Search results for "GEOMETRIA"
showing 10 items of 422 documents
Lipschitz Functions on Submanifolds of Heisenberg Groups
2022
Abstract We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds of Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. We also provide two applications: a Lusin-type approximation of Lipschitz functions on ${\mathbb {H}}$-rectifiable sets and a coarea formula on ${\mathbb {H}}$-rectifiable sets that completes the program started in [18].
Intrinsic rectifiability via flat cones in the Heisenberg group
2022
We give a geometric criterion for a topological surface in the first Heisenberg group to be an intrinsic Lipschitz graph, using planar cones instead of the usual open cones. peerReviewed
Testing the Sobolev property with a single test plan
2020
We prove that in a vast class of metric measure spaces (namely, those whose associated Sobolev space is separable) the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in the weak upper gradient inequality, it suffices to consider the negligible sets of a suitable Borel measure on curves, rather than the ones of the $p$-modulus. Moreover, on $\sf RCD$ spaces we can improve our result, showing that the test plan can be also chosen to be concentrated on an equi-Lipschitz family of curves.
Lipschitz Carnot-Carathéodory Structures and their Limits
2022
AbstractIn this paper we discuss the convergence of distances associated to converging structures of Lipschitz vector fields and continuously varying norms on a smooth manifold. We prove that, under a mild controllability assumption on the limit vector-fields structure, the distances associated to equi-Lipschitz vector-fields structures that converge uniformly on compact subsets, and to norms that converge uniformly on compact subsets, converge locally uniformly to the limit Carnot-Carathéodory distance. In the case in which the limit distance is boundedly compact, we show that the convergence of the distances is uniform on compact sets. We show an example in which the limit distance is not…
Monotonicity Formulas for Harmonic Functions in RCD(0,N) Spaces
2023
We generalize to the RCD(0,N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in RCD spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in RCD(K,N) spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric…
Semigenerated Carnot algebras and applications to sub-Riemannian perimeter
2021
This paper contributes to the study of sets of finite intrinsic perimeter in Carnot groups. Our intent is to characterize in which groups the only sets with constant intrinsic normal are the vertical half-spaces. Our viewpoint is algebraic: such a phenomenon happens if and only if the semigroup generated by each horizontal half-space is a vertical half-space. We call semigenerated those Carnot groups with this property. For Carnot groups of nilpotency step 3 we provide a complete characterization of semigeneration in terms of whether such groups do not have any Engel-type quotients. Engel-type groups, which are introduced here, are the minimal (in terms of quotients) counterexamples. In add…
Poincaré duality for open sets in Euclidean spaces
2016
Todistamme tässä työssä Poincarén dualiteetin Euklidisten avaruuksien avoimille joukoille. Annamme lyhyen johdatuksen differentiaaligeometriaan ja määrittelemme de Rham -kohomologian käsitteen. Itse Poincarén dualiteetin todistuksen aloitamme muutamalla aputuloksella. Näytämme ensin, että Poincarén dualiteetti pätee joukoille, jotka ovat diffeomorfisia avaruuteen R^n . Todis- tamme sitten Poincarén dualiteetin avointen joukkojen yhdisteille erinäisten lisäoletusten vallitessa. Tätä varten esittelemme Mayer–Vietoris jonon de Rham -kohomologialle. Lopulta näytämme Poincarén dualiteetin mielivaltaiselle avoimelle joukolle käytten Whitney-jakoa. Annamme myös havainnollistavan esimerkin Poincaré…
Hyperbolisen geometrian analyyttisiä malleja
2014
Tässä tutkielmassa esitellään viisi erilaista Riemannin monistoa, jotka toimivat hyperbolisen geometrian analyyttisinä malleina. Geometria voidaan karkeasti jakaa kahteen eri tapaukseen, euklidiseen ja epäeuklidiseen. Euklidisessa geometriassa pätee Eukleideen geometrian viides aksiooma, paralleeliaksiooma. Näin ei kuitenkaan ole laita hyperbolisessa geometriassa, joka luokitellaan epäeuklidiseksi geometriaksi. Analyyttisellä geometrialla taas tarkoitetaan koordinaatistoon sidottua geometriaa. Tässä tapauksessa nämä geometrian mallit ovat topologisia 2-ulotteisia pintoja euklidisessa avaruudessa. Lisäksi tutkielman malleissa hyödynnetään näille pinnoille määriteltyä sileää differentiaalirak…
Metric Rectifiability of H-regular Surfaces with Hölder Continuous Horizontal Normal
2022
Two definitions for the rectifiability of hypersurfaces in Heisenberg groups Hn have been proposed: one based on H-regular surfaces and the other on Lipschitz images of subsets of codimension-1 vertical subgroups. The equivalence between these notions remains an open problem. Recent partial results are due to Cole–Pauls, Bigolin–Vittone, and Antonelli–Le Donne. This paper makes progress in one direction: the metric Lipschitz rectifiability of H-regular surfaces. We prove that H-regular surfaces in Hn with α-Hölder continuous horizontal normal, α>0, are metric bilipschitz rectifiable. This improves on the work by Antonelli–Le Donne, where the same conclusion was obtained for C∞-surfaces. In…
Tensor tomography on surfaces
2013
We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case. peerReviewed