On the image of the generalized Gauss map of a complete minimal surface in R4

Chi Chen
1982 Pacific Journal of Mathematics  
The generalized Gauss map of an immersed oriented surface M in Λ 4 is the map which associates to each point of M its oriented tangent plane in G 2A9 the Grassmannian of oriented planes in R\ The Grassmannian G 2)4 is naturally identified with Q 2 , the complex hyperquadric j[zi, z 2 , z 3 , z 4 ] Σ zl=θϊ in P 3 (C) . The normalized Fubini-Study metric on P B (C) with holomorphic curvature 2 induces an invariant metric on Q 2 = G 2A , which corresponds exactly to the metric on the canonical
more » ... n the canonical representation of S 2 (V vΎ) x S 2 (l/ VΊF) in R« as {Xe R* \ x\ + x\ + x\ = (1/2), x\ + xl + x\ -(1/2)}. The product representation above allows us to associate with any map g in Q 2 two canonical projections g u 02 In the case where g is complex analytic map defined on some Riemann surface S Q9 the projections g lf g 2 are complex analytic also. Detailed treatment can be found in the recent work of Hoffman and Osserman.
doi:10.2140/pjm.1982.102.9 fatcat:kkdwdapz6bhezficy6rb2cflem