COMPLETE STATIONARY SURFACES IN ${\mathbb R}^4_1$ WITH TOTAL GAUSSIAN CURVATURE – ∫ KdM = 4π

Abstract

We classify complete, algebraic, spacelike stationary (i.e. zero mean curvature) surfaces in four-dimensional Lorentz space [Formula: see text] with total Gaussian curvature – ∫ K d M = 4π. Such surfaces must be orientable surfaces, congruent to either the generalized catenoids or the generalized Enneper surfaces. The least total Gaussian curvature of a non-orientable algebraic stationary surface is 6π, which can be realized by Meeks' Möbius strip and its deformations (and also by a new class of non-algebraic examples). When the genus of its oriented double covering [Formula: see text] is g, we obtain the lower bound 2(g + 3)π, which is conjectured to be the best lower bound for each g.