The rigidity of embedded constant mean curvature surfaces

Meeks, William H.; Tinaglia, Giuseppe

doi:10.1515/crelle.2011.073

Cited by 5 publications

(14 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Also in [27], we will apply Theorem 1.1 to give a general structure theorem for singular minimal laminations of R 3 with a countable number of singularities. The results described in the next corollary to Theorem 1.1 overlap somewhat with the rigidity results for complete embedded constant mean curvature surfaces by Meeks and Tinaglia described in [41].…”

Section: The Set Of Local Pictures On the Scale Of Topologysupporting

confidence: 85%

The local picture theorem on the scale of topology

Meeks

Pérez

Ros

2018

J. Differential Geom.

Self Cite

View full text Add to dashboard Cite

We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call a minimal parking garage structure on R 3 , whose theory we also develop. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42.

show abstract

Section: The Set Of Local Pictures On the Scale Of Topologysupporting

confidence: 85%

The local picture theorem on the scale of topology

Meeks

Pérez

Ros

2018

J. Differential Geom.

Self Cite

View full text Add to dashboard Cite

show abstract

“…In [24], we prove that if M ⊂ R 3 is a strongly Alexandrov embedded CM C surface with bounded second fundamental form and T (M ) contains a Delaunay surface, then M is rigid. In [27], Smyth and Tinaglia show that if M contains a surface with a plane of Alexandrov symmetry, then M is locally rigid 6 .…”

Section: Remark 311mentioning

confidence: 99%

“…In the special case that M has finite topology 2 , this last result follows from the main theorem in [13], however the full generality of this result is needed in applications in [22,24].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

The dynamics theorem for CMC surfaces in $R^3$

Meeks

Tinaglia²

2010

J. Differential Geom.

Self Cite

View full text Add to dashboard Cite

In this paper, we study the space of translational limits T (M ) of a surface M properly embedded in R 3 with nonzero constant mean curvature and bounded second fundamental form. There is a natural map T which assigns to any surface Σ ∈ T (M ), the set T (Σ) ⊂ T (M ). Among various dynamics type results we prove that surfaces in minimal T -invariant sets of T (M ) are chord-arc. We also show that if M has an infinite number of ends, then there exists a nonempty minimal T -invariant set in T (M ) consisting entirely of surfaces with planes of Alexandrov symmetry. Finally, when M has a plane of Alexandrov symmetry, we prove the following characterization theorem: M has finite topology if and only if M has a finite number of ends greater than one. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42

show abstract

“…There was already a class of results on the isometric indeformability of minimal or constant mean curvature surfaces with topology (see for instance [5,17,23,26,30,32]). Typically, these follow for us from Theorem 1.1 by exhibiting a cycle with nonzero force.…”

Section: Introductionmentioning

confidence: 99%

The number of constant mean curvature isometric immersions of a surface

Smyth¹,

Tinaglia²

2013

Comment. Math. Helv.

Self Cite

View full text Add to dashboard Cite

In classical surface theory there are but few known examples of surfaces admitting nontrivial isometric deformations and fewer still non-simplyconnected ones. We consider the isometric deformability question for an immersion x : M → R 3 of an oriented non-simply-connected surface with constant mean curvature H. We prove that the space of all isometric immersions of M with constant mean curvature H is, modulo congruences of R 3 , either finite or a circle (Theorem 1.1). When it is a circle then, for the immersion x, every cycle in M has vanishing force and, when H = 0, also vanishing torque. Our work generalizes a rigidity result for minimal surfaces [5] to constant mean curvature surfaces (Theorem 5.2). Moreover, we identify closed vector-valued 1-forms whose periods give the force and torque.

show abstract

The rigidity of embedded constant mean curvature surfaces

Cited by 5 publications

References 12 publications

The local picture theorem on the scale of topology

The local picture theorem on the scale of topology

The dynamics theorem for CMC surfaces in $R^3$

The number of constant mean curvature isometric immersions of a surface

Contact Info

Product

Resources

About