Translation surfaces of coordinate finite type

Abstract

We consider translation surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e. their position vector $x$ satisfies the relation ${\Delta }^{III}x=\Lambda x$, where $\Lambda$ is a square matrix of order 3. We show that Sherk's minimal surface is the only translation surface satisfying ${\Delta }^{III}x=\Lambda x$.

arXiv admin note: substantial text overlap with arXiv:1710.07619