Volume 10, Issue 1

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 provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie–Yeroshkin and Kuwae–Li.

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 under suitable bounds on convexity of Ω and Ricci curvature in Ω .

Every integral current in a locally compact metric space X can be approximated by a Lipschitz chain with respect to the normal mass, provided that Lipschitz maps into X can be extended slightly.

The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any 0 < β < α, any compact metric space X of Hausdorff dimension α contains a subset which is biLipschitz equivalent to an ultrametric and has Hausdorff dimension at least β. In this note we present a simple proof of the ultrametric skeleton theorem in doubling spaces using Bartal’s Ramsey decompositions [Bartal 2021]. The same general approach is also used to answer a question of Zindulka [Zindulka 2020] about the existence of “nearly ultrametric” subsets of compact spaces having full Hausdorff dimension.

A charge space ( X , 𝒜, µ ) is a generalisation of a measure space, consisting of a sample space X , a field of subsets 𝒜 and a finitely additive measure µ , also known as a charge . Properties a real-valued function on X may possess include T 1 -measurability and integrability. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations of T 1 -measurability and integrability for a bounded charge space ( µ ( X ) < ∞). These characterisations are convenient for analytic purposes; for example, they facilitate simple proofs that T 1 -measurability is equivalent to conventional measurability and integrability is equivalent to Lebesgue integrability, if ( X , 𝒜, µ ) is a complete measure space. New characterisations of equality almost everywhere of two real-valued functions on a bounded charge space are provided. Necessary and sufficient conditions for the function space L 1 ( X , 𝒜, µ ) to be a Banach space are determined. Lastly, the concept of completion of a measure space is generalised for charge spaces, and it is shown that under certain conditions, completion of a charge space adds no new equivalence classes to the quotient space ℒ p ( X , 𝒜, µ ).

The coarse Ricci curvature of graphs introduced by Ollivier as well as its modification by Lin–Lu– Yau have been studied from various aspects. In this paper, we propose a new transport distance appropriate for hypergraphs and study a generalization of Lin–Lu–Yau type curvature of hypergraphs. As an application, we derive a Bonnet–Myers type estimate for hypergraphs under a lower Ricci curvature bound associated with our transport distance. We remark that our transport distance is new even for graphs and worthy of further study.

In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov–Hausdorff distance.We then construct branching geodesics of the Gromov–Hausdorff distance continuously parameterized by the Hilbert cube, passing through or avoiding sets of all spaces satisfying some of the three properties shown above, and passing through the sets of all infinite-dimensional spaces and the set of all Cantor metric spaces. Our construction implies that for every pair of compact metric spaces, there exists a topological embedding of the Hilbert cube into the Gromov– Hausdorff space whose image contains the pair. From our results, we observe that the sets explained above are geodesic spaces and infinite-dimensional.

Let (𝒳, d , μ ) be a non-homogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. Under assumption that θ and dominating function λ satisfy certain conditions, the authors prove that fractional type Marcinkiewicz integral operator M˜ \tilde M α,lρ,q associated with θ -type generalized fractional kernel is bounded from the generalized Morrey space ℒ r,ϕp/r,κ ( μ ) into space ℒ p,ϕ,κ ( μ ), and bounded from the Lebesgue space L r ( μ ) into space L p ( μ ). Furthermore, the boundedness of commutator M˜ \tilde M α,l,ρq,b generated by b∈RBMO˜(μ) b \in \widetilde {RBMO}\left( \mu \right) and the M˜ \tilde M α,l,ρq,b on space ℒ p ( μ ) and on space ℒ p , ϕ , κ ( μ ) is also obtained.

We propose a model for a growth competition between two subsets of a Riemannian manifold. The sets grow at two different rates, avoiding each other. It is shown that if the competition takes place on a surface which is rotationally symmetric about the starting point of the slower set, then if the surface is conformally equivalent to the Euclidean plane, the slower set remains in a bounded region, while if the surface is nonpositively curved and conformally equivalent to the hyperbolic plane, both sets may keep growing indefinitely.

Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts. To understand their algebraic structure, it is useful to study some examples explicitly. In this work, we provide a list of low-dimensional stratified groups, express their Lie product, and present a basis of left-invariant vector fields, together with their respective left-invariant 1-forms, a basis of right-invariant vector fields, and some other properties. We exhibit all stratified groups in dimension up to 7 and also study some free-nilpotent groups in dimension up to 14.

We show that if there is a smooth function f on a Finsler n -space M satisfying Δ 2 f = − kfg Δ f for a positive constant k , then M is diffeomorphic with the n -sphere 𝕊 n , where g denotes the weighted Riemannian metric. Moreover, we further show that the manifold is isometric to a Finsler sphere if the Ricci curvature is bounded below by ( n − [one.tf]) k and the S -curvature vanishes.

The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary, but transforms the space into a bounded domain. We will show that if the original metric space is a uniform domain with respect to its completion, then the transformed space is also a uniform domain.

We study properties of twisted unions of metric spaces introduced in [Johnson, Lindenstrauss, and Schechtman 1986], and in [Naor and Rabani 2017]. In particular, we prove that under certain natural mild assumptions twisted unions of L 1 -embeddable metric spaces also embed in L 1 with distortions bounded above by constants that do not depend on the metric spaces themselves, or on their size, but only on certain general parameters. This answers a question stated in [Naor 2015] and in [Naor and Rabani 2017]. In the second part of the paper we give new simple examples of metric spaces such that their every embedding into L p , 1 ≤ p < ∞, has distortion at least 3, but which are a union of two subsets, each isometrically embeddable in L p . This extends the result of [K. Makarychev and Y. Makarychev 2016] from Hilbert spaces to L p -spaces, 1 ≤ p < ∞.

Liu and Lu [27] investigated a generalized Gauss curvature flow and obtained an even solution to the dual Orlicz-Minkowski problem under some appropriate assumptions. The present paper investigates a inverse Gauss curvature flow, and achieves the long-time existence and convergence of this flow via a different C 0 -estimate technique under weaker conditions. As an application of this inverse Gauss curvature flow, the present paper first arrives at a non-even smooth solution to the Orlicz Minkowski problem.

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré inequality yet have empty interior. These examples are formed from an iterative process that involves removing well-behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes families of curves, we include a new way of obtaining Poincaré inequalities through the use of relative isoperimetric inequalities, after Korte and Lahti. To do so, we further introduce the notion of of isoperimetric inequalities at given density levels and a way to iterate such inequalities. These tools are presented and apply to general metric measure measures. Our examples subsume the previous results of Mackay, Tyson, and Wildrick regarding non-self similar Sierpiński carpets, and extend them to many more general shapes as well as higher dimensions.

Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.