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:
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- 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:
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