1,720,974 research outputs found
Infinitesimal Hilbertianity of Locally CAT(κ)-Spaces
No full textWe show that, given a metric space (Y,d)(Y,d) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure μμ on YY giving finite mass to bounded sets, the resulting metric measure space (Y,d,μ)(Y,d,μ) is infinitesimally Hilbertian, i.e. the Sobolev space W1,2(Y,d,μ)W1,2(Y,d,μ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at x∈Yx∈Y is the tangent cone at x of YY. The conclusion then follows from the fact that for every x∈Yx∈Y such a cone is a CAT(0)CAT(0) space and, as such, has a Hilbert-like structure.peerReviewe
Existence of p-energy minimizers in homotopy classes and lifts of Newtonian maps
Full text linkWe study the notion of p-quasihomotopy in Newtonian classes of mappings and link it to questions concerning lifts of Newtonian maps, under the assumption that the target space is nonpositively curved. Using this connection we prove that every p-quasihomotopy class of Newtonian maps contains a minimizer of the p-energy if the target has hyperbolic fundamental group.Peer reviewe
Kvasikonformiset ja p-energian minimoivat kuvaukset metristen avaruuksien välillä
Full text linkThis dissertation studies classical questions in the field of geometric analysis in the context of metric spaces.
The dissertation is comprised of three research articles. The first is on the connection of quasiconformal maps and the quasihyperbolic metric. The remaining two concern notions of homotopy classes of Sobolev type maps between metric spaces, comparison with the manifold case, and the existence of minimizers of a p-energy in these homotopy classes.
The unifying theme of all three articles is analysis on metric spaces. That is, all three papers deal with questions concerning maps between metric spaces. The particular type of metric spaces involved is generally referred to as PI-spaces.
PI-spaces satisfy conditions allowing one to extend a large part of classical first order calculus, such as the theory of Sobolev maps and, á posteriori, differentiability of Lipschitz functions.Tässä väitöskirjassa klassisia geometrisen analyysin tutkimuskysymyksiä tarkastellaan uudessa, metristen avaruuksien kontekstissa.
Väitöskirja koostuu kolmesta tutkimusartikkelista. Näistä ensimmäinen käsittelee kvasikonformikuvausten ja kvasihyperbolisen metriikan yhteyttä. Toiset kaksi käsittelevät Sobolev-tyyppisten kuvausten homotopialuokkia metrisissä avaruuksissa, vertailua klassiseen monistojen teoriaan, ja p-energian minimoijien olemassaoloa mainituissa homotopialuokissa.
Artikkeleiden yhteinen teema on analyysi metrisissä avaruuksissa. Kaikki kolme tutkimusasetelmaa koskevat metristen avaruuksien välisten kuvauksien ominaisuuksia. Kyseessä olevia metrisiä avaruuksia kutsutaan usein PI-avaruuksiksi.
PI-avaruudet ovat kattava yleistys avaruuksille, joille klassiset ensimmäisen kertaluvun differentiaalilaskennan periaatteet ovat voimassa. Niille voidaan määritellä esimerkiksi Sobolev-avaruuksia, ja todistaa Lipschitz-funktioiden differentioituvuuslauseita.ei saavutettav
Generalized products and Lorentzian length spaces
Full text linkWe construct a Lorentzian length space with an orthogonal splitting on a
product of an interval and a metric space, and use this framework
to consider the relationship between metric and causal geometry, as well as
synthetic time-like Ricci curvature bounds.
The generalized Lorentzian product naturally has a Lorentzian length
structure but can fail the push-up condition in general. We recover the push-up
property under a log-Lipschitz condition on the time variable and establish
sufficient conditions for global hyperbolicity. Moreover we formulate time-like
Ricci curvature bounds without push-up and regularity assumptions, and obtain a
partial rigidity of the splitting under a strong energy condition.Comment: 31 pages. Comments are welcome
Generalized products and Lorentzian length spaces
No full textWe construct a Lorentzian length space with an orthogonal splitting on a product I×X of an interval and a metric space, and use this framework to consider the relationship between metric and causal geometry, as well as synthetic time-like Ricci curvature bounds. The generalized Lorentzian product naturally has a Lorentzian length structure but can fail the push-up condition in general. We recover the push-up property under a log-Lipschitz condition on the time variable and establish sufficient conditions for global hyperbolicity. Moreover we formulate time-like Ricci curvature bounds without push-up and regularity assumptions, and obtain a partial rigidity of the splitting under a strong energy condition.peerReviewe
Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces
Full text linkUsing the duality of metric currents and polylipschitz forms, we show that a BLD-mappingf : X -> Y between oriented cohomology manifolds X and Y induces a pull-back operator f * : M-k,M-loc(Y) -> M-k,M-loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push forward f(*): M-k,M-loc(X) -> M-k,M-loc(Y). As an application we obtain a non-smooth version of the cohomological boundedness theorem of Bonk and Heinonen for locally Lipschitz contractible cohomology n-manifolds X admitting a BLD-mapping R-n -> X.Peer reviewe
Metric currents and polylipschitz forms
Full text linkWe construct, for a locally compact metric space X, a space of polylipschitz forms Gamma over bar c*(X), which is a pre-dual for the space of metric currents D*(X) of Ambrosio and Kirchheim. These polylipschitz forms may be seen as an analog of differential forms in the metric setting.Peer reviewe
Pull-Back of Metric Currents and Homological Boundedness of BLD-Elliptic Spaces
No full textUsing the duality of metric currents and polylipschitz forms, we show that a BLD-mapping f : X → Y between oriented cohomology manifolds X and Y induces a pull-back operator f* : Mk,loc(Y) → Mk,loc(X) between the spaces of metric k-currents of locally finite mass. For proper maps, the pull-back is a right-inverse (up to multiplicity) of the push-forward f* : Mk,loc(X) → Mk,loc(Y)
Maximal metric surfaces and the Sobolev-to-Lipschitz property
No full textWe 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 Lytchak–Wenger, which satisfies a related maximality condition
Curvewise characterizations of minimal upper gradients and the construction of a Sobolev differential
Full text linkWe represent minimal upper gradients of Newtonian functions, in the range 1≤p<∞, by maximal directional derivatives along “generic” curves passing through a given point, using plan-modulus duality and disintegration techniques. As an application we introduce the notion of p-weak charts and prove that every Newtonian function admits a differential with respect to such charts, yielding a linear approximation along p-almost every curve. The differential can be computed curvewise, is linear, and satisfies the usual Leibniz and chain rules.
The arising p-weak differentiable structure exists for spaces with finite Hausdorff dimension and agrees with Cheeger’s structure in the presence of a Poincaré inequality. In particular, it exists whenever the space is metrically doubling. It is moreover compatible with, and gives a geometric interpretation of, Gigli’s abstract differentiable structure, whenever it exists. The p-weak charts give rise to a finite-dimensional p-weak cotangent bundle and pointwise norm, which recovers the minimal upper gradient of Newtonian functions and can be computed by a maximization process over generic curves. As a result we obtain new proofs of reflexivity and density of Lipschitz functions in Newtonian spaces, as well as a characterization of infinitesimal Hilbertianity in terms of the pointwise norm.peerReviewe
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