Boundary rigidity and filling volume minimality of metrics close to a flat one

We say that a Riemannian manifold (M, g) with a non-empty boundary ∂M is a minimal orientable filling if, for every compact orientable (M̃, g̃) with ∂M̃ = D ∂M, the inequality d_g̃ (x, y) ≥ d_g(x, y) for all x, y ε ∂M implies vol(M̃, g̃) ≥ vol(M, g). We show that if a metric g on a region M ⊂ R_n with a connected boundary is sufficiently C²-close to a Euclidean one, then it is a minimal filling. By studying the equality case vol(M̃, g̃)= vol(M, g) we show that if d_g̃ (x y) = d_g(x, y) for all (x, y) ε ∂M then.(M, g) is isometric to.(M̃, g̃). This gives the first known open class of boundary rigid manifolds in dimensions higher than two and makes a step towards a proof of Michel's conjecture.