Conformal structure of minimal surfaces with finite topology

Abstract

Abstract. In this paper we show that a complete, embedded minimal surface in R 3 , with finite topology and one end, is conformal to a once-punctured compact Riemann surface. Moreover, using this conformal structure and the embeddedness of the surface, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic to that of the helicoid. More precisely, if g is the stereographic projection of the Gauss map, then in a neighborhood of the puncture, g.p/ D exp.i˛z.p/ C F .p//, where˛2 R, z D x 3 C ix 3 is a holomorphic coordinate defined in this neighborhood and F .p/ is holomorphic in the neighborhood and extends over the puncture with a zero there. As a consequence, the end is asymptotic to a helicoid. This completes the understanding of the conformal and geometric structure of the ends of complete, embedded minimal surfaces in R 3 with finite topology.Mathematics Subject Classification (2010). 53A10; 49Q05.

Bounds on the topology and index of minimal surfaces

“…Since the surface Σ(1, 2) is area-minimizing in W , there is only one such an annular graph. A similar description can be made for Σ (1,2) in the ball B 2 . After removing the portion of Σ(1, 2) inside the innermost cylinder in each of these balls, we obtain a connected, non-compact, stable minimal surface Σ(1, 2) whose Gauss map G : Σ(1, 2) → S 2 satisfies that G (∂ Σ(1, 2)) is contained in a small neighborhood of the limiting normal directions of the corresponding forming catenoids C 1 , C 2 .…”

“…This proves the main statement in item 3 of Theorem 2.2. Item 3(a) follows from work of Bernstein and Breiner [1] or Meeks and Pérez [27]. Item 3(b) occurs when the number k of ends of L 1 satisfies 2 ≤ k < ∞, as follows from Collin [9].…”

Stationary isothermic surfaces in Euclidean 3-space

“…Thus, is a C ∞ hypersurface embedded in R N . 3 Now, since is a connected component of ∂ D, we notice that, in view of (1.4), property (1) and the smoothness of imply that for each x ∈ there exists a unique ξ ∈ ∂ satisfying x ∈ ∂ B(ξ, R), (2.4) since ξ − x must be parallel toν(x). Note that ξ = x + Rν(x), and in view of property (1) and (2.4), comparing the principal curvatures at x of with those of the sphere ∂ B(ξ, R) yields that…”

“…is the inverse mapping of the previous diffeomorphism, property (3) holds. Both properties (4) and (5) follow from the fact that J = .…”

“…5 Uniformly dense domains in R 3 : the case = ∂ With the aid of Nitsche's result [27], the theory of embedded minimal surfaces of finite topology in R 3 [3,8,24] gives the following generalization of [16, [3] and [24].…”