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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.15135 |
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| _version_ | 1866912384729219072 |
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| author | Orikasa, Shunichiro |
| author_facet | Orikasa, Shunichiro |
| contents | We study some analytic properties of distance decreasing self-maps onto the complement of a smooth curve $Σ$ in $S^n$. For $n>4$ and $n\equiv 0 \mod 4$, let $Σ$ be an embedded circle in $S^n$ and let $g$ be a complete Riemannian metric on $X=S^n\backslash Σ$ and $f:(X,g)\to (X,g_{std})$ be a 1-contracting diffeomorphism. We verify the sharp estimate $\inf_{x\in X}Sc(g)_x<n(n-1)$ if any real Lipschitz 2-chain $C$ which represents the unit element $[C]$ in $H_2(S^n, W(Σ); \mathbb{R})$ satisfies $Area_g(C)>C(n)\cdot \max_i\{|θ_i|\}$ where $W(Σ)$ is any tubular neighborhood of $Σ$ and $\{e^{2πiθ_i}\}_i$ are the holonomy parameters along $ι^*S^+$ where $S^+$ is the positive spinor bundle over $S^n$. This answers a question in \cite{gromov2018metric}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_15135 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of Contraction Mappings to The Complement of Closed Curves Orikasa, Shunichiro Differential Geometry We study some analytic properties of distance decreasing self-maps onto the complement of a smooth curve $Σ$ in $S^n$. For $n>4$ and $n\equiv 0 \mod 4$, let $Σ$ be an embedded circle in $S^n$ and let $g$ be a complete Riemannian metric on $X=S^n\backslash Σ$ and $f:(X,g)\to (X,g_{std})$ be a 1-contracting diffeomorphism. We verify the sharp estimate $\inf_{x\in X}Sc(g)_x<n(n-1)$ if any real Lipschitz 2-chain $C$ which represents the unit element $[C]$ in $H_2(S^n, W(Σ); \mathbb{R})$ satisfies $Area_g(C)>C(n)\cdot \max_i\{|θ_i|\}$ where $W(Σ)$ is any tubular neighborhood of $Σ$ and $\{e^{2πiθ_i}\}_i$ are the holonomy parameters along $ι^*S^+$ where $S^+$ is the positive spinor bundle over $S^n$. This answers a question in \cite{gromov2018metric}. |
| title | Analysis of Contraction Mappings to The Complement of Closed Curves |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2502.15135 |