Snowballs are Quasiballs

TL;DR

This paper introduces snowballs, 3D analogs of snowflake-bound domains, and constructs quasiconformal maps from these snowballs to the unit ball, extending planar concepts into three dimensions.

Contribution

It defines snowballs as 3D compact sets homeomorphic to the ball and constructs quasiconformal maps from these sets to the unit ball, advancing geometric analysis in three dimensions.

Findings

Snowballs are homeomorphic to the unit ball.

Existence of quasiconformal maps from snowballs to the unit ball.

Extension of planar snowflake domain concepts to 3D.

Abstract

We introduce snowballs, which are compact sets in $\R^3$ homeomorphic to the unit ball. They are 3-dimensional analogs of domains in the plane bounded by snowflake curves. For each snowball $B$ a quasiconformal map $f\colon \R^3\to \R^3$ is constructed that maps $B$ to the unit ball.