Maximal metric surfaces and the Sobolev-to-Lipschitz property
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Calculus of Variations
Maximal metric surfaces and the Sobolev-to-Lipschitz property Paul Creutz1 · Elefterios Soultanis2 Received: 7 February 2020 / Accepted: 31 July 2020 / Published online: 20 September 2020 © The Author(s) 2020
Abstract 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. Mathematics Subject Classification 30L10 · 49Q05 · 53B40
1 Introduction 1.1 Main result A celebrated result due to Bonk and Kleiner [3] states that an Ahlfors 2-regular metric sphere Z is quasisymetrically equivalent to the standard sphere if and only if it is linearly locally connected. Recently, it was shown in [35] that the quasisymmetric homeomorphism 2 (u ), and in this case is unique up u Z : S 2 → Z may be chosen to be of minimal energy E + Z 2 to a conformal diffeomorphism of S . The map u Z gives rise to a measurable Finsler structure on S 2 , defined by the approximate metric differential apmd u Z ; cf. [33] and Sect. 6 below. When Z is a smooth Finsler surface, the approximate metric differential carries all the metric information of Z . In the present
Communicated by J. Jost. The first author was partially supported by the DFG Grant SPP 2026. The second author was partially supported by the Vilho, Yrjö ja Kalle Väisalä Foundation (postdoc pool) and by the Swiss National Science Foundation Grant 182423.
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Elefterios Soultanis [email protected] Paul Creutz [email protected]
1
University of Cologne, Weyertal 86-90, 50931 Cologne, Germany
2
University of Fribourg, Chemin du Musee 23, CH-1700 Fribourg, Switzerland
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generality, however, apmd u Z is defined only almost everywhere and thus does not determine the length of every curve. Definition 1.1 Let Y and Z be Ahlfors 2-regular, linearly locally connected metric spheres. We say that Y and Z are analytically equivalent if there exist energy minimizing parametrizations u Y and u Z such that apmd u Y = apmd u Z
(1.1)
almost everywhere. By the aformentioned uniqueness result, analytic equivalence defines an equivalence relation on the class of linearly locally connected, Ahlfors 2-regular spheres. The main result of this paper states that the equivalence class of such a sphere contains a maximal representative, unique up to an isometry. Theorem 1.2 Let Z be a linearly locally connected, Ahlfors 2-regular sphere. Then there is Z which is analytically and bi-Lipschitz linearly locally connected, Ahlfors 2-regular sphere equivalent to Z and satisfies the following properties. (1) Sobolev-to-Lipschitz property: If f ∈ N 1,2 ( Z ) has weak upper gradient 1, then f has a 1-Lipschitz representative. (2)
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