The Weierstrass-Enneper Representation using hodographic coordinates on a minimal surface

In this paper we obtain the general solution to the minimal surface equation, namely its local Weierstrass-Enneper representation, by using a system of hodographic coordinates. This is done by using the method of solving the Born-Infeld equations by Whitham. We directly compute conformal coordinates on the minimal surface which give the Weierstrass-Enneper representation. From this we derive the hodographic coordinate \rho \in D \subset {\CC} and $\sigma$ its complex conjugate which enables us to write the Weierstrass-Enneper representation in a new way.Comment: 5-pages, semi-expository article, published in Proceedings of the Indian Academy of Sciences, 2003 (an electronic journal

Holomorphic Quillen determinant line bundles on integral compact Kahler manifolds

We show that any compact Kahler manifold with integral Kahler form, parametrizes a natural holomorphic family of Cauchy-Riemann operators on the Riemann sphere such that the Quillen determinant line bundle of this family is isomorphic to a sufficiently high tensor power of the holomorphic line bundle determined by the integral Kahler form. We also establish a symplectic version of the result. We conjecture that an equivariant version of our result is true.Comment: Latex2e, 10 pages, To appear in, Quillen memorial issue, Quarterly J. Mat