The classical plateau problem and the topology of three-dimensional manifolds

Meeks, William H.; Yau, Shing–Tung

doi:10.1016/0040-9383(82)90021-0

Cited by 151 publications

(116 citation statements)

References 37 publications

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“…If ∂B is not assumed to be smooth, then one can use an approximation argument by convex smooth boundaries to obtain the same conclusion (see, e.g., [144]). …”

Section: Definition 221 Any Isolated Point E ∈ E(m ) Is Called a Simentioning

confidence: 99%

The classical theory of minimal surfaces

Meeks¹,

Pérez²

2011

Bull. Amer. Math. Soc.

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Abstract. We present here a survey of recent spectacular successes in classical minimal surface theory. We highlight this article with the theorem that the plane, the helicoid, the catenoid and the one-parameter family {R t } t∈(0,1) of Riemann minimal examples are the only complete, properly embedded, minimal planar domains in R 3 ; the proof of this result depends primarily on work of Colding and Minicozzi, Collin, López and Ros, Meeks, Pérez and Ros, and Meeks and Rosenberg. Rather than culminating and ending the theory with this classification result, significant advances continue to be made as we enter a new golden age for classical minimal surface theory. Through our telling of the story of the classification of minimal planar domains, we hope to pass on to the general mathematical public a glimpse of the intrinsic beauty of classical minimal surface theory and our own perspective of what is happening at this historical moment in a very classical subject.

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“…If ∂B is not assumed to be smooth, then one can use an approximation argument by convex smooth boundaries to obtain the same conclusion (see, e.g., [144]). …”

Section: Definition 221 Any Isolated Point E ∈ E(m ) Is Called a Simentioning

confidence: 99%

The classical theory of minimal surfaces

Meeks¹,

Pérez²

2011

Bull. Amer. Math. Soc.

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“…This hemisphere is a convex three-dimensional manifold in the sense of [13], thus we can apply Theorem E. By this theorem, the minimal surface M 2 which is a solution of the Plateau problem either coincides with the domainḠ on the boundary of M , or its interior lies inside the hemisphere. Since the length of the boundary curve γ is less then 2π, it lies inside a two-dimensional open totally geodesic hemisphere of the boundary ∂M .…”

Section: Proof Of the Theoremmentioning

confidence: 99%

“…Theorem E (W. Meeks and S. T. Yau, [13]). Let M be a convex three-dimensional manifold in the spherical space S 3 , and γ ⊂ ∂M be a closed Jordan curve.…”

Section: Identified Edges As Well As Any Identified Parts Of Edges mentioning

confidence: 99%

Reverse isoperimetric inequality in two-dimensional Alexandrov spaces

Borisenko

2017

Proc. Amer. Math. Soc.

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We prove a reverse isoperimetric inequality for domains homeomorphic to a disc with the boundary of curvature bounded below lying in twodimensional Alexandrov spaces of curvature c. We also study the equality case.Keywords: Alexandrov metric spaces; isoperimetric inequality; λ-convex curve Well-known isoperimetric inequality for the Euclidean plane states that the area F and the length L of the boundary of any plane domain with a rectifiable boundary satisfy the inequalityand equality is attained only for a circle [1]. On the planes of constant curvature there is a similar theorem, and the following inequality holds [2]:where c is the curvature of the plane. In two-dimensional manifolds of bounded curvature for domains homeomorphic to a disc A. D. Alexandrov proved [3] the inequalitywhere ω + is a positive curvature of a domain. In the inequality above equality holds only if the domain is isometric to a lateral surface of a right circular cone with curvature ω + < 2π at the vertex.

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“…If ∂B is not assumed to be smooth, then one can use an approximation argument by convex smooth boundaries (see e.g. [88]) to have the same conclusion.…”

Section: Stable Minimal Surfacesmentioning

confidence: 99%

Conformal properties in classical minimal surface theory

Meeks

Pérez

rez

2004

Surveys in Differential Geometry

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Abstract. This is a survey of recent developments in the classical theory of minimal surfaces in R 3 with an emphasis on the conformal properties of these surfaces such as recurrence and parabolicity. We cover the maximum principle at infinity for properly immersed minimal surfaces in R 3 and some new results on harmonic functions as they relate to the classical theory. We define and demonstrate the usefulness of universal superharmonic functions. We present the compactness and regularity theory of Colding and Minicozzi for limits of sequences of simply connected minimal surfaces and its application by Meeks and Rosenberg in their proof of the uniqueness of the helicoid. Finally, we discuss some recent deep results on the topology and on the index of the stability operator of properly embedded minimal surfaces and give an application of the classical Shiffman Jacobi function to the classification of minimal surfaces of genus zero.

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The classical plateau problem and the topology of three-dimensional manifolds

Cited by 151 publications

References 37 publications

The classical theory of minimal surfaces

The classical theory of minimal surfaces

Reverse isoperimetric inequality in two-dimensional Alexandrov spaces

Conformal properties in classical minimal surface theory

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