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 partial derivative M is a minimal orientable filling if, for every compact orientable. ((M) over tilde, (g) over tilde) with partial derivative(M) over tilde = partial derivative M, the inequality d((g) over tilde)(x, y) >= d(g)(x, y) for all x, y is an element of partial derivative M implies vol((M) over tilde, (g) over tilde) >= vol(M, g). We show that if a metric g on a region M subset of R-n with a connected boundary is sufficiently C-2-close to a Euclidean one, then it is a minimal filling. By studying the equality case vol((M) over tilde, (g) over tilde) = vol(M, g) we show that if d((g) over tilde)(x, y) = d(g)(x, y) for all x, y is an element of partial derivative M then (M, g) is isometric to ((M) over tilde, (g) over tilde). 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.