Results 1  10
of
55
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.
A gap theorem for selfshrinkers of the mean curvature flow in arbitrary codimension
, 2012
"... ..."
(Show Context)
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 ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
(Show Context)
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
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. ..."
Abstract

Cited by 9 (7 self)
 Add to MetaCart
(Show Context)
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.
M.: On the geometry of gradient Einsteintype manifolds
"... Abstract. In this paper we introduce the notion of Einsteintype structure on a Riemannian manifold (M, g), unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasiEinstein manifolds. We show that these general structures can b ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we introduce the notion of Einsteintype structure on a Riemannian manifold (M, g), unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasiEinstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen [8]. 1. Introduction and