Angles between Curves in Metric Measure Spaces

Han, Bang-Xian; Mondino, Andrea

doi:10.1515/agms-2017-0003

Cited by 5 publications

(3 citation statements)

References 42 publications

(69 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remark A.5. A minimal geodesic σ : [− , ] → X in X is said to be strongly nonbranching if a minimal geodesic η : [−δ, δ] → X for some δ < satisfy σ(0) = η(0) and ∠ σ η(0) = 0, then σ = η on [−δ, δ] (see [39,40] for the definition of angles). Then if any minimal geodesic in X is strongly nonbranching, then a Dirichlet domain is a metric measure fundamental domain.…”

Section: Definition A1 (Fundamental Domainmentioning

confidence: 99%

Singular Weyl’s law with Ricci curvature bounded below

Dai,

Honda,

Pan

et al. 2023

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

We establish two surprising types of Weyl’s laws for some compact RCD ⁡ ( K , N ) \operatorname {RCD}(K, N) /Ricci limit spaces. The first type could have power growth of any order (bigger than one). The other one has an order corrected by logarithm similar to some fractals even though the space is 2-dimensional. Moreover the limits in both types can be written in terms of the singular sets of null capacities, instead of the regular sets. These are the first examples with such features for RCD ⁡ ( K , N ) \operatorname {RCD}(K,N) spaces. Our results depend crucially on analyzing and developing important properties of the examples constructed in Pan and Wei [Geom. Funct. Anal. 32 (2022), pp. 676–685], showing them isometric to the α \alpha -Grushin halfplanes. Of independent interest, this also allows us to provide counterexamples to conjectures in Cheeger and Colding [J. Differential Geom. 46 (1997), pp. 406–480] and Kapovitch, Kell, and Ketterer [Math. Z. 301 (2022), pp. 3469–3502].

show abstract

Section: Definition A1 (Fundamental Domainmentioning

confidence: 99%

Singular Weyl’s law with Ricci curvature bounded below

Dai,

Honda,

Pan

et al. 2023

Trans. Amer. Math. Soc. Ser. B

View full text Add to dashboard Cite

show abstract

“…Alexandrov spaces [5], length spaces [6], topological spaces [7], angle spaces [8], proximity spaces [9], etc.…”

Section: Metric Spacesmentioning

confidence: 99%

Space as relation

Marcello¹

2022

Preprint

View full text Add to dashboard Cite

Here we will discuss the philosophical differences between an approach to the deep nature of physical space based on the concept of coordinates and one based on the concept of relation. The philosophical superiority of the second approach will be analysed and maintained, attempting to bring it to its extreme consequences. We will propose considering the concept of spatial dimension completely superfluous in favour of an idea of space as a pure relational structure, capable of including within itself the information necessary to define the properties of the things in the world.

show abstract

“…Even though in the splitting theorem on RCD * (0, N )-spaces established in [G13] the harmonic replacement were not needed (because a priori these spaces do not arise from an approximation), recently harmonic replacement have been considered in RCD-setting in [HM17]. It is worth pointing out that in the proof of harmonic replacement as above, the sharp Laplacian comparison theorem and the maximum principle play a key role.…”

Section: Corollary 45 (Stability Of Harmonic Functions) Letmentioning

confidence: 99%