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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

1 2 FRANCIS BURSTALL, FRANZ PEDIT, AND ULRICH PINKALL hierarchy. These "finite type" tori can then be parameterized by theta functions on some finite genus Riemann surface. As said, this is just a picture and we feel there still is a lot more explaining to be done to become a useful mathematical theory. This present note aims to give a self contained, low technology approach to the above mentioned topics. The natural setting of our discussion is Möbius invariant surface geometry. Since we do not claim to have a complete theory, the exposition will at times be sketchy and leave the Reader, hopefully, with some urge to dwell further on the issues.The most popular integrable hierarchy is the KdV hierarchy. Therefore we begin this paper by deriving this hierarchy as natural flows on holomorphic maps into the Riemann sphere. The basic invariant of a holomorphic map f , its Schwarzian derivative S z (f ) with respect to a coordinate z, turns out to be the function satisfying the equations of the KdV hierarchy. The Schwarzian occurs naturally in a Hill equation2 S z (f )ψ = 0 for some appropriate homogeneous lift [ψ] = f . Notice that under a change of coordinates the Schwarzian transforms aswhich makes it conceptually harder to understand what object the Schwarzian really is (see, however [6]). This approach to KdV is well known [21], but our derivations are closer to the Möbius geometry of conformally immersed Riemann surfaces into the n sphere.

1 4π (W − W * ) = (n + 1)(n(1 − g) − d) + ord H , which generalizes the classical Plücker formula of a complex holomorphic curve [12]. Roughly speaking, ord H counts the singularities of all the higher osculating curves of the Kodaira embedding of L, and W * is the Willmore energy of the dual curve, i.e., the highest osculating curve. For example, a holomorphic curve f in HP n has ord H = 0 if and only if f is a Frenet curve which, for n = 1, simply means that f is immersed. Note that in the complex case, where W = 0, the only Frenet curve is the rational normal curve. This demonstrates again the rather special flavor of the complex theory from our viewpoint. Since W * is nonnegative we obtain the lower boundin which case δ ∈ Γ(KHom + (V, H/V )). Now assume that the flat rank n bundle H has a complex structure J ∈ Γ(End(H)), J 2 = −1. According to (20), the flat connection ∇ on H decomposes intowith ∇ ′′ =∂ + Q a holomorphic structure on H. Then the dual connection ∇ on H −1 decomposes, with respect to the dual complex structure, into

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