From volume cone to metric cone in the nonsmooth setting

TL;DR

This paper extends a classical geometric result to the broader context of RCD spaces, showing that a volume cone condition implies a metric cone structure, thus generalizing Cheeger-Colding's theorem to nonsmooth settings.

Contribution

The paper proves that in RCD spaces, a volume cone condition implies a metric cone structure, extending Cheeger-Colding's result to nonsmooth metric measure spaces.

Findings

Volume cone implies metric cone in RCD spaces

Generalization of Cheeger-Colding theorem to nonsmooth spaces

Advancement in understanding geometric structures of RCD spaces

Abstract

We prove that `volume cone implies metric cone' in the setting of RCD spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding valid in Ricci-limit spaces.