Doc Math J DMV Quaternionic Analysis on Riemann Surfaces and Differential Geometry 1 2 Franz Pedit and Ulrich Pinkall Abstract We present a new approach to the dierential geometry of 3 4 surfaces in R and R that treats this theory as a quaternionied ver sion of the complex analysis and algebraic geometry of Riemann surfaces Mathematics Sub ject Classication C H Introduction Meromorphic functions Let M b e a Riemann surface Thus M is a twodimensional dierentiable manifold equipp ed with an almost complex structure J ie on each tangent space T M we p 2 have an endomorphism J satisfying J making T M into a onedimensional p complex vector space J induces an op eration on forms dened as X JX A map f M C is called holomorphic if df i df 1 A map f M C fg is called meromorphic if at each p oint either f or f is holomorphic Geometrically a meromorphic function on M is just an orientation 2 preserving p ossibly branched conformal immersion into the plane C R or 1 2 rather the sphere CP S 4 Now consider C as emb edded in the quaternions H R Every immersed 4 4 surface in R can b e descib ed by a conformal immersion f M R where M is a suitable Riemann surface In Section we will show that conformality can again b e expressed by an equation like the CauchyRiemann equations df N df 2 3 where now N M S in R Im H is a map into the purely imaginary quaternions of norm In the imp ortant sp ecial case where f takes values in 1 Research supp orted by NSF grants DMS DMS and SFB at TUBerlin 2 Research supp orted by SFB at TUBerlin Documenta Mathematica Extra Volume ICM II Franz Pedit and Ulrich Pinkall 3 R Im H N is just the unit normal vector for the surface f In the dierential 4 geometry of surfaces in R N is called the left normal vector of f Meromorphic 4 4 1 functions f M R fg S HP are dened as in the complex case To summarize from the quaternionic viewp oint i is just one sp ecial imag inary quaternion of norm one The transition from complex analysis to surface theory is done by i leaving the Riemann surface as it is ii allowing the whole of H fg as the target space of meromorphic functions iii writing the Cauchy Riemann equations with a variable i Line bundles A classical metho d to construct meromorphic functions on a Riemann surface M is to take the quotient of two holomorphic sections of a holomorphic line bundle n over M For example if M is realized as an algebraic curve in CP then the n ane co ordinate functions on CP are quotients of holomorphic sections of the inverse of the tautological bundle over M Another common way to construct meromorphic functions is to take quotients of theta functions which also can b e viewed as sections of certain holomorphic line bundles over M In Section we intro duce the notion of a holomorphic quaternionic line bundle L over M Quotients of holomorphic sections of such bundles are meromorphic conformal maps into H and every conformal map can b e obtained as such a quotient in a unique way Every complex holomorphic line bundle E gives rise to a certain holomor phic quaternionic bundle L E E The deviation of a general holomorphic quaternionic bundle L from just b eing a doubled complex bundle can b e globally measured by a quantity Z 2 W jQj called the Willmore functional of L Here Q is a certain tensor eld the Hopf eld On compact surfaces W is up to a constant the Willmore functional in the 4 usual sense of surface theory of f M H R where f is the quotient of any two holomorphic sections of L Abelian differentials A second classical metho d to construct meromorphic functions on a Riemann surface M is to use Ab elian dierentials ie integrals of meromorphic forms In the quaternionic theory there is no go o d analog of the canonical bundle K On the other hand also in the complex case forms often arise as pro ducts of sections 1 of two line bundles E and KE Notably this is the case in situations where the RiemannRo ch theorem is applied This setup carries over p erfectly to the quaternionic case including the RiemannRo ch theorem itself Documenta Mathematica Extra Volume ICM II Quaternionic Analysis on Riemann Surfaces We show that for each holomorphic quaternionic line bundle L there exists 1 a certain holomorphic quaternionic line bundle KL such that any holomorphic 1 section of L can b e multiplied with any holomorphic section of KL the pro duct b eing a closed H valued form that lo cally integrates to a confor mal map f into H df 1 In the case where KL is isomorphic to L itself we call L a spin bundle If is a nowhere vanishing holomorphic section of a spin bundle then df 3 denes a conformal immersion into R This construction is in fact a more intrinsic version of the Weierstrassrepresentation for general surfaces in space that has received much attention in the recent literature just as when expressed in 4 co ordinates gives a representation for surfaces in R The Hopf eld Q mentioned ab ove can b e identied as the Diracp otential or mean curvature halfdensity of the surface f H jdf j Q Here H is the mean curvature and jdf j is the square ro ot of the induced metric Applications The only geometric application discussed in some detail in this pap er is a rigidity 2 3 theorem for spheres if f g S R are two conformal immersions which are not congruent up to scale but have the same mean curvature halfdensity then Z 2 H This inequality is sharp Many other applications to b e discussed in a more elab orate future pap er will concern the geometry of Willmore surfaces critical p oints of the Willmore 3 4 functional b oth in R and R Moreover rudiments of an algebraic geometry of n holomorphic curves in quaternionic pro jective space HP can b e develop ed Conformal surfaces the standard example 3 Let M b e a Riemann surface and f M R a smo oth map The map f is a conformal immersion if i df v is p erp endicular to df J v for any tangent vector v where J is the complex structure on M and ii jdf v j jdf J v j for v Documenta Mathematica Extra Volume ICM II Franz Pedit and Ulrich Pinkall 2 If N M S is the oriented unit normal to f then the conformality condition can b e rephrased as df J v N df v To see the similarity with complex function theory we rewrite this condition using 3 quaternions H R Im H We will always think of R Im H as the imaginary 3 quaternions If x y R then xy x y x y With the notation the conformality condition for f b ecomes For the rest of the article we will take this to b e the dening equation for conformality also in 4 the case of maps not necessarily immersions into R 4 Definition A map f M R H is conformal if there exists a map 2 N M H such that N and df N df At immersed p oints this is equivalent to the usual notion of conformality and 3 f determines N uniquely If f is R valued then N is the oriented unit normal but otherwise N is not normal to f We will call N the left normal to f Moreover 1 1 if f is conformal so is its Mo ebius inversion f with left normal f N f Thus the ab ove denition is Mo ebius invariant and hence denes conformality of maps 4 1 f M S HP H fg Holomorphic quaternionic line bundles A quaternionic line bund le L over a base manifold is a smo oth rank real vector bundle whose b ers have the structure of dimensional quaternionic right vector spaces varying smo othly over the base Two quaternionic line bundles L and L 1 2 are isomorphic if there exists a smo oth bundle isomorphism A L L that is 1 2 quaternionic linear on each b er We adopt the usual notation Hom L L and H 1 2 End L Hom L L etc for the spaces of quaternionic linear maps H H The zero section of a quaternionic line bundle over an oriented surface has co dimension so that transverse sections have no zeros Thus any quaternionic line bundle over a Riemann surface M is smo othly isomorphic to M H Complex quaternionic line bundles Example Given a conformal map f M H with left normal N the quater nionic line bundle L M H also has a complex structure J L L given by J N for L We make this additional complex structure part of our theory Definition A complex quaternionic line bund le over a base manifold is a pair L J where L is a quaternionic line bundle and J End L is a quaternionic H 2 linear endomorphism such that J Documenta Mathematica Extra Volume ICM II Quaternionic Analysis on Riemann Surfaces Put dierently a complex quaternionic line bundle is a rank two left complex vector bundle whose complex structure is compatible with the right quaternionic structure Two complex quaternionic line bundles are isomorphic if the quater nionic linear isomorphism is also left complex linear The dual of a quaternionic line bundle L 1 L f L H quaternionic linearg has a natural structure of a left quaternionic line bundle via 1 1 for H L and L Using conjugation we can regard L as a right quaternionic line bundle If L has a complex structure then the 1 complex structure on L is given by J J 1 so that L is also complex quaternionic Any complex quaternionic line bundle L can b e tensored on the left by a complex line bundle E yielding the complex quaternionic line bundle EL On a Riemann surface M we have the canonical and anticanonical bundles K and K It is easy to see that KL f TM L J g and KL f TM L J g In this way we have split the quaternionic rank bundle Hom T M L which R has a left complex structure given by as a direct sum KL KL of two complex quaternionic