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

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 $\mathcal{L}<sup>{1}$-decay assumption on its Ricci curvature.