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 $\Sigma$ in $S^n$. For $n>4$ and $n\equiv 0 \mod 4$, let $\Sigma$ be an embedded circle in $S^n$ and let $g$ be a complete Riemannian metric on $X=S^n\backslash \Sigma$ 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 $H_2(S^n, W(\Sigma); \mathbb{R})$ satisfies $Area_g(C)>C(n)\cdot \max_i\{|\theta_i|\}$ where $W(\Sigma)$ is any tubular neighborhood of $\Sigma$ and $\{e^{2\pi i\theta_i}\}_i$ are the holonomy parameters along $\iota^*S^+$ where $S^+$ is the positive spinor bundle over $S^n$. This answers a question in \cite{gromov2018metric}.