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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.10975 |
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| _version_ | 1866912072680341504 |
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| author | Beach, Isabel Peruyero, Haydeé Contreras Rotman, Regina Searle, Catherine |
| author_facet | Beach, Isabel Peruyero, Haydeé Contreras Rotman, Regina Searle, Catherine |
| contents | 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 $m\exp(c\exp(G(n,k,v,D))$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_10975 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Linear Bounds for the Lengths of Geodesics on Manifolds With Curvature Bounded Below Beach, Isabel Peruyero, Haydeé Contreras Rotman, Regina Searle, Catherine Differential Geometry 53C22 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 $m\exp(c\exp(G(n,k,v,D))$. |
| title | Linear Bounds for the Lengths of Geodesics on Manifolds With Curvature Bounded Below |
| topic | Differential Geometry 53C22 |
| url | https://arxiv.org/abs/2410.10975 |