Standard realizations of crystal lattices via harmonic maps

Authors:
Motoko Kotani and Toshikazu Sunada

Journal:
Trans. Amer. Math. Soc. **353** (2001), 1-20

MSC (2000):
Primary 58E20, 58E11, 58E30; Secondary 82B99

Published electronically:
August 8, 2000

MathSciNet review:
1783793

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

An Eells-Sampson type theorem for harmonic maps from a finite weighted graph is employed to characterize the equilibrium configurations of crystals. It is thus observed that the mimimum principle frames symmetry of crystals.

**1.**Béla Bollobás,*Graph theory*, Graduate Texts in Mathematics, vol. 63, Springer-Verlag, New York-Berlin, 1979. An introductory course. MR**536131****2.**James Eells Jr. and J. H. Sampson,*Harmonic mappings of Riemannian manifolds*, Amer. J. Math.**86**(1964), 109–160. MR**0164306****3.**N. Ejiri,*The second variation formula of the energy function on the Teichmüller space and the Morse index of compact minimal surfaces in tori*, preprint.**4.**Mikhail Gromov and Richard Schoen,*Harmonic maps into singular spaces and 𝑝-adic superrigidity for lattices in groups of rank one*, Inst. Hautes Études Sci. Publ. Math.**76**(1992), 165–246. MR**1215595****5.**Jürgen Jost,*Equilibrium maps between metric spaces*, Calc. Var. Partial Differential Equations**2**(1994), no. 2, 173–204. MR**1385525**, 10.1007/BF01191341**6.**Jürgen Jost,*Convex functionals and generalized harmonic maps into spaces of nonpositive curvature*, Comment. Math. Helv.**70**(1995), no. 4, 659–673. MR**1360608**, 10.1007/BF02566027**7.**M. Kotani and T. Sunada,*Jacobian tori associated with a finite graph and its abelian covering graphs*, preprint.**8.**M. Kotani and T. Sunada,*Albanese maps and off diagonal long time asymptotics for the heat kernel*, preprint.**9.**Tadashi Nagano and Brian Smyth,*Minimal varieties and harmonic maps in tori*, Comment. Math. Helv.**50**(1975), 249–265. MR**0390974****10.**H. Urakawa,*A discrete analogue of the harmonic morphism and Green kernel comparison theorems*, preprint.**11.**E.A. Wood,*Crystals and Light, An Intorduction to Optical Crystallography*, Dover, 1977.**12.**J. Eells and B. Fuglede,*Harmonic maps between Riemannian polyhedra*, a monograph to be published.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
58E20,
58E11,
58E30,
82B99

Retrieve articles in all journals with MSC (2000): 58E20, 58E11, 58E30, 82B99

Additional Information

**Motoko Kotani**

Affiliation:
Mathematical Institute, Graduate School of Science, Tôhoku University, Aoba, Sendai 980-8578, Japan

Email:
kotani@math.tohoku.ac.jp

**Toshikazu Sunada**

Affiliation:
Mathematical Institute, Graduate School of Sciences, Tôhoku University, Aoba, Sendai 980-8578, Japan

Email:
sunada@math.tohoku.ac.jp

DOI:
http://dx.doi.org/10.1090/S0002-9947-00-02632-5

Keywords:
Crystal lattice,
harmonic map,
Albanese map,
Albanese torus,
abelian covering,
weighted graph

Received by editor(s):
March 8, 1999

Published electronically:
August 8, 2000

Article copyright:
© Copyright 2000
American Mathematical Society