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

Abstract

We study the non Ricci flat gradient steady Kähler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature S, namely lim r→∞ r −1 Br S = 0, and show that it is a quotient of Σ×C n−1−k ×N k , where Σ and N denote the Hamilton's cigar soliton and some compact Kähler Ricci flat manifold respectively. As an application, we prove that any non Ricci flat gradient steady Kähler Ricci soliton with Ric ≥ 0, together with subquadratic volume growth or lim sup r→∞ rS < 1 must have universal covering space isometric to Σ × C n−1−k × N k .

On a dichotomy of the curvature decay of steady Ricci soliton

“…therein). This dichotomy conjecture, once established, will be very useful for the studies of steady soliton due to the classification results of steady solitons under different curvature decay conditions (see [7,13,29,30,31,33,52,15]). In fact, Deng-Zhu [29] (see also [52]) showed that the above dichotomy conjecture is true under nonnegative sectional curvature and linear scalar curvature decay conditions R ≤ C(r + 1) −1 .…”

“…Remark 1.11. If we further assume sectional curvature is nonnegative everywhere on M , then Theorem 1.10 also follows from [29] (see also Theorem 1.2 and [15,52]).…”

“…Steady Ricci solitons with fast curvature decay were extensively studied in [13,15,27,29,33,52]. As an application of Theorems 1.3 and 1.4, by exploiting the real analyticity of the soliton metric, we prove some classification results on steady gradient Ricci soliton with fast curvature decay under milder conditions.…”

“…Remark 1.14. Since 0 = lim inf x→∞ v|Rm|, M doesn't satisfy the lower bound in (15). The upper estimate in (16) doesn't hold as lim sup x→∞ v|Rm| > 0.…”

“…The curvature decays ( 15) and ( 16) can be found in the Bryant expanding soliton and the Kähler expander constructed by Feldman-Ilmanen-Knopf [37] respectively. It will be interesting to see whether or not (15) and ( 16) are the only possible curvature decays of expanding soliton. The upper bound in ( 15) is always satisfied on conical expander.…”