On minimal immersions of $S^2$ in $S^{2m}$

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 ...

Compact Complete Minimal Immersions in R

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 spac...

Spherical Minimal Immersions of Spherical Space Forms

Introduction. A number of authors [C], [DW1], [DW2], [L], [T] have studied minimal isometric immersions of Riemannian manifolds into round spheres, and in particular of round spheres into round spheres. As was observed by T. Takahashi [T], if Φ:M → S(r) ⊂ R is such a minimal immersion, then the components of Φ must be eigenfuctions of the Laplace operator on M for the same eigenvalue. And conve...

Minimal Immersions of Symmetric Spaces into Spheres

Spherical 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...