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We show that the ultralimit of a bounded sequence of Lipschitz maps into pointed metric spaces extends naturally to $p$-bounded sequences of Sobolev maps and that this ultralimit for Sobolev maps enjoys desirable properties. We use this to prove the stability of Dehn functions under ultraconvergence of pointed length spaces, thus resolving a problem posed by several researchers in the field. As an application, we obtain a simpler proof of a recent result of Stadler--Wenger, previously proved in the locally compact case by Lytchak--Wenger, characterizing spaces of curvature bounded above by $κ$ via an isoperimetric inequality for curves.
In this paper we study horizontal curvatures for surfaces embedded in three-dimensional contact sub-Riemannian Lie groups. Using a Riemannian approximation scheme, we derive explicit formulas for horizontal Gauss curvature, horizontal mean curvature, and symplectic distortion for surfaces embedded in three dimensional Lie groups with a sub-Riemannian structure obtained by a contact form. We focus on two primary examples: the Heisenberg group and the affine-additive group. We classify surfaces of revolution within these groups that exhibit constant horizontal curvatures, often expressing their profiles through elementary or elliptic integrals.
This work introduces a surface observable for nonabelian four-dimensional BF theory with a cosmological term. The surface observable yields new 2-knot invariants that may extend beyond known examples such as the Alexander invariant. By BV pushforward, the surface observable induces an electric observable in nonabelian Yang–Mills theory, offering a concrete realization of ’t Hooft operators. An application to self-dual Yang–Mills theory is also discussed.
We provide bounds on the dimension of the abnormal set for rank 2 polarizations on metabelian Lie groups, establishing the Sard property for the end-point map of such groups. We also obtain bounds for the dimension of the Goh-abnormal set for metabelian Lie groups where the codimension of the derived subgroup is at most 2, with no assumption on the rank of the polarization. We thus infer that these polarized groups, equipped with sub-Riemannian structures, satisfy the minimizing Sard property.
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.
