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 $c(n)vo{l}^{\frac{1}{n}}$, 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 $a=\lceil \frac{1}{\sin (\frac{\epsilon }{2})}\rceil +1$. 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 $2n!a^{n}d$. We will also show that there exists a "wide" geodesic loop of length at most $2(n+1){!}^2{a}^{(n+1{)}^3}FillRad\le 2\cdot n(n+1){!}^2{a}^{(n+1{)}^3}vo{l}^{\frac{1}{n}}$. Here $FillRad$ is the Filling Radius of $M^n$.

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