Non-Properly Embedded Minimal Planes in Hyperbolic 3-Space

Coskunuzer, Baris

doi:10.1142/s0219199711004415

Cited by 12 publications

(22 citation statements)

References 10 publications

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“…Example V. Simply-connected bridged examples in H 3 . As in the previous subsection, the minimal laminations in example IV give rise to minimal laminations of H 3 consisting of two stable, complete, simply connected minimal surfaces, one of which is proper and the other one which is not proper in the space, and either one is not totally geodesic or both of them are not totally geodesic, depending on the choice of the Euclidean model surface in Figure 2 (for this existence, one can use Anderson [1] to create the corresponding minimal disks M n as in Example IV, and then use the bridge principle at infinity as described in Coskunuzer [10]). In this case, the proper leaf is the unique limit leaf of the minimal lamination.…”

Section: Minimal Laminations With Limit Leavesmentioning

confidence: 99%

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Local removable singularity theorems for minimal laminations

Meeks¹,

Pérez²,

Ros³

2016

J. Differential Geom.

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In this paper we prove a local removable singularity theorem for certain minimal laminations with isolated singularities in a Riemannian three-manifold. This removable singularity theorem is the key result used in our proof that a complete, embedded minimal surface in R 3 with quadratic decay of curvature has finite total curvature. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42 Definition 2.2 A codimension one lamination of a Riemannian three-manifold N is the union of a collection of pairwise disjoint, connected, injectively immersed surfaces, with a certain local product structure. More precisely, it is a pair (L, A) satisfying:1. L is a closed subset of N ;

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Section: Minimal Laminations With Limit Leavesmentioning

confidence: 99%

“…where ν is the unit exterior conormal vector to M ∩ B(ε|p n |) along its boundary. Changing variables in the second integral in (10) we have…”

Section: Proof By Lemma 43 and Sincementioning

confidence: 99%

Local removable singularity theorems for minimal laminations

Meeks¹,

Pérez²,

Ros³

2016

J. Differential Geom.

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“…By Theorem 9.2, the closure of M has the structure of a minimal lamination of R Remark 9.4. Corollary 9.3 is no longer true if we replace R 3 by H 3 given that Coskunuzer [40] has constructed a complete embedded minimal plane in H 3 which is not proper; see especially the last paragraph in [40] for a discussion of the role of this example in H 3 in relation to Theorem 9.2 above.…”

Section: Non-empty and Disjoint From M And M Is Properly Embedded Imentioning

confidence: 99%

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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“…This is because a nonproperly embedded area minimizing hypersurface in H n+1 would have an intersection of infinite volume with a sufficiently large compact ball in H n+1 , which is impossible for absolutely area minimizing hypersurfaces. Also, in a forthcoming paper [21], the author constructs examples of non-properly embedded minimal planes in H 3 .…”

Section: Properly Embeddednessmentioning

confidence: 99%