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, ξ−surfaces are the generalization of self-shrinker surfaces. In the present article we consider ξ−surfaces in Euclidean spaces. We obtained some results related with rotational surfaces in Euclidean 4−space R 4 to become self-shrinkers. Furthermore, we classify the general rotational ξ−surfaces with constant mean curvature. As an application, we give some examples of self-shrinkers and rotational ξ−surfaces in R 4 .