Complete Minimal Cylinders Properly Immersed in the Unit Ball

Abstract

Abstract. The Calabi-Yau conjecture is one of the main problems in the global theory of complete minimal surfaces in R 3 . Francisco Martin and Santiago Morales have constructed complete proper minimal surfaces in convex bodies of R 3 . In this paper, we modify their technique in the cylindrical case, and construct a complete minimal cylinder properly immersed in the unit ball.

On the spectrum of bounded immersions

“…(iii) Alarcon, Ferrer and Martin in [1] extended the results in [20] and [30] from disks and annuli to open surfaces M with any finite topology;…”

“…Example 1. Consider a bounded, complete minimal annulus f : A → R 3 (either the proper one constructed in [30], or the one in [22]). Then, taking the universal covering π : M → A, ϕ = f • π is a bounded, complete minimal surface with non-empty essential spectrum, as it is for every infinite-sheet covering (for instance, one can note that M has the ball property, see Definition 1 and Corollary 3 below).…”

“…(i) with a highly nontrivial refinement of Nadirashvili's technique, F. Martin and S. Morales in [20] proved that, for each convex domain D ⊆ R 3 not necessarily bounded or smooth, there exists a complete minimal disk B ⊆ C properly immersed into D. Later on, M. Tokuomaru in [30] constructed a complete minimal annulus which is proper in the unit ball of R 3 ;…”

Spectrum Estimates and Applications to Geometry

“…Among the new questions one regards the spectrum of bounded minimal surfaces, and among them he asked whether the spectrum of bounded minimal surfaces of R 3 was discrete. Yau's questions motivated the construction of a large number of exotic examples of minimal surfaces in R 3 that followed from Jorge-Xavier and Nadirashvili's methods, see [2], [3], [4], [63], [92], [93], [99], [100], [101], [102], [132].…”

Mean curvature of hypersurfaces in Killing submersions with bounded shadow

“…For instance, in his celebrated paper [14], Nadirashvili constructed a complete minimal surface inside a round ball in . Later on the construction of minimal immersions inside of bounded domains of the Euclidean space was carried out by Martín, Morales, Tokuomaru, Alarcón, Ferrer and Meeks among others (see [2, 7, 12, 13, 20]). More recently, Alarcón and Forstnerič have proved in [1] that every bordered Riemann surface carries a conformal complete minimal immersion into with bounded image.…”