  Finsler geometric surface model is studied as a coarse-grained model for
membranes of three-component such as DOPC, DPPC and Cholesterol. To understand
the phase separation of liquid ordered (DPPC rich) $L_o$ and the liquid
disordered (DOPC rich) $L_d$, we introduce a variable $\sigma (\in \{1,-1\})$
in the triangulated surface model. We numerically find that there appear two
circulars and stripe domains on the surface and that these two morphologies are
separated by a phase transition. The morphological change from the one to the
other with respect to the variation of the area fraction of $L_o$ is consistent
with existing experimental results. This gives us a clear understanding of the
origin of the line tension energy, which has been used to understand those
morphological changes in the three-component membranes. In addition to these
two circulars and stripe domains, raft-like domain and budding domain are also
observed, and the corresponding several phase diagrams are obtained. Technical
details of the Finsler geometry modeling are also shown.

Soft Condensed Matter;Statistical Mechanics