Helicoid-like minimal disks and uniqueness

Bernstein, Jared; Breiner, Christine

doi:10.1515/crelle.2011.037

Cited by 17 publications

(89 citation statements)

References 22 publications

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“…This theorem together with Theorem 1.3 above, Theorem 1.5 below and with results by Bernstein and Breiner [4] or by Meeks and Pérez [115] lead to a complete understanding of the asymptotic geometry of any annular end of a complete, embedded minimal surface with finite topology in R 3 ; namely, the annular end must be asymptotic to an end of a plane, catenoid or helicoid. For a discussion of the proof of Theorem 1.4 in the case of positive genus and a more general classification result of complete embedded minimal annular ends with compact boundary and infinite total curvature in R 3 , see the monograph [116].…”

Section: Theorem 13 (Collin [38]) If M ⊂ Rmentioning

confidence: 79%

“…An important detail is that this simplification does not need the hypothesis of simply connectedness and works in the case of proper minimal annuli with one compact boundary curve and infinite total curvature, giving that such an annulus intersects transversely some horizontal plane in a single proper arc. For details, see [4,5,115]. Theorem 1.1 solves a long-standing conjecture about the uniqueness of the helicoid among properly embedded, simply connected minimal surfaces in R 3 .…”

Section: Uniqueness Of the Helicoid I: The Proper Casementioning

confidence: 88%

“…In particular, if M has finite topology and more than one end, then M has finite total Gaussian curvature. 4 Collin's Theorem reduces the analysis of properly embedded minimal surfaces of finite topology and more than one end in R 3 to complex function theory on compact Riemann surfaces. This reduction then leads to classification results and to interesting topological obstructions, which we include in Section 3 as well.…”

Section: Theorem 13 (Collin [38]) If M ⊂ Rmentioning

confidence: 99%

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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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Section: Theorem 13 (Collin [38]) If M ⊂ Rmentioning

confidence: 79%

Section: Uniqueness Of the Helicoid I: The Proper Casementioning

confidence: 88%

Section: Theorem 13 (Collin [38]) If M ⊂ Rmentioning

confidence: 99%

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The classical theory of minimal surfaces

Meeks¹,

Pérez²

2011

Bull. Amer. Math. Soc.

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“…The stereotypical infinite multi-valued graph is half of a helicoid, i.e., half of an infinite double-spiral staircase. u [1] u [2] u [3] −3π −π π 3π u(ρ, θ) Figure 1: A right-handed 3-valued graph.…”

Section: An Extrinsic Curvature Estimate For Certain Planar Domainsmentioning

confidence: 99%

“…In the case of embedded minimal disks such a description was given by Colding and Minicozzi in [7]; see also [32,33] for related results. By rescaling arguments this description can be improved upon once one knows that the helicoid is the unique complete, embedded, non-flat minimal surface in R 3 as explained below; see [17] and also [1] for proofs of the uniqueness of the helicoid which are based in part on results in [6,7,8,9,10].…”

Section: Definition 24 (Multi-valued Graph)mentioning

confidence: 99%

On curvature estimates for constant mean curvature surfaces

Tinaglia¹

2012

Geometric Analysis

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Limit lamination theorem for H-disks

Meeks

Tinaglia

2021

Invent. math.

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We describe the lamination limits of sequences of compact disks $$M_n$$ M n embedded in $${\mathbb {R}}^3$$ R 3 with constant mean curvature $$H_n$$ H n , when the boundaries of these disks tend to infinity. This theorem generalizes to the non-zero constant mean curvature case Theorem 0.1 by Colding and Minicozzi (Ann Math 160:573–615, 2004) for minimal disks. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in $${\mathbb {R}}^3$$ R 3 with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi (Ann Math 167:211–243, 2008) for minimal disks.

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Helicoid-like minimal disks and uniqueness

Abstract: We show that for an embedded minimal disk in R 3 , near points of large curvature the surface is bi-Lipschitz with a piece of a helicoid. Additionally, a simplified proof of the uniqueness of the helicoid is provided.

Cited by 17 publications

References 22 publications

The classical theory of minimal surfaces

The classical theory of minimal surfaces

On curvature estimates for constant mean curvature surfaces

Limit lamination theorem for H-disks

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