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We investigate the existence of non-constant uniformly-bounded minimalsolutions of the Allen-Cahn equation on a Gromov-hyperbolic group. We show thatwhenever the Laplace term in the Allen-Cahn equation is small enough, thereexist minimal solutions satisfying a large class of prescribed asymptoticbehaviours. For a phase field model on a hyperbolic group, such solutionsdescribe phase transitions that asymptotically converge towards prescribedphases, given by asymptotic directions. In the spirit of de Giorgi'sconjecture, we then fix an asymptotic behaviour and let the Laplace term go tozero. In the limit we obtain a solution to a corresponding asymptotic Plateauproblem by $\Gamma$-convergence.

Minimisers of the Allen-Cahn equation and the asymptotic Plateau problem on hyperbolic groups