S. Eriksson-Bique

Non-smooth sub-Riemannian manifolds with non-integer dimension and a Poincaré inequality

Sylvester Eriksson-Bique, Enrico Le Donne

For every integer $d \geq 3$ and real number $Q \in (d, \frac{d ( d - 1 ) + 2}{2}]$ , we construct a $Q$ -Ahlfors regular metric measure manifold $(M, d, μ)$ of topological d…

On universal elements for doubling geodesic trees

Sylvester Eriksson-Bique, Manisha Garg

For $n \geq 3$ and $c \in (0, 1)$ , let $G T (n, c)$ denote the class of geodesic metric trees of valence at most $n$ whose branch points are uniformly relatively separated with c…

Embeddability and rectifiability of Lipschitz differentiability spaces

Ivan Caamano, Sylvester Eriksson-Bique, Elefterios Soultanis

We prove that Lipschitz differentiability spaces which bi-Lipschitz embed into an RNP-space are countably rectifiable. In contrast to earlier methods of Cheeger and Kleiner, our ap…

Quantitative non-embeddability theorems and metric embeddings of slit carpets

Sylvester Eriksson-Bique, Niilo Joutsenlahti

We study the bi-Lipschitz embedding problem for a class of metric spaces called slit carpets. First we show that the $n$ th stage $M_{n}$ of the standard slit carpet of Meren…

Cartesian products of Sierpiński carpets do not attain their conformal dimension

Riku Anttila, Sylvester Eriksson-Bique

It is a long-standing open question to determine whether the Sierpiński carpet attains its conformal dimension or not. While this problem remains unresolved, we prove that Cartesia…

Conformal dimension and its attainment on self-similar Laakso-type fractal spaces

Riku Anttila, Sylvester Eriksson-Bique, Lassi Rainio

A general construction of Laakso-type fractal spaces was recently introduced by the first two authors. In this paper, we establish a simple condition characterizing when the Ahlfor…

math.MG2025

Quasisymmetric mappings on two variants of fractal percolation

Roope Anttila, Sylvester Eriksson-Bique, Aleksi Pyörälä

We study quasisymmetric maps on two variants of the classical fractal percolation model: the fat and dense fractal percolations. We show that, almost surely conditioned on non-exti…

math.MG2025

Construction of self-similar energy forms and singularity of Sobolev spaces on Laakso-type fractal spaces

Riku Anttila, Sylvester Eriksson-Bique, Ryosuke Shimizu

We construct self-similar $p$ -energy forms $E_{p}$ on a rich class of \emph{Laakso-type fractal spaces} and study the properties of the associated Sobolev spaces $\mathscr{…

math.MG2025

Duality for the gradient of a $p$ -harmonic function and the existence of gradient curves

Sylvester Eriksson-Bique, Saara Sarsa

Every convex optimization problem has a dual problem. The $p$ -Dirichlet problem in metric measure spaces is an optimization problem whose solutions are $p$ -harmonic functions. What…

math.MG2024

Analytically one-dimensional planes and the Combinatorial Loewner Property

Guy C. David, Sylvester Eriksson-Bique

It is a major problem in analysis on metric spaces to understand when a metric space is quasisymmetric to a space with strong analytic structure, a so-called Loewner space. A conje…

math.MG2024

The Combinatorial Loewner Property and super-multiplicativity inequalities for symmetric self-similar metric spaces

Riku Anttila, Sylvester Eriksson-Bique

This paper introduces a general construction of self-similar metric spaces as limits of discrete graphs. Our framework produces many classical examples, such as the Sierpiński carp…

math.MG2024★ 1 cited

On Constructions of Fractal Spaces Using Replacement and the Combinatorial Loewner Property

Riku Anttila, Sylvester Eriksson-Bique

The combinatorial Loewner property was introduced by Bourdon and Kleiner as a quasisymmetrically invariant substitute for the Loewner property for general fractals and boundaries o…

Sharp Hausdorff content estimates for accessible boundaries of domains in metric measure spaces of controlled geometry

Sylvester Eriksson-Bique, Ryan Gibara, Riikka Korte +1

We give a sharp Hausdorff content estimate for the size of the accessible boundary of any domain in a metric measure space of controlled geometry, i.e., a complete metric space equ…

Equality of different definitions of conformal dimension for quasiself-similar and CLP spaces

Sylvester Eriksson-Bique

We prove that for a quasiself-similar and arcwise connected compact metric space all three known versions of the conformal dimension coincide: the conformal Hausdorff dimension, co…

Traces of Newton-Sobolev functions on the visible boundary of domains in doubling metric measure spaces supporting a $p$ -Poincaré inequality

Sylvester Eriksson-Bique, Ryan Gibara, Riikka Korte +1

We consider the question of whether a domain with uniformly thick boundary at all locations and at all scales has a large portion of its boundary visible from the interior; here, "…

Density of continuous functions in Sobolev spaces with applications to capacity

Sylvester Eriksson-Bique, Pietro Poggi-Corradini

We show that capacity can be computed with locally Lipschitz functions in locally complete and separable metric spaces. Further, we show that if $(X, d, μ)$ is a locally complete and…

math.MG2022

A New Hausdorff Content Bound for Limsup Sets

Sylvester Eriksson-Bique

We give a new Hausdorff content bound for limsup sets, which is related to Falconer's sets of large intersection. Falconer's sets of large intersection satisfy a content bound for…

Nilpotent groups and biLipschitz embeddings into $L^{1}$

Sylvester Eriksson-Bique, Chris Gartland, Enrico Le Donne +2

We prove that if a simply connected nilpotent Lie group quasi-isometrically embeds into an $L^{1}$ space, then it is abelian. We reach this conclusion by proving that every Carnot gr…

Isoperimetric and Poincaré inequalities on non-self-similar Sierpiński sponges: the borderline case

Sylvester Eriksson-Bique, Jasun Gong

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…

Bi-Lipschitz embeddings of quasiconformal trees

Guy C. David, Sylvester Eriksson-Bique, Vyron Vellis

A quasiconformal tree is a doubling metric tree in which the diameter of each arc is bounded above by a fixed multiple of the distance between its endpoints. In this paper we show…

math.MG2021★ 10 cited

Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential

Sylvester Eriksson-Bique, Elefterios Soultanis

We represent minimal upper gradients of Newtonian functions, in the range $1 \leq p < \infty$ , by maximal directional derivatives along "generic" curves passing through a given point,…