Null 2-type surfaces in \(E^ 3\) are circular cylinders (Q1110814)

Let M be a connected (not necessarily compact) surface in a Euclidean 3- space \(E^ 3\). M is called a surface of null 2-type if the position vector x has the following spectral decomposition: \(x=x_ 0+x_ q\), with \(\Delta x_ 0=0\) and \(\Delta x_ q=\lambda x_ q\), for some non- constant maps \(x_ 0\) and \(x_ q\), where \(\lambda\) is a non-zero constant. In this article the author proves that open portions of circular cylinders are the only null 2-type surfaces in \(E^ 3\).