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MINIMAL IMMERSIONS OF SPHERES INTO SPHERES

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PMC
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  • Earth Science
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Abstract

In this paper we announce a qualitative description of an important class of closed n-dimensional submanifolds of the m-dimensional sphere, namely, those which locally minimize the n-area in the same way that geodesics minimize the arc length and are themselves locally n-spheres of constant radius r; those r that may appear are called admissible. It is known that for n = 2 each admissible r determines a unique element of the above class. The main result here is that for each n ≥ 3 and each admissible r ≥ [unk]8 there exists a continuum of distinct such submanifolds.

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