Complete Embedded Minimal Surfaces of Finite Total Curvature

Hoffman, David; Meeks, William H.

doi:10.1007/978-3-662-03484-2_2

Cited by 62 publications

(98 citation statements)

References 53 publications

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“…This does not hold for minimal surfaces in R d , d > 4, because any holomorphic curve (in particular, one with infinite total curvature) in C 2 D R 4 is area-minimizing on compact subsets and therefore has index zero. We refer to Section 7 of [19] for a fuller discussion of the index of minimal surfaces of finite total curvature in R 3 . The work of Fischer-Colbrie, of course, raises the question of obtaining lower bounds on the index of a minimal surface in R 3 in terms of the total curvature.…”

Section: Statement and Discussion Of The Resultsmentioning

confidence: 99%

Comparison between second variation of area and second variation of energy of a minimal surface

Ejiri¹,

Micallef²

2008

Advances in Calculus of Variations

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We show that the difference between the Morse index of a closed minimal surface as a critical point of the area functional and its Morse index as a critical point of the energy is at most the real dimension of Teichmüller space. This enables us to bound the index of a closed minimal surface in an arbitrary Riemannian manifold by the area and genus of the surface, and the dimension and geometry of the ambient manifold. Our method also yields surprisingly good upper bounds on the index of a minimal surface of finite total curvature in Euclidean space of any dimension.

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Section: Statement and Discussion Of The Resultsmentioning

confidence: 99%

Comparison between second variation of area and second variation of energy of a minimal surface

Ejiri¹,

Micallef²

2008

Advances in Calculus of Variations

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“…We only consider surfaces that are regular and connected. For details see [2,4,6,8,10,11,16,17]. Theorem 2.1.…”

Section: Preliminariesmentioning

confidence: 99%

Explicit minimal Scherk saddle towers of arbitrary even genera in $\mathbb{R}^3$

Hancco

Lobos

Batista

2014

Publicacions Matemàtiques

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Starting from works by Scherk (1835) and by Enneper-Weierstraß (1863), new minimal surfaces with Scherk ends were found only in 1988 by Karcher (see [9,10]). In the singly periodic case, Karcher's examples of positive genera had been unique until Traizet obtained new ones in 1996 (see [23]). However, Traizet's construction is implicit and excludes towers, namely the desingularisation of more than two concurrent planes. Then, new explicit towers were found only in 2006 by Martín and Ramos Batista (see [13]), all of them with genus one. For genus two, the first such towers were constructed in 2010 (see [22]). Back to 2009, implicit towers of arbitrary genera were found in [5]. In our present work we obtain explicit minimal Scherk saddle towers, for any given genus 2k, k ≥ 3.2010 Mathematics Subject Classification: 53A10.

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“…By this concept we do not count López-Ros parameters, and that dimension has been the highest in which one succeeds in finding a non-trivial explicit F. To date, there still remain only few such examples, while many F's were obtained from onedimensional period problems (see [3], [5], [7], [6] and [16]). By this concept we do not count López-Ros parameters, and that dimension has been the highest in which one succeeds in finding a non-trivial explicit F. To date, there still remain only few such examples, while many F's were obtained from onedimensional period problems (see [3], [5], [7], [6] and [16]).…”

Section: Introductionmentioning

confidence: 99%

Triply periodic minimal surfaces which converge to the Hoffman–Wohlgemuth example

Simoes

Batista

2010

Advg

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Complete Embedded Minimal Surfaces of Finite Total Curvature

Cited by 62 publications

References 53 publications

Comparison between second variation of area and second variation of energy of a minimal surface

Comparison between second variation of area and second variation of energy of a minimal surface

Explicit minimal Scherk saddle towers of arbitrary even genera in $\mathbb{R}^3$

Triply periodic minimal surfaces which converge to the Hoffman–Wohlgemuth example

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