Conformal Geometry of Surfaces in S4 and Quaternions

Abstract

This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued function theory, whose "meromorphic functions" are conformal maps into H, which extends the classical complex function theory on Riemann surfaces. The first results along these lines were presented at the ICM 98 in Berlin [7]. Basic constructions of complex Riemann surface theory, such as holomorphic line bundles, holomorphic curves in projective space, Kodaira embedding, and Riemann-Roch, carry over to the quaternionic setting. Additionally, an important new invariant of the quaternionic holomorphic theory is the Willmore energy. For quaternionic holomorphic curves in HP 1 this energy is the classical Willmore energy of conformal surfaces. The present paper is based on a course given by one of the authors at the Summer School on Differential Geometry at Coimbra in September, 1999. It centers on Willmore surfaces in the conformal 4-sphere HP 1 . The first three sections introduce linear algebra over the quaternions and the quaternionic projective line as a model for the conformal 4-sphere. Conformal surfaces f : M → HP 1 are identified with the pull-back of the tautological bundle. They are treated as quaternionic line subbundles of the trivial bundle M × H 2 . A central object, explained in section 5, is the mean curvature sphere (or conformal Gauss map) of such a surface, which is a complex structure on M × H 2 . It leads to the definition of the Willmore energy, the critical points of which are called Willmore surfaces. In section 7 we identify the new notions of our quaternionic theory with notions in classical submanifold theory. The rest of the paper is devoted to applications: We classify super-conformal immersions as twistor projections from CP 3 in the sense of Penrose, we construct Bäcklund transformations for Willmore surfaces in HP 1 , we set up a duality between Willmore surfaces in S 3 and certain minimal surfaces in hyperbolic 3-space, and we give a new proof of a recent classification result by Montiel on Willmore 2-spheres in the 4-sphere.1

Sequences of Willmore surfaces

“…This discussion, together with Theorem 2.2, gives a unified explanation of results by various authors [3], [7], [15], [4], [14] on the characterization of Willmore surfaces:…”

“…What distinguishes the mean curvature sphere congruence S among touching sphere congruences is a second order touching condition [4,Thm. 2].…”

“…Since this result is interesting in its own right and relies on the theory of 1-step Bäcklund transforms developed in [4], [12], [13], and [11], we postpone its proof to the last section of this paper.…”

“…We choose the adapted framing (ψ, e) of V = L ⊕ eH with ψ = g 1 a smooth section of L. In this frame the mean curvature sphere congruence S of f is given by [4, Sec. 7.2] Sψ = −ψR and Se = eN − ψH with R, N, H : M → H satisfying R 2 = N 2 = −1 and RH = N R. The conformality and touching conditions * δ = Sδ = δS translate to * dg = N dg = −dgR so that N and R are the left and right normals of g : M → H. That S is the mean curvature sphere congruence along f is expressed by [4,Sec. 7.2]…”

Examples of Non-Conformal Submersions from Spheres with Umbilical Fibers

“…In affine coordinates on HP 1 [1], the images of all points of M for which x 1`i x 2 ‰ 0, will be represented in the form of p1, zq with z P C (similarly one can consider the case when x 3`i x 4 ‰ 0). Embedding C in H, we see that πpM q is a submanifold of πpS 7 q.…”