English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We show there is a global correspondence between branched constant mean curvature (i.e. CMC-) immersions in S3/{±1} and pairs of nonconformal harmonic maps into S2 in the same associated family. Furthermore, we give two applications. Let x be a conformal immersion of a Riemann surface M into the Euclidean 3-sphere S(c) of radius 1/c, or the real projective space P (c). The generalized Gauss map G : M → G2,2 is defined by G(z) = [xu ∧ xv], where G2,2 is the Grassmann manifold of oriented 2-planes in R and z = u + iv is a complex coordinate on M . Since G2,2 is isometric to the product of two unit spheres, G splits into G = (g1, g2) : M → S × S. When x is a non-totally umbilic CMC-H immersion, g1 and g2 are both non-conformal harmonic maps in the same associated S-family (see [1], [5], [8] and [9]). More precisely, the holomorphic quadratic differential φj (j = 1, 2) of gj satisfies the relation φ2 = eφ1 (α := arg(H + ic) ∈ (0, π/2]). In this case, we express g2 = g 1 . Conversely, we prove the following: Theorem. Let g : M → S be a non-conformal harmonic map. Suppose that there exists a real number α ∈ (0, π/2] such that the associated harmonic map g is single-valued on M . Then there exists a branched conformal CMC-H immersion x of M into P (c) with G = (g, g), where H = c cotα. Proof. There exists a CMC-H immersion x : M̃ → S(c) having the generalized Gauss map G̃ = (g ◦ π, g ◦ π) (see, for instance, [1] and [8, §4 (replace H by H1)]), where π : M̃ → M is the universal cover. Let U1 and U2 be domains in M̃ such that π(U1) = π(U2). Then x(U1) is congruent with x(U2) by an isometry Φ of S(c). Since the tangent planes at the corresponding points are parallel to each other because G̃ has the same value, we must have Φ = ±id. Remark. The theorem also follows from Bobenko [2, Theorem 14.1]. The parallel branched CMC-immersion arising as the Bonnet pair has the reversed orientation. Taking a finite cover, we get a branched CMC-H immersion into S(c). However, x is non-branched when M is a torus [5, (4.35)]. Received by the editors April 15, 1998. 2000 Mathematics Subject Classification. Primary 53C42; Secondary 53A10. c ©1999 American Mathematical Society

A global correspondence between CMC-surfaces in $S^3$ and pairs of non-conformal harmonic maps into $S^2$