A refinement of Günther’s candle inequality

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

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Published 12 February 2015