Results 1  10
of
77
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 ..."
Abstract

Cited by 55 (6 self)
 Add to MetaCart
(Show Context)
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.
REMARKS ON NONCOMPACT GRADIENT RICCI SOLITONS
, 905
"... Abstract. In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L pLiouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality ..."
Abstract

Cited by 42 (8 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and L pLiouville type results for the weighted Laplacian associated to the potential may be used to obtain triviality, rigidity results, and scalar curvature estimates for gradient Ricci solitons under L p conditions on the relevant quantities.
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 ..."
Abstract

Cited by 35 (2 self)
 Add to MetaCart
(Show Context)
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.
RIGIDITY OF GRADIENT RICCI SOLITONS
, 2007
"... We define a gradient Ricci soliton to be rigid if it is a flat bundle N×ΓR k where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci soli ..."
Abstract

Cited by 34 (3 self)
 Add to MetaCart
(Show Context)
We define a gradient Ricci soliton to be rigid if it is a flat bundle N×ΓR k where N is Einstein. It is known that not all gradient solitons are rigid. Here we offer several natural conditions on the curvature that characterize rigid gradient solitons. Other related results on rigidity of Ricci solitons are also explained in the last section.
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. ..."
Abstract

Cited by 33 (5 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 27 (0 self)
 Add to MetaCart
(Show Context)
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
Einstein metrics from symmetry and bundle constructions
 in Surveys in Differential Geometry VI: Essays on Einstein Manifolds, (A Supplement to the Journal of Differential Geometry
, 1999
"... ar ..."
(Show Context)
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 ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
(Show Context)
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 have finite topological type
 C. R. Math. Acad. Sci. Paris
"... Abstract. We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its BakryÉmery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded fro ..."
Abstract

Cited by 17 (1 self)
 Add to MetaCart
(Show Context)
Abstract. We show that a complete Riemannian manifold has finite topological type (i.e., homeomorphic to the interior of a compact manifold with boundary), provided its BakryÉmery Ricci tensor has a positive lower bound, and either of the following conditions: (i) the Ricci curvature is bounded from above; (ii) the Ricci curvature is bounded from below and injectivity radius is bounded away from zero. Moreover, a complete shrinking Ricci soliton has finite topological type if its scalar curvature is bounded. 1.