Curvature estimates for steady Ricci solitons

Abstract

We derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. We also give an upper bound for the scalar curvature of noncompact expander when the Ricci curvature is nonpositive and the potential function is proper. We then provide a sufficient condition for the scalar curvature of expanding soliton being nonnegative. We also estimate the curvature of expanding soliton in dimensions three and four. As an application, we prove a gap theorem on three dimensional gradient expander.

Curvature Estimates for Four-Dimensional Gradient Steady Ricci Solitons

“…First of all, we need the following key fact, valid for all 4-dimensional gradient Ricci solitons, due to Munteanu and Wang [46] (see also Lemma 1 in [17]). Proposition 3.1.…”

“…Hence, it follows that 4-dimensional complete gradient expanding solitons with nonnegative Ricci curvature Rc ≥ 0 must have bounded Riemann curvature tensor |Rm| ≤ C. We point out that recent progress on curvature estimates for 4-dimensional gradient Ricci solitons has been led by the important work of Munteanu-Wang [46], in which they proved that any complete gradient shrinking soliton with bounded scalar curvature must have bounded Riemann curvature tensor. More significantly, they showed that the Riemann curvature tensor is controlled by the scalar curvature by |Rm| ≤ CR so that if the scalar curvature R decays at infinity so does the curvature tensor Rm; see also [12] for an extension, and [10,17] for similar estimates in the steady soliton case. Their curvature estimate, together with the uniqueness result of Kotschwar-Wang [42], has played a crucial role in the recent advance of classifying 4-dimensional complete gradient Ricci solitons, as well as in the classification of complex 2-dimensional complete gradient Kähler-Ricci solitons with scalar curvature going to zero at infinity by Conlon-Deruelle-Sun [28].…”

“…Motivated by the work of Munteanu-Wang [46], as well as Cao-Cui [10] and Chan [17,18], it is natural to ask if one could also control the Riemann curvature tensor Rm of a 4-dimensional complete gradient expanding soliton with nonnegative Ricci curvature by its scalar curvature R. Despite some key differences with the shrinking case, notably the lack of scalar curvature quadratic (or polynomial) decay lower bound for expanders, this turns out to be possible. Theorem 1.1.…”

Perelman's entropy on ancient Ricci flows

“…Finally, we remark that P.-Y. Chan [Cha20] recently proved that for any 4-dimensional steady gradient Ricci soliton which is a singularity model, we must have that |Rm| ≤ CR for some constant C (without assuming a curvature decaying condition as in [Cha19]).…”

Gradient steady Kahler Ricci solitons with non-negative Ricci curvature and integrable scalar curvature

“…Let (M, g, f ) be a real m dimensional complete non-Ricci flat gradient steady Ricci soliton with Ric ≥ 0 outside some compact subset of M. Further suppose that S → 0 as r → ∞. Then for all α ∈ (0, 1), there exists D > 0 such that (11) r…”