On 2-dimensional surfaces in 3-dimensional and 4-dimensional Euclidean spaces. Results and unsolved problems

Abstract

We present a survey of the results obtained for 2-dimensional surfaces in $E^3$ and $E^4$ and connected with the Gaussian curvature and Gaussian torsion. In this connection, we consider the Monge –Amp´ere equations, obtain the generalizations of Bernstein’s integral formula, and establish some lower estimates for the exterior diameter of surfaces in $E^3$.