Complete non-compact gradient Ricci solitons with nonnegative Ricci curvature

Abstract

Abstract. In this paper, we give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or steady) gradient Kähler-Ricci solitons with positively pinched Ricci curvature should be Ricci flat. The result answers a question in case of Kähler-Ricci solitons proposed by Chow, Lu and Ni in a book.

Curvature estimates for steady Ricci solitons

“…We show that it is also true for gradient expanding soliton if in addition the potential function f → −∞ as r → ∞. It was proved by Deng-Zhu [17] and Deruelle [21] that any gradient expander with non-negative Ricci curvature must have bounded scalar curvature. Together with Remark 3, we have the following corollary.…”

“…Under the assumptions of Corollary 1 or Theorem 6, it follows from Theorem 3 that the scalar curvature S satisfies (7) provided that M is not flat. In view of the negatively curved Bryant's expander and the Cao's Kähler expander, we see that the quadratic factor r 2 in Corollary 1 and Theorem 6 is sharp (see [3], [1], [14] and [17]).…”

“…The expander itself is also flat if in addition Ric ≥ 0 (see [2] and [12]). Deng and Zhu [17] proved that a Kähler gradient expander with complex dimensions n ≥ 2, Ric ≥ 0 and lim r→∞ r 2 S = 0 is isometric to C n (see [33], [24] and [2] for more results on the gap theorems of general Riemannian manifolds). In either cases, Ricci curvature is assumed to be non-negative.…”

Steady Ricci solitons with horizontally ϵ-pinched Ricci curvature

“…Steady Ricci solitons with ϵ-pinched Ricci curvature have been studied by many people [7,9,17,20]. For example, Ni [20] proved that any steady Ricci soliton with ϵ-pinched Ricci curvature and nonnegative sectional curvature must be flat.…”

Asymptotic behavior of positively curved steady Ricci solitons

“…Let (M, g, f ) be a complete κ-noncollapsed steady Kähler-Ricci soltion with nonnegative bisectional curvature. It is proved that there exists a quilibrium point o of M such that ∇f (o) = 0 in [6]. Let φ t be a family of biholomorphisms generated by −∇f .…”