Minimal surfaces in R3 and algebraic curves

AbstractThere exist a natural correspondence, determined by classical osculation duality, between null holomorphic curves in P3 ≃ C3 ∪ P2 and holomorphic curves in P∗3 that lie on C (Q̃1), the projective cone over a certain quadric curve Q̃1. This facilitates the study of minimal surfaces in R3 in terms of holomorphic curves on C(Q̃1). Algebraic curves on C(Q̃1) generate complete branched minimal surfaces of finite total Gaussian curvature. The ‘end’ structure, branch points and total Gaussian curvature of the minimal surface are determined by features of the corresponding algebraic curve. Natural compactifications of the moduli spaces of null meromorphic curves in C3 are given by linear systems on the Hirzebruch surface S2