Minimal surfaces in minimally convex domains

Abstract

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface M into a minimally convex domain D ⊂ R 3 can be approximated, uniformly on compacts inM = M \bM , by proper complete conformal minimal immersionsM → D (see Theorems 1.1, 1.7, and 1.9). We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in R 3 (see Theorem 1.16), and we characterize the minimal surface hull of a compact set K in R n for any n ≥ 3 by sequences of conformal minimal discs whose boundaries converge to K in the measure theoretic sense (see Corollary 5.6).

New Complex Analytic Methods in the Theory of Minimal Surfaces: A Survey

“…Note that the boundary discs κ(ζ, · ) (ζ ∈ bM ) lie in affine 2-planes. A more precise result is available in dimension n = 3; see [8,Theorem 3.2]. In that case the map κ (5.2) may be chosen of the form…”

“…In the subsequent work [8] of the same authors this result was extended to the substantially bigger class of all minimally convex (also called 2-convex) domains. A domain Ω ⊂ R n is minimally convex if it admits a smooth exhaustion function ρ : Ω → R + such that the smallest two eigenvalues λ 1 (x), λ 2 (x) of its Hessian…”

“…If M is a bordered Riemann surface for which one is able to construct in a standard inductive way a proper conformal minimal immersion into a given domain D ⊂ R n , then one can also construct a complete proper one by the procedure in Lemma 5.5 which enlarges the intrinsic boundary distance within the surface as much as desired by an arbitrarily small displacement of the surface in D. Hence it suffices to focus on the existence of proper conformal minimal immersions. Recent examples by Alarcón et al [8] show that some geometric assumptions on the domain are necessary to obtain positive results. Indeed, there is a bounded simply connected domain D ⊂ R 3 carrying no proper conformal minimal disc D → D passing through a certain point in D, and a bounded domain D ⊂ R 3 admitting no proper minimal surfaces with finite topology and a single end (see [8,Examples 1.13 and 1.14]).…”

“…Recent examples by Alarcón et al [8] show that some geometric assumptions on the domain are necessary to obtain positive results. Indeed, there is a bounded simply connected domain D ⊂ R 3 carrying no proper conformal minimal disc D → D passing through a certain point in D, and a bounded domain D ⊂ R 3 admitting no proper minimal surfaces with finite topology and a single end (see [8,Examples 1.13 and 1.14]). Much earlier, Dor [38] found a bounded domain Ω ⊂ C m for any m ≥ 2 which does not admit any proper holomorphic discs D → Ω.…”

Holomorphic Embeddings and Immersions of Stein Manifolds: A Survey

“…The second method is precisely the opposite -it keeps embeddedness, but does not provide any control of the complex structure since one must cut away pieces of the image manifold to keep it suitably bounded. The first of these methods has recently been applied in the theory of minimal surfaces in R n ; we refer to the papers [4,5,7] and the references therein. On the other hand, ambient automorphisms cannot be applied in minimal surface theory since the only class of self-maps of R n (n > 2) mapping minimal surfaces to minimal surfaces are the rigid affine linear maps.…”