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

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 $Ω_{p,q}M$, 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 $m\exp(c\exp(G(n,k,v,D))$.

Figures (10)

• Figure 1: Illustration of the proof of Lemma \ref{['lemma:net']}.
• Figure 2: An illustration of (a) the curves $\sigma_i$ and (b) the curve $\widetilde{\alpha}$ in the proof of Lemma \ref{['lemma:close_curves_homotopy_bounded_width']}.
• Figure 3: An illustration of (a) the loops $P_i$ and (b) the contraction of each $P_i$ in the proof of Lemma \ref{['lemma:close_curves_homotopy_bounded_width']}.
• Figure 4: An illustration of the curve $\beta_i$ in the proof of Lemma \ref{['lemma:short_hom']}.
• Figure 5: Illustration of the first half of the homotopy $H'(t, \tau)$ of Lemma \ref{['lemma:homotopy_loop_length_bound']}.

Theorems & Definitions (27)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Definition 2.1: Width
• Theorem 2.2
• Remark 2.3
• Definition 3.1
• Lemma 3.2
• proof
• Lemma 3.3