Kichoon Yang

HOLOMORPHIC CURVES IN THE COMPLEX QUADRIC

Holomorphic curves in a complex quadric arise naturally as thecomplex conjugate of the Gauss map of a minimal surface in Euclideanspace. In addition, such holomorphic curves play a central role ingenerating a special class of harmonic maps of surfaces into spheres,complex projective space, and the complex Grassmannians. (See Eells-Wood[6], Bryan [2]t , Chern-Wolfson [51, Ramanathan L141, and many referencescited in these papers.)Received 11 March 1986.Copyright Clearance Centre, Inc. Serial-fee code: 0004-9729/87

Plücker formulae for the orthogonal group

A2.00 + 0.00.

Frenet formulae for holomorphic curves in the two quadric

Plucker formulae for horizontal curves in SO(m) -flag manifolds are derived. These formulae are seen to generalise the usual Plucker formulae for projective space curves. They also have applications in the theory of minimal surfaces in Euclidean sphere and the complex hyperquadric.

Minimal Surfaces: General Theory

We give a complete description of holomorphic curves in the complex two quadric via the method of moving frames. For compact curves a Morse theory type integral formula is derived.

Minimal Surfaces with Finite Total Curvature

This chapter contains a general introduction to minimal surfaces, without boundary, in Euclidean space. In Section 1 we review the Riemannian and conformal geometries of surfaces: The overriding observation in intrinsic surface theory is that isothermal coordinates — which always exist by the Korn-Lichtenstein theorem — provide an intimate link between the two geometries. This observation together with a bit of Hodge theory is explained in Section 1. The next section represents an excursion into the method of moving frames and a review of the geometry of submanifolds. The exposition in Section 2 is on a fairly abstract footing — this was to indicate the generality of our method. Several results of this section are used later when we discuss the Gauss map of a minimal surface. In Section 3 we prove that a conformal immersion from a Riemann surface into R n is minimal if and only if it is harmonic.

Elementary Differential Systems

In this chapter we will deal exclusively with complete minimal surfaces in R 3 with finite total curvature; for the sake of convenience we will call such a surface an algebraic minimal surface. In earlier papers [Y4, Y5, Y6, Y7] the author began a study of the Puncture Number Problem for algebraic minimal surfaces. Given a compact Riemann surface M g of genus g, a positive integer r is called a puncture number of M g if M g can be conformally immersed in R 3 as an algebraic minimal surface with exactly r punctures. The set of all puncture numbers for M g is denoted by P (M g ).

Cartan-Kaehler Theory

The totality of exterior differential forms on an n-dimensional manifold M forms a graded algebra over the ring of smooth functions on M:

Quasi-Linear Pfaffian Differential Systems

{ \wedge ^*}\left( M \right) = \mathop \oplus \limits_{p = 0}^n \;{ \wedge ^P}\left( M \right)