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

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 $\varinjlim_{r \to \infty} r^{-1} \int_{B_r} S = 0$, 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 $\operatorname{Ric} \geq 0$, together with subquadratic volume growth or $\lim \sup_{r \to \infty} rS \lt 1$ must have universal covering space isometric to $\Sigma \times \mathbb{C}^{n-1-k} \times N^k$.

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Received 4 November 2018

Accepted 16 August 2019

Published 29 November 2022