Research Project: Minimal Yüzeyler Teorisinde Geometrik Topoloji Yöntemleri
Loading...
Contributors
Funders
ID
TB.00020
Authors
Coşkunüzer, Barış
Faculty Member
Publications
Area minimizing surfaces in mean convex 3-manifolds
(De Gruyter, 2015) Coşkunüzer, Barış; Bourni, Theodora; Department of Mathematics; Yes; College of Sciences
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 partial derivative M. Our main result is that for any g >= 0, the space of simple closed curves in partial derivative 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 partial derivative 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 gamma(g) in A such that the minimum genus of the absolutely area minimizing surfaces gamma(g) bounds is exactly g. As an application of these results, we further prove that the simple closed curves in partial derivative 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 partial derivative 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 Gamma in partial derivative M which bounds a stable minimal surface which is not embedded. We also gave some applications of these results to the simple closed curves in R-3.
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.
Embedded plateau problem
(American Mathematical Society (AMS), 2012) Coşkunüzer, Barış; Department of Mathematics; Yes; College of Sciences
We show that if Gamma is a simple closed curve bounding an embedded disk in a closed 3-manifold M, then there exists a disk Sigma in M with boundary Gamma such that Sigma minimizes the area among the embedded disks with boundary Gamma. Moreover, Sigma is smooth, minimal and embedded everywhere except where the boundary Gamma meets the interior of Sigma. The same result is also valid for homogeneously regular manifolds with sufficiently convex boundary.
