Flexible exponent of geometric 3-manifolds and Legendrian maps of Seifert spaces
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917505297022976 |
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| author | Duan, Jianru Lin, Jianfeng Wang, Shicheng Wang, Zhongzi Wei, Dongyi |
| author_facet | Duan, Jianru Lin, Jianfeng Wang, Shicheng Wang, Zhongzi Wei, Dongyi |
| contents | A classical question in quantitative topology is to bound the mapping degree $\operatorname{deg}(f)$ in terms of its Lipchitz constant $\operatorname{Lip}(f)$. For a closed, oriented manifold $M$, the flexible exponent $α(M)$ is the infimum of $α\geq 0$ such that $|\operatorname{deg} f|\leq C(\operatorname{Lip} f)^α$ holds for all differentiable map $f:M\to M$. The flexible exponent measures how effectively a manifold can wrap itself through self-maps.
For geometric 3-manifolds $M$ in the sense of Thurston, we give the complete result for $α(M)$:
\[ α(M)= \begin{cases} 3 & M \text{ modeled on } \mathbb S^3,\mathbb E^3,\mathbb S^2\times\mathbb E^1,\\ \frac83 & M \text{ modeled on Nil},\\ 2 & M \text{ modeled on Sol},\\ 1 & M \text{ modeled on }\mathbb H^2\times\mathbb E^1,\\ 0 & M \text{ modeled on } \mathbb H^3,\widetilde{\rm SL_2}. \end{cases} \]
To prove $α(M)=8/3$ for Nil 3-manifold $M$, we construct the so-called Legendrian map: a smooth self-map $f: M\to M$ such that $f$ is homotopic to the identity and $f$ maps all $S^1$-fibers into the orthogonal contact plane field simultaneously. Moreover, we prove that any Legendrian map must not be a diffeomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_17257 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Flexible exponent of geometric 3-manifolds and Legendrian maps of Seifert spaces Duan, Jianru Lin, Jianfeng Wang, Shicheng Wang, Zhongzi Wei, Dongyi Geometric Topology Algebraic Topology Differential Geometry Metric Geometry A classical question in quantitative topology is to bound the mapping degree $\operatorname{deg}(f)$ in terms of its Lipchitz constant $\operatorname{Lip}(f)$. For a closed, oriented manifold $M$, the flexible exponent $α(M)$ is the infimum of $α\geq 0$ such that $|\operatorname{deg} f|\leq C(\operatorname{Lip} f)^α$ holds for all differentiable map $f:M\to M$. The flexible exponent measures how effectively a manifold can wrap itself through self-maps. For geometric 3-manifolds $M$ in the sense of Thurston, we give the complete result for $α(M)$: \[ α(M)= \begin{cases} 3 & M \text{ modeled on } \mathbb S^3,\mathbb E^3,\mathbb S^2\times\mathbb E^1,\\ \frac83 & M \text{ modeled on Nil},\\ 2 & M \text{ modeled on Sol},\\ 1 & M \text{ modeled on }\mathbb H^2\times\mathbb E^1,\\ 0 & M \text{ modeled on } \mathbb H^3,\widetilde{\rm SL_2}. \end{cases} \] To prove $α(M)=8/3$ for Nil 3-manifold $M$, we construct the so-called Legendrian map: a smooth self-map $f: M\to M$ such that $f$ is homotopic to the identity and $f$ maps all $S^1$-fibers into the orthogonal contact plane field simultaneously. Moreover, we prove that any Legendrian map must not be a diffeomorphism. |
| title | Flexible exponent of geometric 3-manifolds and Legendrian maps of Seifert spaces |
| topic | Geometric Topology Algebraic Topology Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2605.17257 |