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Roger Züst

Publications and source records attributed to Roger Züst.

15 recordsLinked to original sources
arXiv2026

A calibration on $\mathbf R^{16}$ and Federer's product question

Building upon the construction of a Cayley calibration adapted to a complex structure, we introduce a calibration $Φ$ in $\bigwedge^8 \mathbf R^{16}$ with $|Φ^2| = 294$. This enables us to show that the product of two orthogonally supported calibrations is not necessarily a calibration, thereby providing a negative answer to a question posed by Federer. Dadok and Harvey developed a general method for constructing calibrations as outer products of two unit spinors in the Clifford algebra. We show that $Φ$ arises from the product of two spinors with norm $1$ and $\sqrt{2}$.

math.DG↗
arXiv2025

The Riemannian hemisphere is almost calibrated in the injective hull of its boundary

An exact differential two-form is constructed in the injective hull of the Riemannian circle, whose comass norm, defined via the inscribed Riemannian area on normed planes, is stationary at every point of the open hemisphere spanned by the circle. As a consequence, in any metric space, the induced Finsler mass of a two-dimensional Ambrosio-Kirchheim rectifiable current with boundary a Riemannian circle of length $2π$ admits a lower bound of $2π$ plus a second-order term in the Hausdorff distance to an isometric copy of the hemisphere. This estimate applies to all oriented Lipschitz surfaces spanning the circle, regardless of their topology, and thus offers positive evidence for Gromov's filling area conjecture.

math.DG↗
arXiv2020

Partial regularity of almost minimizing rectifiable G chains in Hilbert space

We adapt to an infinite dimensional ambient space E.R. Reifenberg's epiperimetric inequality and a quantitative version of D. Preiss' second moments computations to establish that the set of regular points of an almost mass minimizing rectifiable $G$ chain in $\ell_2$ is dense in its support, whenever the group $G$ of coefficients is so that $\{\|g\| : g \in G \}$ is discrete and closed.

math.DG↗
arXiv2020

Functions of bounded fractional variation and fractal currents

Extending the notion of bounded variation, a function $u \in L_c^1(\mathbb R^n)$ is of bounded fractional variation with respect to some exponent $α$ if there is a finite constant $C \geq 0$ such that the estimate \[ \biggl|\int u(x) \det D(f,g_1,\dots,g_{n-1})_x \, dx\biggr| \leq C\operatorname{Lip}^α(f) \operatorname{Lip}(g_1) \cdots \operatorname{Lip}(g_{n-1}) \] holds for all Lipschitz functions $f,g_1,\dots,g_{n-1}$ on $\mathbb R^n$. Among such functions are characteristic functions of domains with fractal boundaries and Hölder continuous functions. We characterize functions of bounded fractional variation as a certain subspace of Whitney's flat chains and as multilinear functionals in the setting of Ambrosio-Kirchheim currents. Consequently we discuss extensions to Hölder differential forms, higher integrability, an isoperimetric inequality, a Lusin type property and change of variables. As an application we obtain sharp integrability results for Brouwer degree functions with respect to Hölder maps defined on domains with fractal boundaries.

math.FA↗
arXiv2019

Space of signatures as inverse limits of Carnot groups

We formalize the notion of limit of an inverse system of metric spaces with $1$-Lipschitz projections having unbounded fibers. The purpose is to use sub-Riemannian groups for metrizing the space of signatures of rectifiable paths in Euclidean spaces, as introduced by Chen. The constructive limit space has the universal property in the category of pointed metric spaces with 1-Lipschitz maps. In the general setting some metric properties are discussed such as the existence of geodesics and lifts. The notion of submetry will play a crucial role. The construction is applied to the sequence of free Carnot groups of fixed rank $n$ and increasing step. In this case, such limit space is in correspondence with the space of signatures of rectifiable paths in $\mathbb R^n$. Hambly-Lyons's result on the uniqueness of signature implies that this space is a geodesic metric tree that brunches at every point with infinite valence. As a particular consequence we deduce that every path in $\mathbb R^n$ can be approximated by projections of some geodesics in some Carnot group of rank $n$, giving an evidence that the complexity of sub-Riemannian geodesics increases with the step.

math.MG↗
arXiv2019

Box-counting by Hölder's traveling salesman

We provide a sufficient Dini-type condition for a subset of a complete, quasiconvex metric space to be covered by a Hölder curve. This implies in particular that if the upper box-counting dimension of a set in a quasiconvex metric space is less or equal to $d \geq 1$, then for any $α< \frac{1}{d}$ the set can be covered by an $α$-Hölder curve. On the other hand, for each $1\leq d <2$ we give an example of a compact set $K$, in the plane, just failing the above Dini-type condition, with lower box-counting dimension equal to zero and upper box-counting dimension equal to $d$ that can not be covered by a countable collection of $\frac{1}{d}$-Hölder curves.

math.MG↗
arXiv2019

Distortion of spheres and surfaces in space

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in $\mathbb R^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $π/2$ is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.

math.MG↗
arXiv2017

Coefficient groups inducing nonbranched optimal transport

In this work we consider an optimal transport problem with coefficients in a normed Abelian group $G$, and extract a purely intrinsic condition on $G$ that guarantees that the optimal transport (or the corresponding minimum filling) is not branching. The condition turns out to be equivalent to the nonbranching of minimum fillings in geodesic metric spaces. We completely characterize finitely generated normed groups and finite-dimensional normed vector spaces of coefficients that induce nonbranching optimal transport plans. We also provide a complete classification of normed groups for which the optimal transport plans, besides being nonbranching, have acyclic support. This seems to initiate a new geometric classifications of certain normed groups. In the nonbranching case we also provide a global version of calibration, i.e. a generalization of Monge-Kantorovich duality.

math.MG↗
arXiv2016

Matchings in metric spaces, the dual problem and calibrations modulo 2

We show that for a metric space with an even number of points there is a 1-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the duality gives a version of global calibrations for 1-chains with coefficients in $\mathbb Z_2$. Finally we extend the results to infinite metric spaces and present a notion of "matching dimension" which arises naturally.

math.MG↗
arXiv2016

A solution of Gromov's Hölder equivalence problem for the Heisenberg group

We show that a map with Hölder exponent bigger than $1/2$ from a quasi-convex metric space with vanishing first Lipschitz homology into the Sub-Riemannian Heisenberg group factors through a tree. In particular, if the domain contains a disk, such a map can't be injective. This gives an answer to a question of Gromov for the simplest nontrivial case. The same tools allow to improve on a result of Borisov and it is shown that an isometric immersion of class $C^{1,α}$ of a Riemannian surface with positive Gauss curvature into $\mathbb{R}^3$ has bounded extrinsic curvature if $α> 1/2$.

math.MG↗
arXiv2015

Some results on maps that factor through a tree

We give a necessary and sufficient condition for a map defined on a simply-connected quasiconvex metric space to factor through a tree. In case the target is the Euclidean plane and the map is Hölder continuous with exponent bigger than 1/2, such maps can be characterized by the vanishing of some integrals over the winding number function. This in particular shows that if the target is the Heisenberg group equipped with the Carnot-Carathéodory metric and the Hölder exponent of the map is bigger than 2/3, the map factors through a tree.

math.MG↗
arXiv2013

Metric stability of trees and tight spans

In this note, we prove optimal extension results for roughly isometric relations between metric (R-)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov-Hausdorff (GH) distance, for certain metric spanning constructions: The GH distance of two metric trees spanned by some subsets is smaller than or equal to the GH distance of these sets. The GH distance of the injective hulls, or tight spans, of two metric spaces is at most twice the GH distance between themselves.

math.MG↗
arXiv2012

Some properties of Hölder surfaces in the Heisenberg group

It is a folk conjecture that for alpha > 1/2 there is no alpha-Hoelder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of R^2 into the Heisenberg group that is Hoelder continuous of order strictly greater than 1/2. The Heisenberg group here is equipped with its Carnot-Caratheodory distance. We show that, in the case that such a surface exists, it cannot be of essential bounded variation and it intersects some vertical line in at least a topological Cantor set.

math.MG↗
arXiv2010

Integration of Hölder forms and currents in snowflake spaces

For an oriented $n$-dimensional Lipschitz manifold $M$ we give meaning to the integral $\int_M f dg_1 \wedge ... \wedge dg_n$ in case the functions $f, g_1, >..., g_n$ are merely Hölder continuous of a certain order by extending the construction of the Riemann-Stieltjes integral to higher dimensions. More generally, we show that for $α\in (\frac{n}{n+1},1]$ the $n$-dimensional locally normal currents in a locally compact metric space $(X,d)$ represent a subspace of the $n$-dimensional currents in $(X,d^α)$. On the other hand, for $n \geq 1$ and $α\leq \frac{n}{n+1}$ the latter space consists of the zero functional only.

math.FA↗