Isoperimetric curves on hyperbolic surfaces

Adams, Colin; Morgan, Frank

doi:10.1090/s0002-9939-99-04778-4

Cited by 17 publications

(16 citation statements)

References 3 publications

(2 reference statements)

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“…Thanks to [24,Theorem 18(c)] and [15], it is enough to prove the following generalization of [1] for the negatively curved exact model. Theorem 5.2.…”

Section: Isoperimetric Profilementioning

confidence: 99%

Existence of CMC-foliations in asymptotically cuspidal manifolds

Arezzo¹,

Corrales²

2018

Preprint

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“…Thanks to [24,Theorem 18(c)] and [15], it is enough to prove the following generalization of [1] for the negatively curved exact model. Theorem 5.2.…”

Section: Isoperimetric Profilementioning

confidence: 99%

Existence of CMC-foliations in asymptotically cuspidal manifolds

Arezzo¹,

Corrales²

2018

Preprint

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“…The eigenvalue λ 1 (M) is called the smallest non-zero eigenvalue of M. When M is non-compact, one defines λ(M) = inf f M grad(f ) 2 dVol n M f 2 dVol n where f runs over all C 1 (M). 1 Since M has finite volume, we require that M f dVol n = 0.…”

Section: Background and Motivationmentioning

confidence: 99%

“…To simplify our presentation, we define a horocusp neighborhood in S to be a neighborhood of an end of S which is covered by a horoball in H 2 . Adams and Morgan provide a more specific characterization for the isoperimetric problem for hyperbolic surfaces [1]: Theorem 5.1. (Adams and Morgan [1, Theorem 2.2]) Let S be a connected, geometrically finite hyperbolic surface.…”

Section: Cheeger Minimizers Of Hyperbolic Surfacesmentioning

confidence: 99%

“…Adams and Morgan prove that there exists a hyperbolic metric on the genus two surface such that all isoperimetric minimizers are disks, annuli, complements of disks, or complements of annuli [1,Theorem 3.1]. Since a Cheeger minimizer must also be an isoperimetric minimizer, it follows from Lemmas 5.6 and 5.7 that their surface must either have a single annulus or an embedded disk at every point having area equal to half the area of the surface.…”

Section: Cheeger Minimizers Of Hyperbolic Surfacesmentioning

confidence: 99%

“…Adams and Morgan give a complete classification of solutions to the isoperimetric problem in geometrically finite hyperbolic surfaces [1]. Using their classification and the relationship between the isoperimetric problem and the Cheeger constant, we give a classification of Cheeger minimizers of geometrically finite hyperbolic surfaces in Section 5.…”

Section: Introductionmentioning

confidence: 99%

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The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces

Benson

2015

Preprint

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We give a brief literature review of the isoperimetric problem and discuss its relationship with the Cheeger constant of Riemannian n-manifolds. For some noncompact, finite area 2-manifolds, we prove the existence and regularity of subsets whose isoperimetric ratio is equal to the Cheeger constant. To do this, we use results of Hass-Morgan for the isoperimetric problem of these manifolds. We also give an example of a finite area 2-manifold where no such subset exists. Using work of Adams-Morgan, we classify all such subsets of geometrically finite, finite area hyperbolic surfaces where such subsets always exist. From this, we provide an algorithm for finding these sets given information about the topology, length spectrum, and distances between the simple closed geodesics of the surface. Finding such a subset allows one to directly compute the Cheeger constant of the surface. As an application of this work suggested by Agol, we give a test for Selberg's eigenvalue conjecture. We do this by comparing a quantitative improvement of Buser's inequality resulting from works of both Agol and the author to an upper bound on the Cheeger constant of these surfaces, the latter given by Brooks-Zuk. As expected, our test does not contradict Selberg's conjecture.where D ⊂ M is a smooth n-submanifold with boundary and 0We describe the procedure of using results about the isoperimetric problem on M in order to prove the existence and regularity of subsetsWe refer to ∂A and A as (n−1)and n-dimensional Cheeger minimizers respectively.Buser introduced this procedure when he proved that Cheeger minimizers exist for all compact Riemannian manifolds [11, Remark 3.3, Lemma 3.4]. Since Buser's explanation of this idea was very brief, we elaborate on it in the setting of compact Riemannian manifolds in Section 3. We also discuss criteria for when these results extend to noncompact, finite volume surfaces.

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Isoperimetric Inequalities in Surfaces

Ritoré

2023

Progress in Mathematics

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Isoperimetric curves on hyperbolic surfaces

Abstract: Abstract. Least-perimeter enclosures of prescribed area on hyperbolic surfaces are characterized.

Cited by 17 publications

References 3 publications

Existence of CMC-foliations in asymptotically cuspidal manifolds

Existence of CMC-foliations in asymptotically cuspidal manifolds

The Cheeger Constant, Isoperimetric Problems, and Hyperbolic Surfaces

Isoperimetric Inequalities in Surfaces

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