Wedge dislocations, three-dimensional gravity, and the Riemann–Hilbert problem

An expression for the free energy of an arbitrary static distribution of wedge dislocations in a solid is proposed. It represents a Euclidean version of $(1+2)$-dimensional gravity interacting with an arbitrary number of point particles. It is shown that the solution of the equilibrium equations leads to the Cauchy prob lem for effective equations determining the form of dislocations, while the problem of finding a metric leads to the Riemann–Hilbert problem for a frame with an $\mathbb{O}(3)$ monodromy representation.