Fibers of monotone maps of finite distortion

Abstract

We study topologically monotone surjective $W^{1,n}$-maps of finite distortion $f:\Omega \to {\Omega }^{\prime }$, where $\Omega ,{\Omega }^{\prime }$ are domains in ${\mathbb{R}}^n$, $n\ge 2$. If the outer distortion function $K_f\in L_{\mathrm{loc}}^p(\Omega )$ with $p\ge n-1$, then any such map $f$ is known to be homeomorphic, and hence the fibers $f^{-1}\{y\}$ are singletons. We show that as the exponent of integrability $p$ of the distortion function $K_f$ increases in the range $1/(n-1)\le p<n-1$, then the fibers $f^{-1}\{y\}$ of $f$ start satisfying increasingly strong homological limitations. We also give a Sobolev realization of a topological example by Bing of a monotone $f:{\mathbb{R}}^3\to {\mathbb{R}}^3$ with homologically nontrivial fibers, and show that this example has $K_f\in L_{\mathrm{loc}}^{1/2-\epsilon }({\mathbb{R}}^3)$ for all $\epsilon >0$.

25 pages, 7 figures

References in corpus (1)

• On the heterogeneous distortion inequality