Research Project: Minimal Surfaces in 3-manifolds
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Contributors
Funders
ID
EC.00006
Authors
Coşkunüzer, Barış
Faculty Member
Publications
Generic uniqueness of area minimizing disks for extreme curves
(Johns Hopkins University (JHU) Press, 2010) Coşkunüzer, Barış; Department of Mathematics; Yes; College of Sciences
We show that for a generic nullhomotopic simple closed curve Γ in the boundary of a compact, orientable, mean convex 3-manifold M with H2(M, Z) = 0, there is a unique area minimizing disk D embedded in M with ∂D = Γ. We also show that the same is true for nullhomologous curves in the absolutely area minimizing surface case.
Foliations of hyperbolic space by constant mean curvature hypersurfaces
(Oxford University Press (OUP), 2009) Coşkunüzer, Barış; Department of Mathematics; Yes; College of Sciences
We show that the constant mean curvature hypersurfaces in Hn+1 spanning the boundary of a star-shaped C1,1 domain in Sn∞ (Hn+1) give a foliation of Hn+1. We also show that if is a closed codimension-1 C2,α submanifold in Sn∞ (Hn+1) bounding a unique constant mean curvature hypersurface H in Hn+1 with ∂∞ H = for any H ∈ (−1, 1), then the constant mean curvature hypersurfaces { H} foliate Hn+1.
Examples of area-minimizing surfaces in 3-manifolds
(Oxford University Press (OUP), 2012) Coşkunüzer, Barış; Department of Mathematics; Yes; College of Sciences
In this paper, we give some examples of area-minimizing surfaces to clarify some wellknown features of these surfaces in more general settings. The first example is about Meeks–Yau’s result on the embeddedness of the solution to the Plateau problem. We construct an example of a simple closed curve in R3 which lies in the boundary of a mean convex domain in R3, but the area-minimizing disk in R3 bounding this curve is not embedded. Our second example shows that 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, unlike the area-minimizing disks case.
Uniqueness of area minimizing surfaces for extreme curves
(European Mathematical Society, 2014) Coşkunüzer, Barış; Etgü, Tolga; Department of Mathematics; Yes; College of Sciences
Let MM be a compact, orientable, mean convex 33-manifold with boundary ?M?M. We show that the set of all simple closed curves in ?M?M which bound unique area minimizing disks in MM is dense in the space of simple closed curves in ?M?M which are nullhomotopic in MM. We also show that the set of all simple closed curves in ?M?M which bound unique absolutely area minimizing surfaces in MM is dense in the space of simple closed curves in ?M?M which are nullhomologous in MM.
Non-properly embedded minimal planes in hyperbolic 3-space
(World Scientific Publishing, 2011) Coşkunüzer, Barış; Department of Mathematics; Yes; College of Sciences
In this paper, we show that there are non-properly embedded minimal surfaces with finite topology in a simply connected Riemannian 3-manifold with non-positive curvature. We show this result by constructing a non-properly embedded minimal plane in H3. Hence, this gives a counterexample to Calabi–Yau conjecture for embedded minimal surfaces in negative curvature case.
