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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.
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.
GEOMETRY OF COMPLETE GRADIENT SHRINKING RICCI SOLITONS
, 2009
"... The notion of Ricci solitons was introduced by Hamilton [24] in mid 1980s. They are natural generalizations of Einstein metrics. Ricci solitons also correspond to selfsimilar solutions of Hamilton’s Ricci flow [22], and often arise as limits of dilations of singularities in the Ricci flow. In this ..."
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Cited by 19 (2 self)
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The notion of Ricci solitons was introduced by Hamilton [24] in mid 1980s. They are natural generalizations of Einstein metrics. Ricci solitons also correspond to selfsimilar solutions of Hamilton’s Ricci flow [22], and often arise as limits of dilations of singularities in the Ricci flow. In this paper, we will focus our attention on complete gradient shrinking Ricci solitons and survey some of the recent progress, including the classifications in dimension three, and asymptotic behavior of potential functions as well as volume growths of geodesic balls in higher dimensions.
Complete gradient shrinking Ricci solitons with pinched curvature
 Math. Ann
"... Abstract. We prove that any n–dimensional complete gradient shrinking Ricci soliton with pinched Weyl curvature is a finite quotient of Rn, R × Sn−1 or Sn. In particular, we do not need to assume the metric to be locally conformally flat. 1. ..."
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Cited by 9 (7 self)
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Abstract. We prove that any n–dimensional complete gradient shrinking Ricci soliton with pinched Weyl curvature is a finite quotient of Rn, R × Sn−1 or Sn. In particular, we do not need to assume the metric to be locally conformally flat. 1.
ANALYTIC AND GEOMETRIC PROPERTIES OF GENERIC RICCI SOLITONS
"... Abstract. The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either S3, R × S2 or R3, under some very weak conditions on the vector field X ge ..."
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Cited by 3 (3 self)
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Abstract. The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either S3, R × S2 or R3, under some very weak conditions on the vector field X generating the soliton structure. In doing so we introduce analytical tools that could be useful in other settings; for instance we prove that the OmoriYau maximum principle holds for the XLaplacian on every generic Ricci soliton, without any assumption on X. 1.
On rigidity of gradient KählerRicci solitons with harmonic Bochner tensor
 Proc. Amer. Math. Soc
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