scholarly journals Area minimizing surfaces in homotopy classes in metric spaces

2021 ◽  
Author(s):  
Elefterios Soultanis ◽  
Stefan Wenger
Keyword(s):  
Metric Spaces ◽  
Homotopy Classes ◽  
Area Minimizing

scholarly journals Energy and area minimizers in metric spaces

2017 ◽  
Vol 10 (4) ◽  
pp. 407-421 ◽  
Author(s):  
Alexander Lytchak ◽  
Stefan Wenger
Keyword(s):  
Metric Spaces ◽  
Classical Case ◽  
Plateau Problem ◽  
Hölder Exponents ◽  
Area Minimizing ◽  
Definition Of ◽  
Regularity Results ◽  
Quadratic Isoperimetric Inequality ◽  
Energy Minimizers ◽  
Quasi Convexity

AbstractWe show that in the setting of proper metric spaces one obtains a solution of the classical 2-dimensional Plateau problem by minimizing the energy, as in the classical case, once a definition of area has been chosen appropriately. We prove the quasi-convexity of this new definition of area. Under the assumption of a quadratic isoperimetric inequality we establish regularity results for energy minimizers and improve Hölder exponents of some area-minimizing discs.

scholarly journals Area Minimizing Discs in Metric Spaces

2016 ◽  
Vol 223 (3) ◽  
pp. 1123-1182 ◽  
Author(s):  
Alexander Lytchak ◽  
Stefan Wenger
Keyword(s):  
Metric Spaces ◽  
Area Minimizing

Proximate topology and shape theory

1995 ◽  
Vol 125 (3) ◽  
pp. 595-615 ◽  
Author(s):  
Zvonko Čerin
Keyword(s):  
Metric Spaces ◽  
Topological Spaces ◽  
Shape Theory ◽  
Homotopy Classes ◽  
New Approach ◽  
Compact Metric Spaces ◽  
Shape Category

Most of the development of shape theory was in the so-called outer shape theory, where the shape of spaces is described with the help of some outside objects.This paper belongs to the so-called inner shape theory, in which the shape of spaces is described intrinsically without the use of any outside gadgets. We give a description of shape theory that does not need absolute neighbourhood retracts. We prove that the category ℋN whose objects are topological spaces and whose morphisms are proximate homotopy classes of proximate nets is naturally equivalent to the shape category h. The description of the category ℋN for compact metric spaces was given earlier by José M. R. Sanjurjo. We also give three applications of this new approach to shape theory.

scholarly journals Dehn functions and Hölder extensions in asymptotic cones

2020 ◽  
Vol 2020 (763) ◽  
pp. 79-109
Author(s):  
Alexander Lytchak ◽  
Stefan Wenger ◽  
Robert Young
Keyword(s):  
Metric Spaces ◽  
Broad Class ◽  
Dehn Function ◽  
Closed Curves ◽  
Asymptotic Cones ◽  
Sobolev Mappings ◽  
Area Minimizing ◽  
Existence And Regularity Results ◽  
Definition Of ◽  
Regularity Results

AbstractThe Dehn function measures the area of minimal discs that fill closed curves in a space; it is an important invariant in analysis, geometry, and geometric group theory. There are several equivalent ways to define the Dehn function, varying according to the type of disc used. In this paper, we introduce a new definition of the Dehn function and use it to prove several theorems. First, we generalize the quasi-isometry invariance of the Dehn function to a broad class of spaces. Second, we prove Hölder extension properties for spaces with quadratic Dehn function and their asymptotic cones. Finally, we show that ultralimits and asymptotic cones of spaces with quadratic Dehn function also have quadratic Dehn function. The proofs of our results rely on recent existence and regularity results for area-minimizing Sobolev mappings in metric spaces.

scholarly journals Area minimizing surfaces of bounded genus in metric spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Martin Fitzi ◽  
Stefan Wenger
Keyword(s):  
Isoperimetric Inequality ◽  
Riemannian Manifolds ◽  
Metric Spaces ◽  
Classical Problem ◽  
Regularity Of Solutions ◽  
Finite Collection ◽  
Jordan Curves ◽  
Area Minimizing ◽  
Quadratic Isoperimetric Inequality ◽  
Special Case

AbstractThe Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

Crystallography in two-dimensional metric spaces*

1969 ◽  
Vol 130 (1-6) ◽  
pp. 277-303 ◽  
Author(s):  
Aloysio Janner ◽  
Edgar Ascher
Keyword(s):  
Metric Spaces ◽  
Two Dimensional

Solution of Volterra integral inclusion in b-metric spaces via new fixed point theorem

2016 ◽  
Vol 2017 (1) ◽  
pp. 17-30 ◽  
Author(s):  
Muhammad Usman Ali ◽  
◽  
Tayyab Kamran ◽  
Mihai Postolache ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Metric Spaces ◽  
Integral Inclusion
Sign in / Sign up

Export Citation Format

Share Document