arXiv subjects

Elefterios Soultanis

Publications and source records attributed to Elefterios Soultanis.

27 records · Page 2Linked to original sources

arXiv2020

Area minimizing surfaces in homotopy classes in metric spaces

We introduce and study a notion of relative 1-homotopy type for Sobolev maps from a surface to a metric space spanning a given collection of Jordan curves. We use this to establish the existence and local Hölder regularity of area minimizing surfaces in a given relative 1-homotopy class in proper geodesic metric spaces admitting a local quadratic isoperimetric inequality. If the underlying space has trivial second homotopy group then relatively 1-homotopic maps are relatively homotopic. We also obtain an analog for closed surfaces in a given 1-homotopy class. Our theorems generalize and strengthen results of Lemaire, Jost, Schoen-Yau, and Sacks-Uhlenbeck.

math.DG↗

arXiv2020

Maximal metric surfaces and the Sobolev-to-Lipschitz property

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 Lytchak--Wenger, which satisfies a related maximality condition.

math.MG↗

arXiv2019

Metric currents and polylipschitz forms

We construct, for a locally compact metric space $X$, a space of polylipschitz forms $\barΓ^*_c(X)$, which is a pre-dual for the space of metric currents of $\mathscr{D}_*(X)$ Ambrosio and Kirchheim. These polylipschitz forms may be seen as a substitute of differential forms in the metric setting.

math.MG↗

arXiv2019

Pull-back of metric currents and homological boundedness of BLD-elliptic spaces

Using the duality of metric currents and polylipschitz forms, we show that a BLD-mapping $f\colon X\to Y$ between oriented cohomology manifolds $X$ and $Y$ induces a pull-back operator $f^\ast \colon M_{k,loc}(Y) \to M_{k,loc}(X)$ 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 $f_\ast \colon M_{k,loc}(X)\to M_{k,loc}(Y)$. 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 $\mathbb{R}^n \to X$.

math.MG↗

arXiv2018

Infinitesimal Hilbertianity of locally CAT($κ$)-spaces

We show that, given a metric space $(Y,d)$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $μ$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,μ)$ is infinitesimally Hilbertian, i.e. the Sobolev space $W^{1,2}(Y,d,μ)$ 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 $x\in Y$ is the tangent cone at $x$ of $Y$. The conclusion then follows from the fact that for every $x\in Y$ such a cone is a CAT(0)-space and, as such, has a Hilbert-like structure.

math.MG↗

arXiv2018

Differential of metric valued Sobolev maps

We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is $\mathbb{R}$.

math.FA↗

arXiv2017

Path and quasihomotopy for Sobolev maps between manifolds

We study the relationship between quasihomotopy and path homotopy for Sobolev maps between manifolds. We employ singular integrals on manifolds to show that, in the critical exponent case, path homotopy implies quasihomotopy - and observe the rather surprising fact that $n$-quasihomotopic maps need not be path homotopic. We also study the case where the target is an aspherical manifold, e.g. a manifold with nonpositive sectional curvature, and the contrasting case of the target being a sphere.

math.FA↗

arXiv2016

Homotopy classes of Newtonian spaces

We study notions of homotopy in the Newtonian space $N^{1,p}(X;Y)$ of Sobolev type maps between metric spaces. After studying the properties and relations of two different notions we prove a compactness result for sequences in homotopy classes with controlled homotopies.

math.MG↗

arXiv2016

Existence of $p$-energy minimizers in homotopy classes and lifts of Newtonian maps

We 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.

math.DG↗