Total curvature of branched minimal surfaces

Abstract

An intrinsic, and much simpler, proof of a generalization of Jorge and Meeks' total curvature formula for complete minimal surfaces is given. Let X: M -? j3 be a complete minimal surface with finite total curvature. Osserman proved that conformally M = Sk {P1i., Pn }, n > 1, where Sk is a closed Riemann surface of genus k. See, for example, [4, Theorem 9.1, page 81]. Each Pi corresponds to an end Ei of M. Jorge and Meeks [2] proved that there is an integer Ii > 1 corresponding to Ei, such that the total curvature of M is given by ( l) JoKdA = 27 X%(M)-E Ii) where %(M) = 2(1 k) n is the Euler characteristic of M. The proof of Jorge and Meeks involves a detailed study of the behaviour of the image X(M) at each end. Since the Gauss curvature is an intrinsic quantity, it is natural to look for an intrinsic proof. In this note we give such an intrinsic, and much simpler, proof of a generalization of (1). The Enneper-Weierstrass representation of a branched complete minimal surface of finite total curvature X: M -? 1R3 is given by (2) X(p) =Re ( (192)) (1I+ g2),) 7, where g: M = Sk {Pi1, ,Pn} IC U {xo} is a meromorphic function and i7 is a holomorphic 1-form on M. Both g and q can be extended to Sk as a meromorphic function and 1-form respectively; see [4, Theorem 9.1, page 81]. Note that since the proof given there only involves the neighbourhoods of the punctures Pi, it works for branched minimal surfaces as well. Locally, 7 = f (z)dz, where z = x + iy. The metric induced by X is given by (3) ds2 = A2(dx2 + dy2), Received by the editors November 28, 1994. 1991 Mathematics Subject Classification. Primary 53A10. Supported by Australian Research Council grant A69131962. ?1996 American Mathematical Society 1895 This content downloaded from 157.55.39.215 on Wed, 31 Aug 2016 04:16:59 UTC All use subject to http://about.jstor.org/terms