A refinement of G\"unther's candle inequality

TL;DR

This paper introduces a refined curvature bound called root-Ricci curvature, which improves G"unther's inequality on the candle function of a Riemannian manifold, bridging it closer to Bishop's inequality.

Contribution

The paper develops a new curvature bound called root-Ricci curvature and demonstrates its implications for G"unther's inequality, enhancing understanding of curvature constraints.

Findings

Root-Ricci curvature bounds imply G"unther's inequality.

The new bound bridges the gap between sectional and Ricci curvature bounds.

Special cases relate to previous work by Osserman and Sarnak.

Abstract

We analyze an upper bound on the curvature of a Riemannian manifold, using "root-Ricci" curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of root-Ricci curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our root-Ricci bound implies G\"unther's inequality on the candle function of a manifold, thus bringing that inequality closer in form to the complementary inequality due to Bishop.