The local picture theorem on the scale of topology

Abstract

We prove a descriptive theorem on the extrinsic geometry of an embedded minimal surface of injectivity radius zero in a homogeneously regular Riemannian three-manifold, in a certain small intrinsic neighborhood of a point of almost-minimal injectivity radius. This structure theorem includes a limit object which we call a minimal parking garage structure on R 3 , whose theory we also develop. Mathematics Subject Classification: Primary 53A10, Secondary 49Q05, 53C42.

A New Golden Age of Minimal Surfaces

“…The answer is no, as shown in 2007 by Hauswirth and Pacard, who used gluing techniques 4 to merge a Hoffman-Meeks minimal surface (we mentioned these surfaces when describing examples in ℳ ( , 3) in the subsection "Complete Minimal Surfaces with Finite Topology") with two halves of a Riemann minimal surface ℛ. In Figure 7 (left) we can see a schematic representation of one of the examples by Hauswirth and Pacard, when the central surface to be 4 …”

“…It should also be noted that in the above description, the set̂where the Gaussian curvatures of the disks blow up consists of not only the singularities of the limit lamination ℒ but also possibly embedded arcs of class 1,1 around which ℒ is a local foliation, as in Figure 11 We have said that in order to apply the theory of Colding-Minicozzi to a sequence of embedded minimal surfaces, we need to assume that the are compact disks with boundaries in ambient spheres. This condition is not really a restriction, as it can be naturally obtained by a rescaling argument so that the injectivity radius function of the rescaled minimal surfaces is uniformly bounded away from zero (Meeks, Pérez, and Ros [4]). …”

Structure theorems for singular minimal laminations

“…The second result, Theorem 4.1, describes the structure of a singular minimal lamination of R 3 whose singular set is countable. Both results depend on the local theory of embedded minimal surfaces and minimal laminations developed in [23,30,31], and on the previously mentioned work of Colding and Minicozzi. For the definition and the general theory of minimal laminations, see for instance [20,26,27,31,32,33].…”

“…We will next describe both the limit object of the surfaces M n ∩ B(p, 1/k) as n → ∞ and the surfaces themselves for n large; this description relies on Colding-Minicozzi theory and is adapted from a similar description in [23]; we have include it here as well for the sake of completeness.…”

Triply periodic constant mean curvature surfaces

“…Either M n has locally bounded norm of the second fundamental form in R 3 or not. If M n does NOT have locally bounded norm of the second fundamental form in R 3 then by Theorem 1.5 in [22], the surfaces M n converge on compact subsets of R 3 to a minimal parking garage structure of R 3 with two oppositely oriented columns, see Figure 1 and see for instance [12] for a detailed description of this limit object. Figure 1: Parking garage structure: in this picture, the sequence of surfaces on the left hand side converge smoothly away from the union S 1 ∪ S 2 or two straight lines orthogonal to the foliation of horizontal planes described on the right hand side.…”