On Surfaces in R 4

Abstract

We provide answers (Theorem C) to some questions concerning surfaces in R4 and maps into the quadric Q2 raised by D. Hoffman and R. Osserman. Let S be an oriented surface immersed in R4. The Gauss map of S is the map G of S into G(2, 4), the Grassmannian of oriented two-planes in R4, given by G( p) = TpS. G(2,4) can be identified with Q2, the complex quadric in CP3, and in turn Q2 is biholomorhic to CP' x CP1. If we give CP3 the Fubini-Study metric of constant holomorphic sectional curvature 2, then the induced metric on Q2 is given by 21dw, 12/(1 + IW1 I)2 + 21dW2,2/(1 + IW21 ) where (w1, w2) are coordinates on C x C, viewed as local coordinates on CP' x CP' [1]. The metric 21dw12/(1 + Iwi2)2 is the metric on C induced by the map of C onto S2(1/ 4) c R3 given by w al1(J2iw), where a-1 is inverse stereographic projection (with the sphere sitting on the xy-plane). Thus, Q2 is isometric to S2(1/ x/) X S2(1/ v'I). In particular, if z is a local conformal parameter on S, then any map G of S into Q2 splits into a pair of maps G(z) = (f1(z), f2(z)), where w, = fi(z) as above. Now define the following quantities on S for i = 1, 2: F,2:=' T,(z)= (f1)2 wheref, O with the usual z and z derivative notation. The following results are from [1, 2]. THEOREM A. For the Gauss map G of an oriented surface S immersed in R4, write G = (f1(z), f2(z)) as above. Then we necessarily have (1) |1-I219 and (2) Im{T1+T2} O THEOREM B. Let SO be a simply connected Riemann surface (here and subsequently), let G = (f1(z), f2(z)) be some map of SO into Q2, and define Fi and Ti as before, where z is a conformal parameter on SO. (i) If F1 = F2 0, then G is the Gauss map of a minimal surface in R4, provided SO is not compact. Received by the editors April 19, 1984. 1980 Mathemnatics Subject Classification. Primary 53A05; Secondary 53A10. ?1985 American Mathematical Society 0002-9939/85 $1.00 + $.25 per page