Two-dimensional theta functions and crystallization among Bravais lattices (Q2821657)

A Bravais lattice of \(\mathbb R^2\) is \(L=\mathbb Zu\oplus\mathbb Zv\), where \(u,v\) are such that \(\|u\|\leq \|v\|\) and \((\widehat{u,v})\in [{\pi\over 3},{\pi\over 2}]\). In the paper, minimization problems among Bravais lattices for finite energy per point are studied. A sufficient condition for the minimality of the triangular lattice among Bravais lattices of fixed density is found. An example of convex decreasing positive interacting potential is given, for which the triangular lattice is not a minimizer for a class of densities. The optimality of the triangular lattice at high fixed densities or without density constraint is obtained.