Compact Complete Minimal Immersions in R
نویسنده
چکیده
In this paper we find, for any arbitrary finite topological type, a compact Riemann surface M, an open domain M ⊂ M with the fixed topological type, and a conformal complete minimal immersion X : M → R3 which can be extended to a continuous map X : M → R3, such that X|∂M is an embedding and the Hausdorff dimension of X(∂M) is 1. We also prove that complete minimal surfaces are dense in the space of minimal surfaces spanning a finite set of closed curves in R3, endowed with the topology of the Hausdorff distance.
منابع مشابه
Compact Complete Minimal Immersions In
ANTONIO ALARC´ON ABSTRACT. In this paper we construct complete minimal surfaces with boundary in R 3 of arbitrary finite topology. 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 minimal immersion X : M → R 3 which can be extended to a continuous map X : M → R 3 , such that X |∂M is an ...
متن کامل. D G ] 3 0 A pr 2 00 8 COMPACT COMPLETE PROPER MINIMAL IMMERSIONS IN STRICTLY CONVEX BOUNDED REGULAR DOMAINS
Consider a strictly convex bounded regular domain C of R. In this paper we construct complete properly immersed in C minimal surfaces with boundary. Our examples have arbitrary finite topology. We also prove that complete properly immersed in C minimal surfaces with boundary are dense in the space of properly immersed in C minimal surfaces with boundary, endowed with the topology of the Hausdor...
متن کاملSpherical Minimal Immersions of the 3-sphere
In 1966 Takahashi [11] proved that a minimal isometric immersion f : S(1) → S(r) of round spheres exists iff r = √ m/λp, where λp is the pth eigenvalue of the Laplacian on S; in this case, the components of f are spherical harmonics on S of order p. This immersion is unique up to congruence on the range and agrees with the generalized Veronese map if m = 2 as was shown in 1967 by Calabi [1]. In...
متن کاملOn the Large-Scale Structure of the Moduli of Eigenmaps and Spherical Minimal Immersions
Minimal immersions of a compact Riemannian homogeneous manifold into round spheres, or spherical minimal immersions for short, or “spherical soap bubbles,” belong to a fast growing and fascinating area between algebra and geometry. This theory has rich interconnections with a variety of mathematical disciplines such as representation theory, convex geometry, harmonic maps, minimal surfaces, and...
متن کاملOn the Structure of Convex Sets with Applications to the Moduli of Spherical Minimal Immersions
We study the properties of certain affine invariant measures of symmetry associated to a compact convex body L in a Euclidean vector space. As functions of the interior of L, these measures of symmetry are proved or disproved to be concave in specific situations, notably for the reduced moduli of spherical minimal immersions. MSC 2000: 53C42
متن کامل