Porosity of the branch set of discrete open mappings with controlled linear dilatation

Abstract

Assume that $X$ and $Y$ are locally compact and locally doubling metric spaces, which are also generalized $n$-manifolds, that $X$ is locally linearly locally $n$-connected, and that $Y$ has bounded turning. In this paper, addressing Heinonen's ICM 02 talk, we study the geometry of the branch set ${\mathcal{B}}_f$ of a quasiregular mapping between metric $n$-manifolds. In particular, we show that ${\mathcal{B}}_f\cap \{x\in X:H_f(x)<\infty \}$ is countably porous, as is its image $f({\mathcal{B}}_f\cap \{x\in X:H_f(x)<\infty \})$. As a corollary, ${\mathcal{B}}_f\cap \{x\in X:H_f(x)<\infty \}$ and its image are null sets with respect to any locally doubling measures on $X$ and Y, respectively. Moreover, if either $H_f(x)\le H$ or $H_f^{\ast }(x)\le H^{\ast }$ for all $x\in X$, then both ${\mathcal{B}}_f$ and $f\left({\mathcal{B}}_f\right)$ are countably $\delta$-porous, quantitatively, with a computable porosity constant. When further metric and analytic assumptions are placed on $X$, $Y$, and $f$, our theorems generalize the well-known Bonk--Heinonen theorem and Sarvas' theorem to a large class of metric spaces. Moreover, our results are optimal in terms of the underlying geometric structures. As a direct application, we obtain the important Väisälä's inequality in greatest generality. Applying our main results to special cases, we solve an open problem of Heinonen--Rickman and an open question of Heinonen--Semmes.

27 pages, the paper is substantially shortened and several in-rigorous statements in the initial version are made precise

References in corpus (1)

• Uniformization of two-dimensional metric surfaces

Cited by in corpus (1)

• Equivalence of quasiregular mappings on subRiemannian manifolds via the Popp extension