Remarks on non-compact gradient Ricci solitons

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 $L^p$-Liouville 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.

The main changes over the previous version are that Theorem 2 has been improved by removing the pointwise growth assumption on $|\nabla f|$ in the case $p=1$, and that in Theorem 3, a shrinking gradient Ricci soliton is shown to be isometric to ${\mathbb{R}}^m$ in the endpoint case $S_{\ast }=0$

References in corpus (4)

• Noncompact Shrinking 4-Solitons with Nonnegative Curvature
• Comparison Geometry for the Bakry-Emery Ricci Tensor
• Rigidity of gradient Ricci Solitons
• On complete gradient shrinking Ricci solitons

Cited by in corpus (5)

• On Type I Singularities in Ricci flow
• On the conditions to extend Ricci flow(II)
• On the injectivity radius and tangent cones at infinity of gradient Ricci solitons
• Complete gradient shrinking Ricci solitons with pinched curvature
• Volume estimates and the asymptotic behavior of expanding gradient Ricci solitons