Minimal Immersions of Spheres into Spheres

• DOI: 10.2307/1970752
• Publication Date: 2006-04-24T11:51:18Z
• Authors: Manfredo P. Do Carmo; Nolan R. Wallach

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.