Wide short geodesic loops on closed Riemannian manifolds

Abstract

It is not known whether or not the lenth of the shortest periodic geodesic on a closed Riemannian manifold $M^n$ can be majorized by $c(n) vol^{ 1 \over n}$, or $\tilde{c}(n)d$, where $n$ is the dimension of $M^n$, $vol$ denotes the volume of $M^n$, and $d$ denotes its diameter. In this paper we will prove that for each $\epsilon >0$ one can find such estimates for the length of a geodesic loop with with angle between $\pi-\epsilon$ and $\pi$ with an explicit constant that depends both on $n$ and $\epsilon$. That is, let $\epsilon > 0$, and let $a = \lceil{ {1 \over {\sin ({\epsilon \over 2})}}} \rceil+1 $. We will prove that there exists a "wide" (i.e. with an angle that is wider than $\pi-\epsilon$) geodesic loop on $M^n$ of length at most $2n!a^nd$. We will also show that there exists a "wide" geodesic loop of length at most $2(n+1)!^2a^{(n+1)^3} FillRad \leq 2 \cdot n(n+1)!^2a^{(n+1)^3} vol^{1 \over n}$. Here $FillRad$ is the Filling Radius of $M^n$.