Non-properly Embedded H-Planes in Hyperbolic 3-Space
For any H in [0,1), we construct complete, non-proper, stable, simply-connected surfaces with constant mean curvature H embedded in hyperbolic 3-space.
math.DG↗
arXiv subjects
Baris Coskunuzer
Publications and source records attributed to Baris Coskunuzer.
$H$-Surfaces with Arbitrary Topology in Hyperbolic 3-Space
In this paper, we show that any open orientable surface S can be properly embedded in H^3 as a minimizing H-surface for any 0<=H<1. We obtained this result by proving a version of the bridge principle at infinity for H-surfaces. We also show that any open orientable surface S can be nonproperly embedded in H^3 as a minimal surface, too.
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Examples of Area Minimizing Surfaces in 3-manifolds
In this paper, we give some examples of area minimizing surfaces to clarify some well-known features of these surfaces in more general settings. The first example is about Meeks-Yau's result on embeddedness of solution to the Plateau problem. We construct an example of a simple closed curve in R^3 which lies in the boundary of a mean convex domain in R^3, but the area minimizing disk in R^3 bounding this curve is not embedded. Our second example shows that Brian White's boundary decomposition theorem does not extend when the ambient space has nontrivial homology. Our last examples show that there are properly embedded absolutely area minimizing surfaces in a mean convex 3-manifold M such that while their boundaries are disjoint, they intersect each other nontrivially.
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Uniqueness of Area Minimizing Surfaces for Extreme Curves
Let M be a compact, orientable, mean convex 3-manifold with boundary. We show that the set of all simple closed curves in the boundary of M which bound unique area minimizing disks in M is dense in the space of simple closed curves in the boundary of M which are nullhomotopic in M. We also show that the set of all simple closed curves in the boundary of M which bound unique absolutely area minimizing surfaces in M is dense in the space of simple closed curves in the boundary of M which are nullhomologous in M.
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Area minimizing surfaces in mean convex 3-manifolds
In this paper, we give several results on area minimizing surfaces in strictly mean convex 3-manifolds. First, we study the genus of absolutely area minimizing surfaces in a compact, orientable, strictly mean convex 3-manifold M bounded by a simple closed curve in the boundary of M. Our main result is that for any g>=0, the space of simple closed curves in the boundary of M where all the absolutely area minimizing surfaces they bound in M has genus >=g is open and dense in the space A of nullhomologous simple closed curves in the boundary of M. For showing this we prove a bridge principle for absolutely area minimizing surfaces. Moreover, we show that for any g>=0, there exists a curve in A such that the minimum genus of the absolutely area minimizing surfaces it bounds is exactly g. As an application of these results, we further prove that the simple closed curves in the boundary of M bounding more than one minimal surface in M is an open and dense subset of A. We also show that there are disjoint simple closed curves in the boundary of M bounding minimal surfaces in M which are not disjoint. This allows us to answer a question of Meeks, by showing that for any strictly mean convex 3-manifold M, there exists a simple closed curve Γin the boundary of M which bounds a stable minimal surface which is not embedded.
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Non-properly Embedded Minimal Planes in Hyperbolic 3-Space
In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with nonpositive curvature. We show this result by constructing a non-properly embedded minimal plane in hyperbolic 3-space. Hence, this gives a counterexample to Calabi-Yau conjecture for embedded minimal surfaces in the negative curvature case.
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Embedded Plateau Problem
We show that if C is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk D in M with boundary C such that D minimizes the area among the embedded disks with boundary C. Moreover, D is smooth, minimal and embedded everywhere except where the boundary C meets the interior of D. The same result is also valid for homogenously regular manifolds with sufficiently convex boundary.
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Asymptotic Plateau Problem
This is a survey of old and recent results about the asymptotic Plateau problem. Our aim is to give a fairly complete picture of the field, and present the current situation.
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On the Number of Solutions to Asymptotic Plateau Problem
We give a simple topological argument to show that the number of solutions of the asymptotic Plateau problem in hyperbolic space is generically unique. In particular, we show that the space of codimension-1 closed submanifolds of sphere at infinity, which bounds a unique absolutely area minimizing hypersurface in hyperbolic n-space, is dense in the space of all codimension-1 closed submanifolds at infinity. In dimension 3, we also prove that the set of uniqueness curves in asymptotic sphere for area minimizing planes is generic in the set of Jordan curves at infinity. We also give some nonuniqueness results for dimension 3, too.
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Nonuniqueness of the solutions to the asymptotic Plateau problem
Because of the relevance of the results, this paper is merged into the paper titled "On the Number of Solutions to Asymptotic Plateau Problem" (arXiv:math.DG/0505593) as a new section.
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Foliations of Hyperbolic Space by Constant Mean Curvature Hypersurfaces
We show that the constant mean curvature hypersurfaces in the hyperbolic n-space spanning the boundary of a star shaped C^{1,1} domain in the asymptotic sphere give a foliation of the hyperbolic n-space. We also show that if C is a closed codimension-1 C^{2,a} submanifold in the asymptotic sphere bounding a unique constant mean curvature hypersurface S_H in the hyperbolic n-space with asymptotic boundary C for any -1<H<1, then the constant mean curvature hypersurfaces {S_H} foliates the hyperbolic n-space.
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Generic Uniqueness of Area Minimizing Disks for Extreme Curves
We show that for a generic nullhomotopic simple closed curve C in the boundary of a compact, orientable, mean convex 3-manifold M with trivial second homology, there is a unique area minimizing disk D embedded in M where the boundary of D is C. We also show that the same is true for absolutely area minimizing surfaces.
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Generic uniqueness of least area planes in hyperbolic space
We study the number of solutions of the asymptotic Plateau problem in H^3. By using the analytical results in our previous paper, and some topological arguments, we show that there exists an open dense subset of C^3 Jordan curves in S^2_{infty}(H^3) such that any curve in this set bounds a unique least area plane in H^3.
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Number of Least Area Planes in Gromov Hyperbolic 3-Spaces
We show that for a generic simple closed curve C in the asymptotic boundary of a Gromov hyperbolic 3-space with cocompact metric X, there exist a unique least area plane P in X with asymptotic boundary C. This result has interesting topological applications for constructions of canonical 2-dimensional objects in 3-manifolds.
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Least Area Planes in Hyperbolic 3-Space are Properly Embedded
We show that if P is an embedded least area (area minimizing) plane in hyperbolic 3-space whose asymptotic boundary is a simple closed curve with at least one smooth point, then P is properly embedded.
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Properly Embedded Least Area Planes in Gromov Hyperbolic 3-Spaces
We show that for any simple closed curve in the sphere at infinity of a Gromov hyperbolic 3-space with cocompact metric, there exist a properly embedded least area plane in the space spanning the given curve. This gives a positive answer to a conjecture of Gabai. Soma has already proven this conjecture earlier. Our technique here is simpler and more general, and it can be applied to many similar settings.
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Minimizing Constant Mean Curvature Hypersurfaces in Hyperbolic Space
We study the constant mean curvature (CMC) hypersurfaces in hyperbolic space whose asymptotic boundaries are closed codimension-1 submanifolds in sphere at infinity. We consider CMC hypersurfaces as generalizations of minimal hypersurfaces. We naturally generalize some notions of minimal hypersurfaces like being area minimizing, convex hull property, exchange roundoff trick to the CMC hypersurface context. We also give a generic uniqueness result for CMC hypersurfaces in hyperbolic space.
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Mean Convex Hulls and Least Area Disks spanning Extreme Curves
We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes spanning that curve. Applying this to quasi-Fuchsian manifolds, we show that for any quasi-Fuchsian manifold, there exist a canonical mean convex core capturing all essential minimal surfaces. On the other hand, we also show that for a generic C^3-smooth curve in the boundary of C^3-smooth mean convex domain in R^3, there exist a unique least area disk spanning the curve.
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