Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

Creutz, Paul

doi:10.4171/rmi/1309

Rev. Mat. Iberoam.

2021

DOI: 10.4171/rmi/1309

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Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

Paul Creutz

Abstract: Our main result gives an improved bound on the filling areas of curves in Banach spaces which are not closed geodesics. As applications we show rigidity of Pu's classical systolic inequality and investigate the isoperimetric constants of normed spaces. The latter has further applications concerning the regularity of minimal surfaces in Finsler manifolds.

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2024

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Cited by 3 publications

References 43 publications

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Escape from compact sets of normal curves in subfinsler Carnot groups

Le Donne,

Paddeu

2024

ESAIM: COCV

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In the setting of subFinsler Carnot groups, we consider curves that satisfy the normal equation coming from the Pontryagin Maximum Principle. We show that, unless it is constant, each such a curve leaves every compact set, quantitatively. Namely, the distance between the points at time 0 and time t grows at least of the order t^(1/s), where s denotes the step of the Carnot group. In particular, in subFinsler Carnot groups there are no periodic normal geodesics.

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Escape from compact sets of normal curves in subfinsler Carnot groups

Le Donne,

Paddeu

2024

ESAIM: COCV

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The Branch Set of Minimal Disks in Metric Spaces

Creutz

Romney

2022

International Mathematics Research Notices

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We study the structure of the branch set of solutions to Plateau’s problem in metric spaces satisfying a quadratic isoperimetric inequality. In our 1st result, we give examples of spaces with isoperimetric constant arbitrarily close to the Euclidean isoperimetric constant $(4\pi )^{-1}$ for which solutions have large branch set. This complements recent results of Lytchak–Wenger and Stadler stating, respectively, that any space with Euclidean isoperimetric constant is a CAT($0$) space and solutions to Plateau’s problem in a CAT($0$) space have only isolated branch points. We also show that any planar cell-like set can appear as the branch set of a solution to Plateau’s problem. These results answer two questions posed by Lytchak and Wenger. Moreover, we investigate several related questions about energy-minimizing parametrizations of metric disks: when such a map is quasisymmetric, when its branch set is empty, and when it is unique up to a conformal diffeomorphism.

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An approach to metric space-valued Sobolev maps via weak* derivatives

Creutz,

Evseev

2024

Analysis and Geometry in Metric Spaces

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We give a characterization of metric space-valued Sobolev maps in terms of weak* derivatives. More precisely, we show that Sobolev maps with values in dual-to-separable Banach spaces can be defined in terms of classical weak derivatives in a weak* sense. Since every separable metric space X X embeds isometrically into ℓ ∞ {\ell }^{\infty } , we conclude that Sobolev maps with values in X X can be characterized by postcomposition with such embedding and the mentioned weak gradients. A slight variation on our definition was proposed previously by Hajłasz and Tyson. However, we show that their definition does not work in the sense that for technical reasons the arising Sobolev space is essentially empty.

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Rigidity of the Pu inequality and quadratic isoperimetric constants of normed spaces

Cited by 3 publications

References 43 publications

Escape from compact sets of normal curves in subfinsler Carnot groups

Escape from compact sets of normal curves in subfinsler Carnot groups

The Branch Set of Minimal Disks in Metric Spaces

An approach to metric space-valued Sobolev maps via weak* derivatives

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