Maximal surfaces with conelike singularities
A space-like surface S in a 3-dimensional Minkowski space \(L^ 3=({\mathbb{R}}^ 3,dx^ 2+dy^ 2-dz^ 2)\) is said to be maximal if the mean curvature vanishes identically. Any such surface is represented locally by \(Z=u(x,y)\) and S is maximal if u satisfies the equation: \((1- u^ 2_ x)u_{yy}+2u_ xu_ yu_{xy}+(1-u^ 2_ y)u_{xx}=0\). Maximal surfaces S in \(L^ 3\) have singularities which are of different kinds from those appearing in minimal surfaces in the Euclidean space. In this paper a conelike singularity is defined and the following theorem is proved: Let S be a complete maximal surface in \(L^ 3\), with at least one conelike singularity. Suppose that the Gauss map of S is one-to-one. Then S is congruent to the surface defined by \(\sqrt{x^ 2+y^ 2}+a \sinh (z/a)=0,\) where a is a nonzero real constant.
scientific article; zbMATH DE number 3874113