Volume 7, Issue 1

We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved vertices. Those results seem to be the first assuming non-constant Bakry-Émery curvature assumptions on graphs.

Let G be a group and let S be a generating set of G . In this article,we introduce a metric d C on G with respect to S , called the cardinal metric.We then compare geometric structures of ( G , d C ) and ( G , d W ), where d W denotes the word metric. In particular, we prove that if S is finite, then ( G , d C ) and ( G , d W ) are not quasiisometric in the case when ( G , d W ) has infinite diameter and they are bi-Lipschitz equivalent otherwise. We also give an alternative description of cardinal metrics by using Cayley color graphs. It turns out that colorpermuting and color-preserving automorphisms of Cayley digraphs are isometries with respect to cardinal metrics.

We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness. Similarly to the previous work on the short-scale case, we show that this idleness function is concave and piecewise linear with at most 3 linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long-scale curvature for the Cartesian product of two regular graphs.

We use continuous and discrete unification to prove that standard triple bubbles in ℝ 2 and 𝕊 2 are the minimizers of perimeter, among all clusters (Definition 2.3) enclosing the same triple of areas. Unification defines a unified measurement that allows all configurations, regardless of areas, to compete together. Continuous unification proves that if a unified minimizer were better than expected, it would have to have at least one interior bubble component. Discrete unification proves there can only be one interior bubble and that it must be connected. This leaves only the “daisy” configurations: one interior bubble surrounded by an even number of “petals.” A more careful analysis also eliminates these, leaving only the standard triple bubbles as minimizers. The result on the sphere is new; the result in the plane is due to Wichiramala [11]. The double bubble in the sphere was done by Masters [6].

The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space of marked groups is uniformly a-T-menable. We furthermore extend this result to ones with other target spaces. By combining our main results with constructions of Osajda and Arzhantseva–Osajda, we construct two systems of markings of a certain sequence of finite groups with two opposite extreme behaviors of the resulting two disjoint unions: With respect to one marking, the space has property A. On the other hand, with respect to the other, the space does not admit fibred coarse embeddings into Banach spaces with non-trivial type (for instance, uniformly convex Banach spaces) or Hadamard manifolds; the Cayley limit group is, furthermore, non-exact.

We study a family of spheres with constant mean curvature (CMC) in the Riemannian Heisenberg group H 1 . These spheres are conjectured to be the isoperimetric sets of H 1 . We prove several results supporting this conjecture. We also focus our attention on the sub-Riemannian limit.

We provide a general strategy to construct multilinear inequalities of Brascamp–Lieb type on compact homogeneous spaces of Lie groups. As an application we obtain sharp integral inequalities on the real unit sphere involving functions with some degree of symmetry.

The aim of this note is to generalize to the class of non collapsed RCD( K , N ) metric measure spaces the volume bound for the effective singular strata obtained by Cheeger and Naber for non collapsed Ricci limits in [13]. The proof, which is based on a quantitative differentiation argument, closely follows the original one. As a simple outcome we provide a volume estimate for the enlargement of Gigli-DePhilippis’ boundary ([20, Remark 3.8]) of ncRCD( K , N ) spaces.

The theory of boundary regularity for p -harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p -Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

We show that if a CD ( K , n ) space ( X , d , f ℋ n ) with n ≥ 2 has curvature bounded above by κ in the sense of Alexandrov then f is constant.

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f * : M k,loc ( Y ) → M k,loc ( X ) between the spaces of metric k -currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f * : M k,loc ( X ) → M k,loc ( Y ). As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n -manifolds X admitting a BLD-mapping ℝ n → X .

In this short note, we prove that the stochastic order of Radon probability measures on any ordered topological space is antisymmetric. This has been known before in various special cases. We give a simple and elementary proof of the general result.