Analysis of Contraction Mappings to The Complement of Closed Curves
Shunichiro Orikasa
TL;DR
The paper extends Llarull-type rigidity to distance-decreasing maps from $X=S^n\setminus\Sigma$ to $S^n\setminus\Sigma$ with $\Sigma$ a smooth circle, proving $\inf_{x} Sc(g)_x< n(n-1)$ under refined geometric/topological conditions when $n>4$ and $n\equiv0\pmod{4}$. The approach combines spin geometry, the Lichnerowicz formula, a collapsing map $\epsilon_\delta$, and a relative index theorem to produce a nonzero index that contradicts a curvature bound, using a pullback bundle $E=h^*E_0$ built from holonomy data along $\Sigma$. A universal constant $C(n)\sim 2^n$ bounds the holonomy-contribution via a Lipschitz $2$-chain representing the unit element in $H_2(S^n, W(\Sigma); \mathbb{R})$, solving a question of Gromov on metric bounds for $X$ with nontrivial obstacle $\Sigma$. The results cover both trivial and nontrivial monodromy cases, and the paper provides concrete constructions (e.g., 'good triplets' and 'nice' embeddings) to realize the index arguments in the noncompact setting. Overall, the work advances quantitative relations between scalar curvature, area of Lipschitz chains, and holonomy data in the presence of codimension-2 obstacles on the sphere.
Abstract
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}.