Geodesics and soap bubbles in surfaces

Hass, Joel; Morgan, Frank

doi:10.1007/pl00004560

Cited by 18 publications

(10 citation statements)

References 12 publications

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“…Proof of Lemma 2.2. The existence and regularity results of [HM,3.1,3.3] extend immediately to surfaces with convex boundary and provide a minimizer, possibly with bumping as asserted, C 1 away from ∂S, constant curvature κ 0 away from bumping. (As long as the boundary is convex, it does not interfere with the proof, a convexification argument.)…”

Section: Lemmamentioning

confidence: 79%

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Untitled

2000

Trans. Amer. Math. Soc.

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Section: Lemmamentioning

confidence: 79%

“…Lemma 2.2. Existence without an a priori bound on the number of boundary components follows as in [HM,Thm. 3.4].…”

Section: Upper Bound On Least Perimetermentioning

confidence: 99%

Untitled

2000

Trans. Amer. Math. Soc.

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“…Joel Hass and Frank Morgan prove existence and regularity for smooth closed Riemannian surfaces [5,Theorem 3.4]. Geometric measure theory provides a proof to cover all n-dimensional manifolds that are of finite volume or are compact under their isometries [7, Theorem 2.1], [9, pp.…”

Section: Arc Normal To Boundarymentioning

confidence: 99%

The Isoperimetric Problem on Euclidean, Spherical, and Hyperbolic Surfaces

Simonson¹

2011

Journal of the Korean Mathematical Society

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Abstract. We solve the isoperimetric problem, the least-perimeter way to enclose a given area, on various Euclidean, spherical, and hyperbolic surfaces, sometimes with cusps or free boundary. On hyperbolic genustwo surfaces, Adams and Morgan characterized the four possible types of isoperimetric regions. We prove that all four types actually occur and that on every hyperbolic genus-two surface, one of the isoperimetric regions must be an annulus.In a planar annulus bounded by two circles, we show that the leastperimeter way to enclose a given area is an arc against the outer boundary or a pair of spokes. We generalize this result to spherical and hyperbolic surfaces bounded by circles, horocycles, and other constant-curvature curves. In one case the solution alternates back and forth between two types, a phenomenon we have yet to see in the literature.We also examine non-orientable surfaces such as spherical Möbius bands and hyperbolic twisted chimney spaces.

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“…Пуанкаре ввел ограниче-ния Гаусса-Бонне: минимизируется длина всех вложенных замкнутых кривых, которые делят поверхность на две части, на каждой из которых интеграл кривизны одинаков (это в точно-сти утверждение теоремы Гаусса-Бонне для геодезической линии). Полное (и красивое) дока-зательство по способу, предложенному Пуанкаре, было выведено только в 1994 г. Ж. Хассом и Ф. Морганом [17].…”

Section: заключениеunclassified

A note by Poincare

Chenciner¹

2005

Nelin. Dinam.

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Получено 10 сентября 2004 г.30 ноября 1896 года в журнале "Comptes rendus de l' Acad´emie des Sciences" Анри Пуанкаре опубликовал за-метку, озаглавленную «О периодических решениях и принципе наименьшего действия». В ней он предложил искать периодические решения плоской задачи трех тел, минимизируя лагранжево действие по петлям в конфигурационном пространстве, которые удовлетворяют заданным ограничениям (эти ограничения эквивалентны тому, что мы задаем их класс гомологий). В случае ньютоновского потенциала, обратно пропорционального расстоянию, «задача столк-новения» не позволила Пуанкаре найти решение, поэтому он решил заменить этот потенциал на «потенциал сильного взаимодействия», который обратно пропорционален квадрату расстояния.В этой работе объясняется природа трудностей, с которыми столкнулся Пуанкаре и показано как эти труд-ности, спустя столетие, были частично разрешены для ньютоновского потенциала, что привело к открытию новых поразительных семейств периодических решений плоской и пространственной задач n-тел.Ключевые слова: Пуанкаре, задача трех тел, минимизирующие действие периодические ре-шения. A. Chenciner A note by Poincar´eOn November 30th 1896, Poincar´e published a note entitled "On the periodic solutions and the least action principle" in the "Comptes rendus de l' Academie des Sciences". He proposed to find periodic solutions of the planar Three-Body Problem by minimizing the Lagrangian action among loops in the configuration space which satisfy given constraints (the constraints amount to fixing their homology class). For the Newtonian potential, proportional to the inverse of the distance, the "collision problem" prevented him from realizing his program; hence he replaced it by a "strong force potential" proportional to the inverse of the squared distance.In the lecture, the nature of the difficulties met by Poincar´e is explained and it is shown how, one century later, these have been partially resolved for the Newtonian potential, leading to the discovery of new remarkable families of periodic solutions of the planar or spatial n-body problem.

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Geodesics and soap bubbles in surfaces

Cited by 18 publications

References 12 publications

Untitled

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The Isoperimetric Problem on Euclidean, Spherical, and Hyperbolic Surfaces

A note by Poincare

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