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ON COMPLETE GRADIENT SHRINKING RICCI SOLITONS
, 2009
"... In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the we ..."
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Cited by 55 (6 self)
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In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the wellknown theorem of Bishop that a complete noncompact Riemannian manifold with nonnegative Ricci curvature has at most Euclidean volume growth.
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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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 isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in [8, 10]. 1. The results A complete Riemannian metric gij on a smooth manifold M n is called a gradient Ricci soliton 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 Ricci soliton. Clearly, when f is a constant a gradient Ricci soliton is simply a Einstein manifold. Thus
On rigidity of gradient KählerRicci solitons with harmonic Bochner tensor
 Proc. Amer. Math. Soc
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ON TYPEII SINGULARITIES IN RICCI FLOW ON R n+1
"... Abstract. For n+1 ≥ 3, we construct complete solutions to Ricci flow on R n+1 which encounter global singularities at a finite time T. The singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t) −2λ for λ ≥ 1. Near the origin, blowups of such ..."
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Abstract. For n+1 ≥ 3, we construct complete solutions to Ricci flow on R n+1 which encounter global singularities at a finite time T. The singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t) −2λ for λ ≥ 1. Near the origin, blowups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blowups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on R n+1 whose blowups near the origin converge uniformly to the Bryant soliton. 1.
3 GENERALIZED RICCI FLOW II: EXISTENCE FOR NONCOMPACT COMPLETE MANIFOLDS
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ON TYPEII SINGULARITIES IN RICCI FLOW ON Rn+1
"... Abstract. In each dimension n+1 ≥ 3 and for each real number λ ≥ 1, we construct complete solutions to Ricci flow on Rn+1 which encounter global singularities at a finite time T. The singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t)−(λ+1) ..."
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Abstract. In each dimension n+1 ≥ 3 and for each real number λ ≥ 1, we construct complete solutions to Ricci flow on Rn+1 which encounter global singularities at a finite time T. The singularities are forming arbitrarily slowly with the curvature blowing up arbitrarily fast at the rate (T − t)−(λ+1). Near the origin, blowups of such a solution converge uniformly to the Bryant soliton. Near spatial infinity, blowups of such a solution converge uniformly to the shrinking cylinder soliton. As an application of this result, we prove that there exist standard solutions of Ricci flow on Rn+1 whose blowups near the origin converge uniformly to the Bryant soliton. 1.