CONNECTEDNESS OF THE SPACE OF MINIMAL 2-SPHERES IN S2m(l)

Loo's theorem asserts that the space of all branched minimal 2spheres of degree d in 54(1) is connected. The main theorem in this paper is that the assertion is still true for S2m (1). It is shown that any branched minimal 2-sphere in S2m (1) can be deformed, preserving its degree, to a meromorphic function. 0. Introduction After the celebrated research on minimal 2-spheres in the unit sphere 5^(1) by Calabi [Ca], there was much attention given not only to the study of individual minimal spheres but also to the structure of the space of all minimal 2-spheres in SN( 1). Calabi proved that if a minimal 2-sphere is immersed fully in SN(l), then N must be even. The simplest case is the space of all minimal 2-spheres of degree d in S2(l). This space has two connected components. One component is identified with the space of all meromorphic functions of degree d; the other is its conjugate. These two components are connected in the space of all minimal 2-spheres of degree d in S3(l). Recently, Loo [L] determined the space of all minimal 2-spheres S2 of degree d in the unit 4-sphere 54(1). In particular, he proved that this space is connected. In this paper, we prove that the space of all branched minimal 2-spheres S2 of degree d in the unit TV-sphere 5^(1) is connected for N > 3. We shall see that any branched minimal 2-sphere g : S2 —► 52m(l) of degree d can be deformed to a nonfull minimal sphere of degree d . By repeating this process, g is deformed eventually to a ± meromorphic function S2 —► 5*2(1) of degree d . In other words, every element in the space of all minimal spheres S2 —► S2m(l) of degree d is connected to a ± meromorphic function of degree d. Every two ± meromorphic functions are connected as we noted above. From these facts, we can prove that the space of all g : S2 —► 52m(l) of degree d is connected. Received by the editors June 4, 1991 and, in revised form, June 22, 1992. 1991 Mathematics Subject Classification. Primary 49F10; Secondary 58E20.