Local Estimate on Convexity Radius and decay of injectivity radius in a Riemannian manifold

Abstract

In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold $M$: 1) the convexity radius of $p$, $\mathrm{conv}(p)\ge \min \{\frac{1}{2}\mathrm{inj}(p),\mathrm{foc}(B_{\mathrm{inj}(p)}(p))\}$, where $\mathrm{inj}(p)$ is the injectivity radius of $p$ and $\mathrm{foc}(B_r(p))$ is the focal radius of open ball centered at $p$ with radius $r$; 2) for any two points $p,q$ in $M$, $\mathrm{inj}(q)\ge \min \{\mathrm{inj}(p),\mathrm{conj}(q)\}-d(p,q),$ where $\mathrm{conj}(q)$ is the conjugate radius of $q$; 3) for any $0<r<\min \{\mathrm{inj}(p),\frac{1}{2}\mathrm{conj}(B_{\mathrm{inj}(p)}(p))\}$, any (not necessarily minimizing) geodesic in $B_r(p)$ has length $\le 2r$. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.

19 pages. Final version to appear in Communications in Contemporary Mathematics. The only change is that we use "concentric convexity radius" instead of "pointed convexity radius"