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.

Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry–Emery Ricci curvature

“…This has been already known by Ni [30] (see the proof of Corollary 1.3 in Ni [30]). As was pointed out by the referee in April 2011, it has been also known by [34] and [35].…”

“…See Theorem 1 in[35] or Proposition 2 in[34]. Actually, this has been proved in the proof of Corollary 1.3 in[30].…”

Maximum principles and gradient Ricci solitons

“…Let u ∈ C 2 (M) be a function with u * = sup M u < +∞ and (x k ) a sequence of points on M satisfying (2.1)(i) and (2.1)(iii). It is shown in the proof of [13,Theorem 1.9…”

“…We recall that a Riemannian manifold (M, g) is said to be stochastically complete if, for some (and hence for any) (x, t) ∈ M × (0, +∞) one has M p(x, y, t) dy = 1, where p(x, y, t) is the minimal heat kernel of the Laplace-Beltrami operator . It is well known that any Riemannian manifold satisfying the Omori-Yau maximum principle is stochastically complete [13]. In fact, stochastic completeness is equivalent to the Riemannian manifold to satisfy the weak maximum principle, that is (2.1)(i) and (2.1)(iii).…”

“…First of all, we have the following Proof. We will need the following sufficient conditions for a Riemannian manifold to satisfy the Omori-Yau maximum principle [13].…”