Steady soliton with $\mathcal{L}^{1}$ decay curvature

TL;DR

This paper proves a compactness theorem for gradient Ricci solitons with scalar curvature bounds, demonstrating smoothness of limits and showing steady solitons are asymptotically cylindrical under an -decay Ricci curvature condition.

Contribution

It establishes a new compactness result for gradient Ricci solitons and characterizes steady solitons with -decay Ricci curvature as asymptotically cylindrical.

Findings

Limit of noncollapsed gradient Ricci solitons with bounded Ricci curvature is smooth.

Steady gradient Ricci solitons with -decay Ricci curvature are asymptotically cylindrical.

Abstract

In this paper, we establish a compactness theorem for gradient Ricci solitons with scalar curvature bounds and uniform lower bounds of harmonic coordinates. Our approach is to bootstrap regularity in harmonic coordinates by exploiting the soliton equation. As an application, we show that the regular part of any noncollapsed limit of gradient Ricci solitons with bounded Ricci curvature is smooth. Further, we show that a steady gradient Ricci soliton is asymptotically cylindrical under an $\mathcal{L}^{1}$-decay assumption on its Ricci curvature.