Complete surfaces of finite total curvature
It is well known that the total curvature of a complete minimal surface in \(E^ n\) is an integral multiple of \(2\pi\) (or of \(4\pi\) if \(n=3)\). Furthermore the Gauss map extends continuously to the end compactification \(\bar M\) where \(M=\bar M\setminus \{p_ 1,...,p_ k\}\). The proof relies heavily on complex analysis. In the present paper the author assumes the weaker condition that the total norm square of the second fundamental form is finite. Again he obtains that the total curvature is an integral multiple of \(2\pi\) (or of \(4\pi\) if \(n=3)\). Furthermore he gives a sufficient condition for the continuous extension of the Gauss map to \(\bar M.\) An example shows that this is not true in general.
scientific article; zbMATH DE number 4026254