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

Abstract

Applying the general theory about complete spacelike stationary (i.e. zero mean curvature) surfaces in 4-dimensional Lorentz space R 4 1 , we classify those regular algebraic ones with total Gaussian curvature − KdM = 4π. Such surfaces must be oriented and be congruent to either the generalized catenoids or the generalized enneper surfaces. For non-orientable stationary surfaces, we consider the Weierstrass representation on the oriented double covering M (of genus g) and generalize Meeks and Oliveira's Möbius bands. The total Gaussian curvature are shown to be at least 2π(g + 3) when M → R 4 1 is algebraic-type. We conjecture that there do not exist non-algebraic examples with − KdM = 4π.

Complete stationary surfaces inR14with total Gaussian curvature−∫KMa1 2013Differential Geometry and its ApplicationsSelf Cite1010View full textAdd to dashboardCite

“…Such an immersed surface x : M → R 4 1 is called a stationary surface. Based on this general theory, in [5] we classified those algebraic ones with least possible total (Gaussian) curvature − KdM = 4π. This extends a classical result of Osserman that a complete minimal surface in R 3 with total curvature 4π is either the catenoid or the Enneper surface [8].…”

“…Another work in [5] is that we generalized the theory of non-orientable minimal surfaces in R 3 [6,7,3,4] to our setting. This is done by the same idea, namely, lifting everything to the oriented double covering surface M .…”

“…On the other hand, new examples with a single good singular end are shown to exist.By Gauss-Bonnet type formulas established in [2], we know that complete algebraic stationary surfaces with − KdM = 6π must be non-orientable. On the other hand, we obtained a lower bound in [5] for any g ≥ 0 as below:(1)Thus it is natural to study complete, algebraic, non-orientable stationary surfaces with total curvature 6π, which is the least possible value. By (1) and the generalized Jorge-Meeks formula (see Theorem 2.7), we know such surface M must be homomorphic to a projective plane with one or two ends.…”

“…In particular, the topological type of a projective plane with two punctures does not occur.New examples appear in R 4 1 . We have constructed complete stationary Möbius strips with total curvature 6π in [5]. Among them, one family is a generalization of Meeks and Olivaira's examples; another family has an essential singularity at the end.…”

“…By Gauss-Bonnet type formulas established in [2], we know that complete algebraic stationary surfaces with − KdM = 6π must be non-orientable. On the other hand, we obtained a lower bound in [5] for any g ≥ 0 as below:…”