that if the curvature operator and the diameter of (M, g(t)) are uniformly bounded along the flow, then we have a sequential convergence of the flow toward the s ..."
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Cited by 7 (0 self)
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We will consider a τflow, given by the equation d dtgij = −2Rij + 1 τ gij on a closed manifold M, for all times t ∈ [0, ∞). We will prove that if the curvature operator and the diameter of (M, g(t)) are uniformly bounded along the flow, then we have a sequential convergence of the flow toward
Monotone volume formulas for geometric flows, arXiv:0905.2328
"... We consider a closed manifold M with a Riemannian metric gij(t) evolving by ∂t gij = −2Sij where Sij(t) is a symmetric twotensor on (M, g(t)). We prove that if Sij satisfies the tensor inequality D(Sij, X) ≥ 0 for all vector fields X on M, where D(Sij, X) is defined in (1.6), then one can construc ..."
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Cited by 10 (1 self)
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construct a forwards and a backwards reduced volume quantity, the former being nonincreasing, the latter being nondecreasing along the flow ∂t gij = −2Sij. In the case where Sij = Rij, the Ricci curvature of M, the result corresponds to Perelman’s wellknown reduced volume monotonicity for the Ricci flow
On the Completeness of Gradient Ricci
, 807
"... Abstract A gradient Ricci soliton is a triple (M, g, f) satisfying Rij + ∇i∇jf = λgij for some real number λ. In this paper, we will show that the completeness of the metric g implies that of the vector field ∇f. 1. ..."
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Abstract A gradient Ricci soliton is a triple (M, g, f) satisfying Rij + ∇i∇jf = λgij for some real number λ. In this paper, we will show that the completeness of the metric g implies that of the vector field ∇f. 1.
CERNTH.7187/94 The WuYang Ambiguity Revisited*
, 1994
"... Several examples are given of continuous families of SU(2) vector potentials Aa i (x) in 3 space dimensions which generate the same magnetic field B ai (x) (with det B ̸ = 0). These WuYang families are obtained from the Einstein equation Rij = −2Gij derived recently via a local map of the gauge fie ..."
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Several examples are given of continuous families of SU(2) vector potentials Aa i (x) in 3 space dimensions which generate the same magnetic field B ai (x) (with det B ̸ = 0). These WuYang families are obtained from the Einstein equation Rij = −2Gij derived recently via a local map of the gauge
Results 1  10
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31