Analysis of Contraction Mappings to The Complement of Closed Curves

We study some analytic properties of distance decreasing self-maps onto the complement of a smooth curve $\mathrm{\Sigma }$ in $S^n$. For $n>4$ and $n\equiv 0\mathrm{m}\mathrm{o}\mathrm{d}4$, let $\mathrm{\Sigma }$ be an embedded circle in $S^n$ and let $g$ be a complete Riemannian metric on $X=S^n\mathrm{\backslash }\mathrm{\Sigma }$ 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 $H_2(S^n,W(\mathrm{\Sigma });\mathbb{R})$ satisfies $Are{a}_g(C)>C(n)\cdot {\max }_i\{\mathrm{\mid }{\theta }_i\mathrm{\mid }\}$ where $W(\mathrm{\Sigma })$ is any tubular neighborhood of $\mathrm{\Sigma }$ and $\{e^{2\pi i{\theta }_i}{\}}_i$ are the holonomy parameters along ${\iota }^{\ast }{S}^{+}$ where $S^{+}$ is the positive spinor bundle over $S^n$. This answers a question in \cite{gromov2018metric}.