All orientable metric surfaces are Riemann surfaces and admit
global conformal parameterizations. Riemann surface structure
is a fundamental structure and governs many natural physical
phenomena, such as heat diffusion, electric-magnetic fields on
the surface. Good parameterization is crucial for simulation
and visualization. This paper gives an explicit method for
finding optimal global conformal parameterizations of arbitrary
surfaces. It relies on certain holomorphic differential forms
and conformal mappings from differential geometry and Riemann
surface theories. Algorithms are developed to modify topology,
locate zero points, and determine cohomology types of differential
forms. The implementation is based on finite dimensional optimization
method. The optimal parameterization is intrinsic to the geometry,
preserving angular structure, and can play an important role in
various applications including texture mapping, remeshing, morphing
and simulation. The method is demonstrated by visualizing the Riemann
surface structure of real surfaces represented as triangle meshes.