A Weierstrass representation for linear Weingarten spacelike surfaces of maximal type in the Lorentz-Minkowski space.
The authors extend the Weierstrass representation for maximal spacelike surfaces in the 3-dimensional Lorentz-Minkowski space \({\mathbf L}^3\) to a conformal representation of the spacelike surfaces whose mean curvature \(H\) is proportional to its Gaussian curvature \(K= -2aH+ bK= 0\), \(a\), \(b\) being two real constants with \(a\neq 0\) (``linear Weingarten surfaces of maximal type'', shortly: LWM-spacelike surfaces; see Theorem 9). This representation is used in order to study the Gaussian curvature and the Gauss map of such surfaces when the immersion is complete (it is shown that for any \(b\neq 0\) there exist nonflat complete LWM-spacelike surfaces) proving that the surface is a plane or the supremum of its Gaussian curvature is a negative constant \((=4a^2/b^2)\) and its Gauss map is a diffeomorphism onto the hyperbolic plane (Theorem 12). The rotation LWM-spacelike surfaces can be described explicitly as follows (Theorem 20): every rotation LWM-spacelike surface in \({\mathbf L}^3\) is isometric to a piece of one of the following: (i) plane; (ii) LWM-catenoid with vertical axis; (iii) LWM-catenoid with horizontal axis; (iv) LWM-Enneper surface.
linear Weingarten surface of maximal type
scientific article; zbMATH DE number 1970505