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 ${\underline{\lim }}_{r\to \infty }{r}^{-1}{\int }_{B_r}S=0$, and show that it is a quotient of $\Sigma \times {\mathbb{C}}^{n-1-k}\times N^k$, where $\Sigma$ 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 $Ric\ge 0$, together with subquadratic volume growth or ${\mathrm{limsup}}_{r\to \infty }rS<1$ must have universal covering space isometric to $\Sigma \times {\mathbb{C}}^{n-1-k}\times N^k$.

References in corpus (4)

• Existence of Gradient Kahler-Ricci Solitons
• Classification of expanding and steady Ricci solitons with integral curvature decay
• 3d steady Gradient Ricci Solitons with linear curvature decay
• Classification of gradient steady Ricci solitons with linear curvature decay

Cited by in corpus (1)

• On a dichotomy of the curvature decay of steady Ricci soliton