Nonlinear partial differntial equations related to geometric variational problems

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The head investigator researches the geometric visualization of standard realized crystal lattices, which related with harmonic maps as an example of geometric variational problems. The standard realization of crystal lattices is defined by Kotani-Sunada, and it is considered as a realization of real crystals in nature. The definition of crystal lattice is an abelian covering of finite graph. The covering transformation group and/or the first fundamental group of a finite graph is its the first homology group and it is abelian. So, the crystal lattice is an abelian convering graph of a finite graph. The standard realization of a crystal lattices is defined using harmonic maps into Albanese Torus. Hence the definition is very abstract.The head investigator construct an application to visualize the standard realization of crystal lattices. Sunada constructs K4 Crystal Lattice as the standard realization of K4 Graph in 3-dimensional Euclidean space. Our application plays important role in calculations of the K4 real crystal with Carbon atoms