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International audience; The Gauss map g of a surface ⌃ in R 4 takes its values in the Grassmannian of oriented 2-planes of R 4 : G + (2, 4). We give geometric criteria of stability for minimal surfaces in R 4 in terms of g. We show in particular that if the spherical area of the Gauss map |g(⌃)| of a minimal surface is smaller than 2⇡ then the surface is stable by deformations which fix the boundary of the surface. This answers the question of [BDC3] in R 4 .

We show that the Dirichlet problem for the minimal hypersurface equation defined on arbitrary C2 bounded domain Ω of an arbitrary complete Riemannian manifold M is solvable if the oscillation of the boundary data is bounded by a function \({\mathcal{C}}\) that is explicitely given and that depends only on the first and second derivatives of the boundary data as well as the second fundamental form of the boundary \({\partial\Omega}\) and the Ricci curvature of the ambient space M. This result extends Theorem 2 of Jenkins-Serrin (J Reine Angew Math 229:170–187,1968) about the solvability of the Dirichlet problem for the minimal hypersurface equation defined on bounded domains of the Euclidean space. We deduce that the Dirichlet problem for the minimal hypersurface equation is solvable for any continuous boundary data on a mean convex domain. We also show existence and uniqueness of the Dirichlet problem with boundary data at infinity—exterior Dirichlet problem—on Hadamard manifolds.