arXiv subjects

Denis Marti

Publications and source records attributed to Denis Marti.

6 recordsLinked to original sources

arXiv2026

Systolic inequalities for metric surfaces via filling minimality

We prove optimal systolic inequalities for length spaces homeomorphic to a torus of genus one or a real projective plane. In both cases, the optimal constant coincides with the constant from the (reversible) Finsler setting. This generalizes the classical results for Riemannian and Finsler surfaces. The proof of the inequality for the torus relies on an analysis of the asymptotic volume growth of the universal cover, together with a strengthening of the previously known area minimality of two-dimensional normed planes. For the inequality of the real projective plane, we similarly extend a minimality result of hemispheres. These results build upon works of Burago-Ivanov and Ivanov, respectively. In their proofs we apply recent uniformization theorems for metric disks and the theory of area-minimizing disks in metric spaces due to Lytchak-Wenger.

math.MG↗

arXiv2026

The metric fundamental class of non-orientable manifolds and manifolds with boundary

We introduce the metric fundamental class for metric spaces that are homeomorphic to compact, non-orientable, smooth manifolds with (possibly empty) boundary. This is an integer rectifiable current that provides an analytic representation of the topological fundamental class of the space. Under certain weak geometric conditions, we show the existence of such a current, extending earlier results for orientable, closed manifolds obtained in collaboration with Basso and Wenger. As an application, we present new rectifiability results.

math.MG↗

arXiv2026

One-dimensional and codimension one homology of metric manifolds

We compare singular homology and homology via integral currents in metric spaces that are homeomorphic to smooth manifolds. For such spaces, we provide sufficient conditions that guarantee the existence of a surjective homomorphism from the codimension one homology group via integral currents to the codimension one singular homology group. Moreover, we show that a one-dimensional isoperimetric inequality for integral currents implies that the one-dimensional homology groups coincide.

math.MG↗

arXiv2025

The Lipschitz-volume rigidity problem for metric manifolds

We prove a Lipschitz-volume rigidity result for $1$-Lipschitz maps of non-zero degree between metric manifolds (metric spaces homeomorphic to a closed oriented manifold) and Riemannian manifolds. The proof is based on degree theory and recent developments of Lipschitz-volume rigidity for integral currents.

math.DG↗

arXiv2024

Characterization of metric spaces with a metric fundamental class

We consider three conditions on metric manifolds with finite volume: (1) the existence of a metric fundamental class, (2) local index bounds for Lipschitz maps, and (3) Gromov--Hausdorff approximation with volume control by bi-Lipschitz manifolds. Condition (1) is known for metric manifolds satisfying the LLC condition by work of Basso--Marti--Wenger, while (3) is known for metric surfaces by work of Ntalampekos--Romney. We prove that for metric manifolds with finite Nagata dimension, all three conditions are equivalent and that without assuming finite Nagata dimension, (1) implies (2) and (3) implies (1). As a corollary we obtain a generalization of the approximation result of Ntalampekos--Romney to metric manifolds of dimension $n\ge 2$, which have the LLC property and finite Nagata dimension.

math.MG↗

arXiv2023

Geometric and analytic structures on metric spaces homeomorphic to a manifold

We study metric spaces homeomorphic to a closed oriented manifold from both geometric and analytic perspectives. We show that such spaces (which are sometimes called metric manifolds) admit a non-trivial integral current without boundary, provided they satisfy some weak assumptions. The existence of such an object should be thought of as an analytic analog of the fundamental class of the space and can also be interpreted as giving a way to make sense of Stokes' theorem in this setting. Using our existence result, we establish that Riemannian manifolds are Lipschitz-volume rigid among certain metric manifolds and we show the validity of (relative) isoperimetric inequalities in metric $n$-manifolds that are Ahlfors $n$-regular and linearly locally contractible. The former statement is a generalization of a well-known Lipschitz-volume rigidity result in Riemannian geometry and the latter yields a relatively short and conceptually simple proof of a deep theorem of Semmes about the validity of Poincar\'e inequalities in these spaces. Finally, as a further application, we also give sufficient conditions for a metric manifold to be rectifiable.

math.MG↗