General rotational ξ-surfaces in euclidean spaces

Abstract

The general rotational surfaces in the Euclidean 4-space R-4 was first studied by Moore (1919). The Vranceanu surfaces are the special examples of these kind of surfaces. Self-shrinker flows arise as special solution of the mean curvature flow that preserves the shape of the evolving submanifold. In addition, xi-surfaces are the generalization of self-shrinker surfaces. In the present article we consider xi-surfaces in Euclidean spaces. We obtained some results related with rotational surfaces in Euclidean xi- space R-4 to become self-shrinkers. Furthermore, we classify the general rotational xi-surfaces with constant mean curvature. As an application, we give some examples of self-shrinkers and rotational xi-surfaces in R-4.