Preservation of Bounded Geometry under Sphericalization and Flattening

The sphericalization of a metric space produces a bounded metric space from an unbounded metric space, while the flattening procedure produces an unbounded metric space from a bounded metric space. This corresponds to obtaining the Riemann sphere from the complex plane, and obtaining the complex plane from the Riemann sphere. In this paper, we show that sphericalization and flattening procedures on a complete metric measure space preserve properties such as Ahlfors regularity and doubling property. We also show that if the metric space has a doubling measure and is in addition quasiconvex and annular quasiconvex, then the sphericalization and flattening procedures preserve the property of supporting a p-Poincare inequality.