We develop general approaches that parameterize brain anatomical
surfaces with Riemann surface structure. All metric orientable surfaces
are Riemann surfaces and admit conformal structure. With harmonic
energy minimization, holomorphic 1-form and the Ricci flow methods, we
can parameterize brain surfaces with various canonical surfaces such as
sphere, Euclidean plane and punched hole disks. The resulting surface
subdivision and the parameterizations of the components are intrinsic
and stable. Our parameterization scheme offers a way to explicitly
match landmark curves in anatomical surfaces such as the cortex,
providing a surface-based framework to compare anatomy statistically and
to generate grids on surfaces for PDE-based signal processing. Various
applications of our research will also be discussed.