Linear Bounds for the Lengths of Geodesics on Manifolds With Curvature Bounded Below

TL;DR

This paper establishes explicit bounds on the lengths of geodesics between points on manifolds with curvature, volume, and diameter constraints, providing a computable function for homotopy and geodesic existence.

Contribution

It introduces a computable rational function that bounds the length of geodesics and homotopies on manifolds with curvature and size constraints, extending previous qualitative results.

Findings

Existence of at least m geodesics of bounded length between points

Explicit bounds on homotopies of curves with controlled length

A computable function G(n,k,v,D) for length estimates

Abstract

Let $M$ be a simply connected Riemannian manifold in $\mathscr{M}_{k,v}^D(n)$, the space of closed Riemannian manifolds of dimension $n$ with sectional curvature bounded below by $k$, volume bounded below by $v$, and diameter bounded above by $D$. Let $c$ be the smallest positive real number such that any closed curve of length at most $2d$ can be contracted to a point over curves of length at most $cd$, where $d$ is the diameter of $M$. In this paper, we show that under these hypotheses there exists a computable rational function, $G(n,k,v,D)$, such that any continuous map of $S^l$ to $\Omega_{p,q}M$, the space of piecewise differentiable curves on $M$ connecting $p$ and $q$, is homotopic to a map whose image consists of curves of length at most $\exp(c\exp(G(n,k,v,D))$. In particular, for any points $p,q \in M$ and any integer $m>0$ there exist at least $m$ geodesics connecting $p$ and $q$ of length at most $m\exp(c\exp(G(n,k,v,D))$.