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92
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.
ON LOCALLY CONFORMALLY FLAT GRADIENT STEADY Ricci Solitons
, 2009
"... In this paper, we prove that a complete noncompact nonflat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton. ..."
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Cited by 42 (6 self)
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In this paper, we prove that a complete noncompact nonflat conformally flat gradient steady Ricci soliton is, up to scaling, the Bryant soliton.
Smooth metric measure spaces with nonnegative curvature
 Comm. Anal. Geom
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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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Cited by 30 (0 self)
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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.
Existence of Ricci flows of incomplete surfaces
 Comm. Partial Differential Equations
"... We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases. 1 ..."
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Cited by 29 (7 self)
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We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases. 1
Sharp logarithmic Sobolev inequalities on gradient solitons and applications
, 2008
"... We show that gradient solitons, expanding, shrinking or steady, for the Ricci flow have potentials leading to suitable reference probability measures on the manifold. Under suitable conditions these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by ..."
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Cited by 27 (0 self)
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We show that gradient solitons, expanding, shrinking or steady, for the Ricci flow have potentials leading to suitable reference probability measures on the manifold. Under suitable conditions these reference measures satisfy sharp logarithmic Sobolev inequalities with lower bounds characterized by the geometry of the manifold. In the proof various useful volume growth estimates are also established for gradient shrinking and expanding solitons. 1
Rigidity of QuasiEinstein Metrics
 Differential Geom. Appl
"... Abstract. We call a metric quasiEinstein if themBakryEmery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We stud ..."
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Cited by 26 (3 self)
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Abstract. We call a metric quasiEinstein if themBakryEmery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasiEinstein metrics and prove several rigidity results. We also give a splitting theorem for some Kähler quasiEinstein metrics. 1.
Rotational symmetry of selfsimilar solutions to the Ricci flow
 Invent. Math
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Evolution of the Weyl Tensor under Ricci Flow
, 2010
"... We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally ..."
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Cited by 23 (11 self)
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We compute the evolution equation of the Weyl tensor under the Ricci flow of a Riemannian manifold and we discuss some consequences for the classification of locally conformally