Linear Bounds for the Lengths of Geodesics on Manifolds With Curvature Bounded Below
| Autor: | Beach, Isabel, Peruyero, Haydeé Contreras, Rotman, Regina, Searle, Catherine |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $m\exp(c\exp(G(n,k,v,D))$. Comment: 22 pages, 10 figures |
| Databáze: | arXiv |
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