Integral Ricci curvature bounds along geodesics for nonexpanding gradient Ricci solitons

Abstract

Following Li and Yau (Acta Math 156:153–201 1986) and similar to Perelman (The entropy formula for the Ricci flow and its geometric applications), we define an energy functional $${\mathcal{J}}$$ associated to a smooth function $${\phi}$$ on a complete Riemannian manifold. As an application, we deduce integral Ricci curvature upper bounds along modified geodesics for complete steady and shrinking gradient Ricci solitons.