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Université de Fribourg

Dehn functions and Hölder extensions in asymptotic cones

Lytchak, Alexander ; Wenger, Stefan ; Young, Robert

In: Journal für die reine und angewandte Mathematik, 2020, vol. 2020, no. 763, p. 79–109

The Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we...

Université de Fribourg

Self-avoiding walk on $\mathbb{Z}^{2}$ with Yang–Baxter weights : Universality of critical fugacity and 2-point function

Glazman, Alexander ; Manolescu, Ioan

In: Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 2020, vol. 56, no. 4, p. 2281–2300

We consider a self-avoiding walk model (SAW) on the faces of the square lattice Z2. This walk can traverse the same face twice, but crosses any edge at most once. The weight of a walk is a product of local weights: each square visited by the walk yields a weight that depends on the way the walk passes through it. The local weights are parametrised by angles θ∈[π3,2π3] and satisfy the...

Université de Fribourg

Maximal metric surfaces and the Sobolev-to-Lipschitz property

Creutz, Paul ; Soultanis, Elefterios

In: Calculus of Variations and Partial Differential Equations, 2020, vol. 59, no. 5, p. 177

We find maximal representatives within equivalence classes of metric spheres. For Ahlfors regular spheres these are uniquely characterized by satisfying the seemingly unrelated notions of Sobolev-to-Lipschitz property, or volume rigidity. We also apply our construction to solutions of the Plateau problem in metric spaces and obtain a variant of the associated intrinsic disc studied by...

Université de Fribourg

Nondefective stationary discs and 2-jet determination in higher codimension

Bertrand, Florian ; Meylan, Francine

In: The Journal of Geometric Analysis, 2020, p. -

We discuss the links between stationary discs, the defect of analytic discs, and 2-jet determination of CR automorphisms of generic nondegenerate real submanifolds of CN of class C4.

Université de Fribourg

Morrey’s 𝜖-conformality lemma in metric spaces

Fitzi, Martin ; Wenger, Stefan

In: Proceedings of the American Mathematical Society, 2020, vol. 148, no. 10, p. 4285–4298

We provide a simpler proof and slight strengthening of Morrey's famous lemma on $ \varepsilon $-conformal mappings. Our result more generally applies to Sobolev maps with values in a complete metric space, and we obtain applications to the existence of area minimizing surfaces of higher genus in metric spaces. Unlike Morrey's proof, which relies on the measurable Riemann mapping theorem, we...

Université de Fribourg

Open manifolds with non-homeomorphic positively curved souls

González-Álvaro, David ; Zibrowius, Marcus

In: Mathematical Proceedings of the Cambridge Philosophical Society, 2020, vol. 169, no. 2, p. 357–376

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively- curved manifolds that are tangentially homotopy equivalent but not homeomorphic, and we deduce that an open manifold may admit a pair of non-homeomorphic simply connected and positively-curved souls. Examples of such pairs are...

Université de Fribourg

Overcommuting pairs in groups and 3-manifolds bounding them

Liechti, Livio ; Marché, Julien

In: Journal of the London Mathematical Society, 2020, p. -

We introduce the notions of overcommutation and overcommutation length in groups, and show that these concepts are closely related to representations of the fundamental groups of 3-manifolds and their Heegaard genus. We give many examples including translations in the affine group of the line and provide upper bounds for the overcommutation length in SL2, related to the Steinberg relation.