Chord arc properties for constant mean curvature disks

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Tinaglia²

2016

Self Cite

In this paper we prove a theorem concerning lamination limits of sequences of compact disks M n embedded in R 3 with constant mean curvature 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 [8]. We apply this theorem to prove the existence of a chord arc result for compact disks embedded in R 3 with constant mean curvature; this chord arc result generalizes Theorem 0.5 by Colding and Minicozzi in [9] for minimal disks.

Section: Applications Of the Main Theorem 41 Chord-arc Propertymentioning

confidence: 99%

“…We begin by making the following definition. [23]). There exists a δ 1 ∈ (0, 1 2 ) such that the following holds.…”

Section: Applications Of the Main Theorem 41 Chord-arc Propertymentioning

confidence: 99%

Limit lamination theorems for H-surfaces

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Tinaglia²

2016

Self Cite

“…In this paper we apply results in [22][23][24] to obtain (after passing to a subsequence) minimal lamination limits for any sequence of compact disks M n embedded in R 3 with constant mean curvature H n , when the boundaries of these disks tend to infinity; see Theorem 1.1 below. This theorem is inspired by, and generalizes to the non-zero constant mean curvature setting, Theorem 0.1 by Colding and Minicozzi [7] and it is related to work in [3,14,15,26].…”

Section: Introductionmentioning

confidence: 99%

“…The proofs of the results described in this paper depend in an essential manner on the existence of extrinsic curvature estimates for disks embedded in R 3 of non-zero constant mean curvature that appear in [23], as well as on a key extrinsic one-sided curvature estimate obtained in [24] and a weak cord arc result derived in [22]; these results from [22][23][24] are described in Sect. 2.…”

Section: Introductionmentioning

confidence: 99%

Limit lamination theorem for H-disks

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.

“…|A M | denotes the norm of the second fundamental form of M .3. The radius of a Riemannian n-manifold with boundary is the supremum of the intrinsic distances of points in the manifold to its boundary.The next two results are contained in [20], see also [18,19,21,22,23].Theorem 2.2 (Radius Estimates) There exists an R ≥ π such that any H-disk in R 3 has radius less than R/H.…”

mentioning

confidence: 99%

Triply periodic constant mean curvature surfaces

Advances in Mathematics

Tinaglia

2018

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Given a closed flat 3-torus N , for each H > 0 and each non-negative integer g, we obtain area estimates for closed surfaces with genus g and constant mean curvature H embedded in N . This result contrasts with the theorem of Traizet [33], who proved that every flat 3-torus admits for every positive integer g with g = 2, connected closed embedded minimal surfaces of genus g with arbitrarily large area. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42.We now recall several key notions and theorems from [20] that we will apply in the next section. Definition 2.1 Let M be an H-surface, possibly with boundary, in a complete oriented Riemannian 3-manifold N . 1. M is an H-disk if M is diffeomorphic to a closed disk in the complex plane. 2. |A M | denotes the norm of the second fundamental form of M .3. The radius of a Riemannian n-manifold with boundary is the supremum of the intrinsic distances of points in the manifold to its boundary.The next two results are contained in [20], see also [18,19,21,22,23].Theorem 2.2 (Radius Estimates) There exists an R ≥ π such that any H-disk in R 3 has radius less than R/H.