The existence of minimal immersions of two-spheres

Abstract

In this article we announce a series of results on the existence of harmonic maps from surfaces to Riemannian manifolds and, as corollaries of these results, obtain theorems on the existence of minimal immersions of 2-spheres.Let N be a compact connected Riemannian manifold and, for convenience, assume that N is isometrically imbedded in R k for some sufficiently large k. Let M be a closed Riemann surface with any metric compatible with its conformaiHarmonic maps satisfy an Euler-Lagrange equationAs + A(s)(ds f ds) = 0 in a weak sense, where A is the second fundamental form of the imbedding N C R fc . It then follows from regularity theorems that harmonic maps are C°°. If s is harmonic and a conformai immersion, it is also an extremal for the area integral.Proving the existence of harmonic maps of M into N by direct methods from global analysis such as Morse theory or Ljusternik-Schnirelman theory applied to E defined on some function space manifold is difficult, because E is invariant under the conformai group of M y and the extremal maps of E form a noncompact set when M = S 2 . In particular, E does not satisfy condition C of Palais-Smale. However, for a > 1, a slightly different integral, AMS (MOS) subject classifications (1970). Primary 53A10, 58E05.

Minimal immersions of closed Riemann surfaces

“…In a previous paper we obtained results on minimal immersions of the two-sphere into compact Riemannian manifolds [18]. Here we extend some of these results to surfaces of higher topological type.…”

“…In §2 we state some convergence properties of sequences of harmonic maps proved in [18] and extend earlier results to the case of variable conformai structures on the domain surface in Theorem 2.3. In §3 we prove a version of our main result for the special case of genus one, where the argument is technically somewhat different, but the result, Theorem 3.3, is essentially the same as the main Theorem 4.4, which is proved in §4.…”

“…In §1 we discuss the existence theory and properties of harmonic maps and recall results from our earlier paper [18]. In §2 we state some convergence properties of sequences of harmonic maps proved in [18] and extend earlier results to the case of variable conformai structures on the domain surface in Theorem 2.3.…”

“…In this section we outline the results on harmonic maps that we shall be using in the rest of the paper. A more detailed account and proofs are contained in [18].…”

“…Let s: M -» TV be a conformai branched immersion with respect to the conformai structure ¡i on M. Then s is harmonic with respect to p if and only if s is a branched minimal immersion. Theorem 1.2 [18,Theorem 1.8]. Let s: M -» TV be harmonic with respect to the conformai structure p and suppose that p is a critical point of E(s, ■) with respect to all C°° variations of p. Then s is a conformai branched minimal immersion of M into N.…”