Project Description:
Variational method is a useful mathematical tool in various areas of research. Recently, solving variational problems on surfaces has become an important research topic. In this paper, we describe an explicit method to solve variational problems on general Riemann surfaces, using the conformal parameterization and covariant derivatives defined on the surface. To simplify the computation, the surface is firstly mapped conformally to the two dimensional rectangular domains, by computing the holomorphic 1-form on the surface. It is well known that the Jacobian of a conformal map is simply the scalar multiplication of the conformal factor. Therefore with the conformal parameterization, the covariant derivatives on the parameter domain are similar to the usual Euclidean differential operators, except for the scalar
multiplication. As a result, any variational problem on the surface can be formulated to a 2D problem with a simple formula and efficiently solved by well developed numerical scheme on the 2D domain.  To examine the algorithm more systematically, we have presented the numerical error analysis.

Publication:

• L.M. Lui, X. Gu, T.F. Chan and S.-T. Yau, Variational Method on Riemann Surfaces using Conformal Parameterization and its Applications to Image Processing, Journal of Methods and Applications of Analysis, Volume 15, Number 4 (2008), 513-538
  
• L.M. Lui, Y. Wang, T.F. Chan, " PDE on manifold using global conformal parametrization", Variational, Geometric, and Level Set Methods in Computer Vision: Third International Workshop, VLSM 2005, Beijing, China, Oct. 16, 2005, pp. 307-319
  

Result: