From volume cone to metric cone in the nonsmooth setting

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 Riccilimit spaces. Contents4 Variants 48

Noncompact RCD(0,N) spaces with linear volume growth

“…We remark that recently Gigli and De Philippis have proved 'volume cone implies metric cone' in the setting of RCD(K, N ) spaces in a preprint [30]. The proof is very involved and lengthy, and it relies on the tools and results recently developed in [26], [27], [28], [31] etc.…”

“…There are many other researches on RCD(K, N ) spaces, see [32], [38], [34], [51], [24], [35], [33], [30], and so on.…”

“…The proof is very involved and lengthy, and it relies on the tools and results recently developed in [26], [27], [28], [31] etc. As is indicated in [30], the techniques in [30] can be adapted to other setting. In Section 5, we will prove that the strongly minimal volume growth condition implies that there is a measure preserving map from (b −1 ((r 0 , ∞)), m) to some (X ′ ×R + , m ′ ⊗L 1 ) (see Proposition 5.3).…”

“…Hence Theorem 1.4 is essentially a 'volume cone implies metric cone'-type theorem in the setting of noncompact RCD(0, N ) spaces. Most of our remaining arguments in the proof of Theorem 1.4 follow the strategy in [30] closely. Since [30] has provided a complete and clear proof for the 'volume cone implies metric cone'-type theorem, in this paper we just highlight some detailed calculations due to different backgrounds and also concentrate on the geometric outcome for the case we are dealing with, the readers can refer to [30] for more details of the proof.…”

“…Most of our remaining arguments in the proof of Theorem 1.4 follow the strategy in [30] closely. Since [30] has provided a complete and clear proof for the 'volume cone implies metric cone'-type theorem, in this paper we just highlight some detailed calculations due to different backgrounds and also concentrate on the geometric outcome for the case we are dealing with, the readers can refer to [30] for more details of the proof. Since Theorem 1.4 is a kind of local version of splitting theorem on RCD(0, N ) spaces, some of the calculations here share the similarity to those in [28] [29].…”