Pré-Publication, Document De Travail Année : 2023

Uniqueness and non-uniqueness for the asymptotic Plateau problem in hyperbolic space

Résumé

We prove a number of results on the number of solutions to the asymptotic Plateau problem in H^3. In the direction of non-uniqueness, we construct an example of a quasicircle that is the asymptotic boundary of uncountably many pairwise distinct stable minimal disks. Moreover, we discuss criteria that ensure uniqueness. Given a Jordan curve Λ in the asymptotic boundary of H^3 , we show that uniqueness of the minimal surfaces with asymptotic boundary Λ is equivalent to uniqueness in the smaller class of stable minimal disks, and, when Λ is invariant by a Kleinian group, to uniqueness in the even smaller class of group invariant stable minimal disks. Finally, we show that if a quasicircle (or more generally, a Jordan curve of finite width) Λ is the asymptotic boundary of a minimal surface Σ with principal curvatures less than or equal to 1 in absolute value, then uniqueness holds. Contents

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Dates et versions

hal-04235517 , version 1 (11-10-2023)
hal-04235517 , version 2 (31-10-2025)

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Zheng Huang, Ben Lowe, Andrea Seppi. Uniqueness and non-uniqueness for the asymptotic Plateau problem in hyperbolic space. 2023. ⟨hal-04235517v1⟩
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