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arXiv · 1403.2876

Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems

A. I. Bobenko·Yu. B. Suris
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Abstract

Pluri-Lagrangian systems are variational systems with the multi-dimensional consistency property. This notion has its roots in the theory of pluriharmonic functions, in the Z-invariant models of statistical mechanics, in the theory of variational symmetries going back to Noether and in the theory of discrete integrable systems. A $d$-dimensional pluri-Lagrangian problem can be described as follows: given a $d$-form $L$ on an $m$-dimensional space, $m > d$, whose coefficients depend on a function $u$ of $m$ independent variables (called field), find those fields $u$ which deliver critical points to the action functionals $S_Σ=\int_ΣL$ for any $d$-dimensional manifold $Σ$ in the $m$-dimensional space. We investigate discrete 2-dimensional linear pluri-Lagrangian systems, i.e. those with quadratic Lagrangians $L$. The action is a discrete analogue of the Dirichlet energy, and solutions are called discrete pluriharmonic functions. We classify linear pluri-Lagrangian systems with Lagrangians depending on diagonals. They are described by generalizations of the star-triangle map. Examples of more general quadratic Lagrangians are also considered.

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A. I. Bobenko, Yu. B. Suris. 2014-03-12. Discrete pluriharmonic functions as solutions of linear pluri-Lagrangian systems. https://doi.org/10.1007/s00220-014-2240-5

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