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Given a C2-domain with compact boundary in an arbitrary complete Riemannianmanifold, we search for smallness conditions on the boundary data for which theDirichlet problem for the minimal hypersurface equation is solvable. We obtainan extension to Riemannian manifolds of an existence result of G. H. Williams (J. Reine Angew. Math. 354:123-140, 1984).

The Dirichlet problem for the minimal hypersurface equation with Lipschitz continuous boundary data on domains of a Riemannian manifold