The number of constant mean curvature isometric immersions  of a surface

Smyth, Brian; Tinaglia, Giuseppe

doi:10.4171/cmh/281

Cited by 15 publications

(16 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In particular, if D ∩ B(ε R) = Ø, then H ≤ C a R . Next, we recall the notion of flux of an H -surface; see for instance [11,12,25] for further discussions of this invariant. The flux of a 1-cycle in an H -surface M is a homological invariant and we say that M has zero flux if the flux of any 1-cycle in M is zero; in particular, since the first homology group of a disk is zero, an H -disk has zero flux.…”

Section: Corollary 26mentioning

confidence: 99%

Limit lamination theorem for H-disks

Meeks

Tinaglia

2021

Invent. math.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Corollary 26mentioning

confidence: 99%

Limit lamination theorem for H-disks

Meeks

Tinaglia

2021

Invent. math.

Self Cite

View full text Add to dashboard Cite

show abstract

“…The following is essential for the proof of Theorem 5. For its proof we adopt techniques used in [19,49,53]. Proof: (i) We claim that for any σ ∈ D in the group of deck transformations of the universal coverπ :M → M, the surfacesf ± θ :M → Q 4 c in the associated family off = f •π andf ± θ • σ are congruent for any θ ∈ S 1 .…”

Section: The Moduli Space Of Non-simply-connected Surfacesmentioning

confidence: 99%

On the Moduli Space of Isometric Surfaces with the Same Mean Curvature in 4-Dimensional Space Forms

Polymerakis

Vlachos

2018

J Geom Anal

View full text Add to dashboard Cite

We study the moduli space of congruence classes of isometric surfaces with the same mean curvature in 4-dimensional space forms. Having the same mean curvature means that there exists a parallel vector bundle isometry between the normal bundles that preserves the mean curvature vector fields. We prove that if both Gauss lifts of a compact surface to the twistor bundle are not vertically harmonic, then there exist at most three nontrivial congruence classes. We show that surfaces with a vertically harmonic Gauss lift possess a holomorphic quadratic differential, yielding thus a Hopf-type theorem. We prove that such surfaces allow locally a oneparameter family of isometric deformations with the same mean curvature. This family is trivial only if the surface is superconformal. For such compact surfaces with non-parallel mean curvature, we prove that the moduli space is the disjoint union of two sets, each one being either finite, or a circle. In particular, for surfaces in R 4 we prove that the moduli space is a finite set, under a condition on the Euler numbers of the tangent and normal bundles.We give an application to Lagrangian surfaces in R 4 . Let J be a canonical complex structure on R 4 which is compatible with the orientation, i.e., for orthonormal vectors e 1 , e 2 ∈ R 4 , the oriented orthonormal basis {e 1 , e 2 , Je 1 , Je 2

show abstract

“…We next describe the notion of flux of an H-surface in R 3 , see for instance [6,7,32] for further discussion of this invariant.…”

Section: Thenmentioning

confidence: 99%

Triply periodic constant mean curvature surfaces

Meeks

Tinaglia

2018

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

The number of constant mean curvature isometric immersions of a surface

Cited by 15 publications

References 28 publications

Limit lamination theorem for H-disks

Limit lamination theorem for H-disks

On the Moduli Space of Isometric Surfaces with the Same Mean Curvature in 4-Dimensional Space Forms

Triply periodic constant mean curvature surfaces

Contact Info

Product

Resources

About