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# Asian Journal of Mathematics

## Volume 19 (2015)

### Number 1

### A refinement of Günther’s candle inequality

Pages: 121 – 134

DOI: https://dx.doi.org/10.4310/AJM.2015.v19.n1.a5

#### Authors

#### Abstract

We analyze an upper bound on the curvature of a Riemannian manifold, using “$\sqrt{\mathrm{Ric}}$” curvature, which is in between a sectional curvature bound and a Ricci curvature bound. (A special case of $\sqrt{\mathrm{Ric}}$ curvature was previously discovered by Osserman and Sarnak for a different but related purpose.) We prove that our $\sqrt{\mathrm{Ric}}$ bound implies Günther’s inequality on the candle function of a manifold, thus bringing that inequality closer in form to the complementary inequality due to Bishop.

#### Keywords

Günther-Bishop Theorem, Riemannian manifold, Ricci curvature, candle function, volume bounds, curvature bounds

#### 2010 Mathematics Subject Classification

53B20

Published 12 February 2015