On the fundamental group of steady gradient Ricci solitons with nonnegative sectional curvature

TL;DR

This work establishes a dichotomy for complete steady gradient Ricci solitons with nonnegative sectional curvature: their fundamental group $pi1(M)$ is either trivial or infinite, and in the $\,kappa$-noncollapsed setting the manifold must be diffeomorphic to $R^n$. The authors leverage a splitting theorem for the universal cover and a detailed analysis of the covering (deck) transformation group, showing that nontrivial finite quotients are impossible under these curvature assumptions. A key technical step is proving that, when the universal cover splits as $N\times R^k$ with $N$ positively curved, any finite group action cannot be free on the $R^k$ factor, forcing triviality of the deck group. As a consequence, every $n$-dimensional complete $\,kappa$-noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature is diffeomorphic to $R^n$, contributing to the topology and rigidity theory of steady solitons and their quotients.

Abstract

In this paper, we study the fundamental group of the complete steady gradient Ricci soliton with nonnegative sectional curvature. We prove that the fundamental group of such a Ricci soliton is either trivial or infinite. As a corollary, we show that an $n$-dimensional complete $κ$-noncollapsed steady gradient Ricci soliton with nonnegative sectional curvature must be diffeomorphic to $\mathbb{R}^n$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2: Theorem 1.1. bamler_fundamental_2021
• Theorem 1.3
• Corollary 1.4
• Definition 2.1: Covering automorphism groups of Riemannian covering maps
• Proposition 2.2: lee_introduction_2012petersen_riemannian_2006
• Theorem 2.3: Corollary 12.9. lee_introduction_2011
• Proposition 2.4: Corollary 2.33. lee_introduction_2018
• Theorem 3.1: Theorem 1.1. guan_rigidity_nodate
• Proposition 3.2