Analysis of Contraction Mappings to The Complement of Closed Curves

Abstract

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 $n\equiv 0\mathrm{mod}4$, 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 $Are{a}_g(C)>C(n)\cdot {\max }_i\{|{\theta }_i|\}$ where $W(\Sigma )$ is any tubular neighborhood of $\Sigma$ and $\{e^{2\pi i{\theta }_i}{\}}_i$ are the holonomy parameters along ${\iota }^{\ast }{S}^{+}$ where $S^{+}$ is the positive spinor bundle over $S^n$. This answers a question in \cite{gromov2018metric}.

26 pages