Fine structure of the space of spherical minimal immersions
نویسندگان
چکیده
منابع مشابه
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...
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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...
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Eigenmaps and spherical minimal immersions of round spheres into round spheres form an interesting subject that has been studied by many authors. The components of an eigenmap are eigenfunctions of the Laplace-Beltrami operator on Sm, the Euclidean m-sphere, m ≥ 2; and the eigenfunctions correspond to a fixed eigenvalue λk, k ≥ 1. Given m ≥ 2 and k ≥ 1, these eigenmaps can be parametrized by a ...
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The boundary of a DoCarmo-Wallach moduli space parametrizing (harmonic) eigenmaps between spheres or spherical minimal immersions carries a natural stratification. In this paper we study the critical points of the distance function on the boundary strata. We show that the critical points provide a natural generalization of eigenmaps with L-orthonormal components. We also point out that many cla...
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...
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ژورنال
عنوان ژورنال: Transactions of the American Mathematical Society
سال: 1996
ISSN: 0002-9947,1088-6850
DOI: 10.1090/s0002-9947-96-01588-7