Plateau probleminin çözümünün gömülü olma durumu

In this thesis we will study minimal surfaces and the Plateau problem. We will rstgive the proof of Douglas solution to Plateau problem. Then we will study the paperof Meeks and Yau in which they prove the embeddedness of the solution to the Plateauproblem under some conditions. We will give the proofs of most of the theorems indetail.Minimal surfaces received their name according to the property to minimize areafor prescribed boundary values. Minimal surface theory is a branch of dierentialgeometry which studies problems related to minimal surfaces. The basic problem thatleads to minimal surface theory is the Plateau problem. The Plateau problem asks theexistence of an area minimizing disk for a given simple closed curve in a manifold M.The existence was proven in 1930, by Douglas.After the existence is proven, the regularity was also shown. In the following years,the question of embeddedness of the solution has been studied. It is not necessarily truethat for any Jordan curve, any area minimizing surface is embedded. So under whatconditions the solution to Plateau problem is embedded was an interesting question.In their paper Meeks and Yau proved that if the Jordan curve is on the boundaryof a convex manifold and is contractible, then the solution is embedded. They usedtopological techniques to solve the problem.