On willmore surfaces

In this thesis, willmore surfaces were examined as an application of rotational surfaces and solutions of the Willmore rotational surface equation were sought. . The concept of rotational surface is explained by mentioning some basic definitions and theorems that will be used primarily in the thesis. In addition, Gaussian curvature and minimal surfaces on the Euclidean surface have been mentioned and studies on this subject have been briefly mentioned. Then, the profile curve is mentioned as a parametric rotational surfaces. Information on how to find parametric equations of these rotational surfaces has been given about some examples and their solutions. The basic form coefficients of the surface were obtained by mentioning the Willmore function in Euclidean space. The first and second basic form coefficients were obtained by mentioning the Willmore rotational surface and the mean curvature vector fields were obtained from the Willmore rotational surface equation. Willmore surface equation is written as a differential and numerical solution of this differential equation is sought.