Twist or theory of immersions of surfaces in four-dimensional spheres and hyperbolic spaces

Abstract

Let f : S → S(^4) be an immersion of a Riemann surface in the 4-sphere. The thesis begins with a study of the adapted moving frame of / in order to produce conditions for certain naturally defined lifts to SO(5)/U(2) and S0(5)T(^2) to be conformal, harmonic and holomorphic with respect to two different but naturally occuring almost complex structures. This approach brings together the results of a number of authors regarding lifts of conformal, minimal immersions including the link with solutions of the Toda equations. Moreover it is shown that parallel mean curvature immersions have haj-monic lifts into S0(5)/U(2).A certain natural lift of / into CP(^3), the twistor space of S(^4), is studied more carefully via an explicit description and in the case of / being a conformal immersion this gives a beautiful and simple formula for the lift in terms of a stereographic co-ordinate associated to /. This involves establishing explicitly the two-to-one correspondence between elements of the matrix groups Sp(2) and SO(5) and working with quaternions. The formula enables properties of such lifts to be explored and in particular it is shown that the harmonic sequence of a harmonic lift is either finite or satisfies a certain symmetry property. Uniqueness properties of harmonic lifts are also proved. Finally, the ideas are extended to the hyperbolic space H(^4) and after an exposition of the twistor fibration for this case, a method for constructing superminimal immersions of surfaces into H'^ from those in S"' is given.