Translation and homothetical TH-surfaces in the 3-dimensional Euclidean space E 3 and Lorentzian-Minkowski space E 3 1

In the 3-dimensional Euclidean space E 3 and Lorentzian-Minkowski space E 3 1 , a translation and homothetical TH-surface is parameterized z ( u , v ) = A ( f ( u ) + g ( v ) ) + B f ( u ) g ( v ) , where f and g are smooth functions and A , B are non-zero real numbers. In this paper, we define TH-surfaces in the 3-dimensional Euclidean space E 3 and Lorentzian-Minkowski space E 3 1 and completely classify minimal or flat TH-surfaces.