Curvature estimates for constant mean curvature surfaces

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Pérez

Ros

2019

Self Cite

AbstractWe apply the local removable singularity theorem for minimal laminations [W. H. Meeks III, J. Pérez and A. Ros, Local removable singularity theorems for minimal laminations, J. Differential Geom. 103 (2016), no. 2, 319–362] and the local picture theorem on the scale of topology [W. H. Meeks III, J. Pérez and A. Ros, The local picture theorem on the scale of topology, J. Differential Geom. 109 (2018), no. 3, 509–565] to obtain two descriptive results for certain possibly singular minimal laminations of {\mathbb{R}^{3}}. These two global structure theorems will be applied in [W. H. Meeks III, J. Pérez and A. Ros, Bounds on the topology and index of classical minimal surfaces, preprint 2016] to obtain bounds on the index and the number of ends of complete, embedded minimal surfaces of fixed genus and finite topology in {\mathbb{R}^{3}}, and in [W. H. Meeks III, J. Pérez and A. Ros, The embedded Calabi–Yau conjectures for finite genus, preprint 2018] to prove that a complete, embedded minimal surface in {\mathbb{R}^{3}} with finite genus and a countable number of ends is proper.

Section: Example 23 Colding and Minicozzimentioning

confidence: 97%

Section: Conjecture 15 (Fundamental Singularity Conjecture)mentioning

confidence: 99%

See 1 more Smart Citation

Structure theorems for singular minimal laminations

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

Pérez

Ros

2019

Self Cite

“…Based on the proof of this result, Meeks and Rosenberg [15] showed that complete, connected minimal surfaces with positive injectivity radius embedded in R 3 are proper. Meeks and Tinaglia [16] then extended both results by proving that complete surfaces with constant mean curvature embedded in R 3 are proper if they have finite topology or positive injectivity radius. It is natural to ask to what extent similar properness results hold for complete surfaces of finite topology in other ambient spaces, where the surfaces are not necessarily embedded or have constant mean curvature.…”

mentioning

confidence: 95%

Properly immersed surfaces in hyperbolic $3$-manifolds

Ramos

2019

J. Differential Geom.

We study complete finite topology immersed surfaces Σ in complete Riemannian 3-manifolds N with sectional curvature K N ≤ −a 2 ≤ 0, such that the absolute mean curvature function of Σ is bounded from above by a and its injectivity radius function is not bounded away from zero on each of its annular end representatives. We prove that such a surface Σ must be proper in N and its total curvature must be equal to 2πχ(Σ). If N is a hyperbolic 3-manifold of finite volume and Σ is a properly immersed surface of finite topology with nonnegative constant mean curvature less than 1, then we prove that each end of Σ is asymptotic (with finite positive multiplicity) to a totally umbilic annulus, properly embedded in N .Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42. Key words and phrases: Calabi-Yau problem, hyperbolic 3-manifolds, asymptotic injectivity radius, bounded mean curvature, isoperimetric inequality. Definition 1.1. Let e be an end of a complete Riemannian surface Σ whose injectivity radius function we denote by I Σ : Σ → (0, ∞). Let E(e) be the collection of proper subdomains E ⊂ Σ, with compact boundary, that represents e. We define the (lower) asymptotic injectivity radius of e byNote that if Σ has an end e which admits a one-ended representative E, then I ∞ Σ (e) = lim inf E I Σ | E . If Σ has finite topology, then every end of Σ has a one-ended representative which is an annulus (i.e., a surface with the topology of S 1 × [0, ∞)), hence this simpler definition can be used. Moreover, Lemma 5.1 in the Appendix shows that if Σ has nonpositive Gaussian curvature and an end e has a representative E which is an annulus, then for every divergent sequence of points {p n } n∈N on E,We define the mean curvature function of an immersed two-sided surface Σ with a given unit normal field in a Riemannian 3-manifold to be the pointwise average of its principal curvatures; note that if Σ does not have a unit normal field, then the absolute value |H Σ | of the mean curvature function of Σ still makes sense because a unit normal field locally exists on Σ and under a change of this local choice, the principal curvatures change sign.Theorem 1.2. Let N be a complete 3-manifold with sectional curvature K N ≤ −a 2 ≤ 0, for some a ≥ 0. Let ϕ : Σ → N be an isometric immersion of a complete surface Σ with finite topology, whose mean curvature function satisfies |H ϕ | ≤ a. Then Σ has nonpositive Gaussian curvature and the following hold:A. If N is simply connected, then I ∞ Σ (e) = ∞ for every end e of Σ.B. If N has positive injectivity radius Inj(N ) = δ > 0, then every end e of Σ satisfies I ∞ Σ (e) ≥ δ. In particular, Σ has positive injectivity radius.C. If I Σ is bounded, then Σ has finite total curvature Σ K Σ = 2πχ(Σ), where χ(Σ) is the Euler characteristic of Σ. Furthermore, for each annular end representative E of Σ, the induced map ϕ * : π 1 (E) → π 1 (N ) on fundamental groups is injective.

“…|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

Self 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.