The topology of complete minimal surfaces of finite total Gaussian curvature

The classical theory of minimal surfaces

“…In this setting, the Gauss map g has a well-defined finite degree on M . A direct consequence of (3) is that the total curvature of an M as in Theorem 2.11 is −4π times the degree of its Gauss map g. It turns out that this degree can be computed in terms of the genus of the compactification M and the number of ends, by means of the Jorge-Meeks formula 12 [89]. Rather than stating here this general formula for an immersed surface M as in Theorem 2.11, we will emphasize the particular case when all the ends of M are embedded: (9) deg(g) = genus(M ) + #(ends) − 1.…”

Flat fronts in hyperbolic 3-space

“…In these two cases, as in ours, equality implies the embeddedness of ends. (See [8,5] for the minimal surface case and [12] for the hyperbolic case.) To prove that equality implies the ends are embedded, a criterion for embeddedness of ends given in [4] will be applied.…”

The geometry of minimal surfaces of finite genus II; nonexistence of one limit end examples

“…Collin's Theorem reduces the question of topological obstructions for M ∈ M of finite topology and more than one end to the question of topological obstructions for complete embedded minimal surfaces of finite total curvature in R 3 . For example, if M is a complete embedded minimal surface in R 3 with finite total curvature, genus g and k ends, then M is properly embedded in R 3 and the Jorge-Meeks formula [20] calculates its total curvature to be −4π(g + k − 1). The first topological obstructions for complete embedded minimal surfaces M of finite total curvature were given by Jorge and Meeks [20], who proved that if M has genus zero, then M does not have 3, 4 or 5 ends.…”

“…For example, if M is a complete embedded minimal surface in R 3 with finite total curvature, genus g and k ends, then M is properly embedded in R 3 and the Jorge-Meeks formula [20] calculates its total curvature to be −4π(g + k − 1). The first topological obstructions for complete embedded minimal surfaces M of finite total curvature were given by Jorge and Meeks [20], who proved that if M has genus zero, then M does not have 3, 4 or 5 ends. Later this result was generalized by López and Ros [22] who proved that the plane and the catenoid are the only genus zero minimal surfaces of finite total curvature in M. About the same time, Schoen [37] proved that a complete embedded minimal surface of finite total curvature and two ends must be a catenoid.…”

“…Thus, one obtains the following corollary to Theorem 1. It is known that for any integer k ≥ 2, the k-noid defined by Jorge and Meeks [20] has genus zero, k catenoid type ends and index 2k − 3 (Montiel-Ros [33] and Ejiri-Kotani [13]). Also, there exist examples of complete immersed minimal surfaces of genus zero with a finite number of parallel catenoidal ends which satisfy the Jorge-Meeks total curvature formula, but which have arbitrarily large index of stability.…”