## 5-Point CAT(0) Spaces after Tetsu Toyoda

Published in Analysis and Geometry in Metric Spaces

We give another proof of Toyoda’s theorem that describes 5-point subspaces in CAT(0) length spaces.

Published in Analysis and Geometry in Metric Spaces

We give another proof of Toyoda’s theorem that describes 5-point subspaces in CAT(0) length spaces.

Published in Analysis and Geometry in Metric Spaces

The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any 0

Published in Analysis and Geometry in Metric Spaces

Assuming that there exists a translating soliton u∞ with speed C in a domain Ω and with prescribed contact angle on ∂Ω, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to u∞ + Ct as t →∞. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings ...

Published in Analysis and Geometry in Metric Spaces

We establish the Bonnet–Myers theorem, Laplacian comparison theorem, and Bishop–Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with ϵ-range introduced in our previous paper, that p...

Published in Analysis and Geometry in Metric Spaces

We consider vector valued mappings defined on metric measure spaces with a measurable differentiable structure and study both approximations by nicer mappings and regular extensions of the given mappings when defined on closed subsets. Therefore, we propose a first approach to these problems, largely studied on Euclidean and Banach spaces during th...

Published in Analysis and Geometry in Metric Spaces

We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov’s book [4]. We obtain a complete answer for this question.

Published in Analysis and Geometry in Metric Spaces

In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell’s lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space un...

Published in Analysis and Geometry in Metric Spaces

We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains (Y, dY). We say that a metric space (Y, dY) is a quasiconformal Jordan domain if the completion ̄Y of (Y, dY) has finite Hausdorff 2-measure, the boundary ∂Y = ̄Y \ Y is homeomorphic to 𝕊1, and there exists a homeomorphism ϕ: 𝔻 →(Y, dY) that is quasiconformal in...

Published in Analysis and Geometry in Metric Spaces

In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricc...

Published in Analysis and Geometry in Metric Spaces

Let p(·) : ℝn → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝn introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces Hp(·)(Θ) via the radial grand maximal function and then obtain its atomic decomposition, which g...