Integral Ricci curvature bounds for possibly collapsed spaces with Ricci curvature bounded from below

TL;DR

This paper establishes bounds on the local $L^p$ norm of Ricci curvature for complete Riemannian manifolds with a lower Ricci bound, especially in collapsed volume scenarios, for $p<1/2$.

Contribution

It introduces new integral Ricci curvature bounds that depend solely on dimension and adapt to volume collapse, extending previous pointwise bounds.

Findings

Bounds on local $L^p$ Ricci curvature depending on dimension

Bounds improve with volume collapse

Applicable for $p<1/2$ in Ricci curvature analysis

Abstract

Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for $p< 1/2$ we show the existence of bounds on the local $L^p$ norm of the Ricci curvature that depend only on the dimension and which improve with volume collapse.