Compact complete proper minimal immersions in strictly convex bounded regular domains of R[sup 3]

Abstract

Consider a strictly convex bounded regular domain C of R 3 . For any arbitrary finite topological type we find a compact Riemann surface M, an open domain M ⊂ M with the fixed topological type, and a conformal complete proper minimal immersion X : M → C which can be extended to a continuous map X : M → C. 2000 Mathematics Subject Classification 53A10 · 53C42 · 49Q10 · 49Q05

Every bordered Riemann surface is a complete conformal minimal surface bounded by Jordan curves: Figure. 5.1.

“…It may be viewed as a version of Theorem 1.1 in which we additionally ensure that the boundary curves of a minimal surface are contained in the boundary of the domain, at the cost of losing topological embeddedness of these curves. A partial result in this direction can be found in [3] where the first named author constructed compact complete proper minimal immersions of surfaces with arbitrary finite topology into smoothly bounded strictly convex domains in R 3 , but without control of the conformal structure on the surface or the flux of the immersion.…”

Complete Non-Orientable Minimal Surfaces in ℝ3 and Asymptotic Behavior

“…In the orientable setting, compact complete minimal immersions of the disc into R 3 were constructed by Martín and Nadirashvili [10]; examples with arbitrary finite topology were given later by Alarcón [1]. Other related results can be found in [8,2]. The construction methods used in [10,1] are refinements of Nadirashvili's technique for constructing complete bounded minimal surfaces in R 3 ; see [14].…”

Minimal surfaces in minimally convex domains

“…. , m}, then F (D) ⊂ Co(Γ j ) by the maximum principle, a contradiction since Co(Γ j ) does not intersect the ball 1 2 B. If on the other hand Λ(F ) ⊂ 2S, there is a point ζ 1 ∈ D with F (ζ 1 ) ∈ S. Pick j ∈ {1, .…”

“…Since F is complete, the minimal surface F ( M ) ⊂ D has infinite area, and hence its boundary F (bM ) is necessarily non-rectifiable in view of the isoperimetric inequality. The corresponding result for smoothly bounded strongly convex domains in R n for any n ≥ 3 is [2, Theorem 1.2]; see also [1] for a previous partial result in this line. As in the latter result, we are unable to achieve that F be a topological embedding on bM , so F (bM ) needs not consist of Jordan curves.…”

“…Example 1.13. We exhibit a simply connected domain D ⊂ R 3 of the form D = 2B \ K, where B is the unit ball of R 3 and K is a compact set contained in a thin shell around the unit sphere S = bB, such that the image of every proper conformal minimal disc F : D → D avoids the ball 1 2 B ⊂ D. Clearly, Theorem 1.1 fails in this example even without the completeness condition. Note however that D admits complete properly immersed minimal surfaces normalized by any given bordered Riemann surface in view of Theorem 1.7, applied to the strongly convex boundary component 2S ⊂ bD.…”