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 . f M ; Q g/ with @ f M D @M , the inequality d Q g .x; y/ d g .x; y/ for all x; y 2 @M implies vol. f M ; Q g/ vol.M; g/: We show that if a metric g on a region M 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. f M ; Q g/ D vol.M; g/ we show that if d Q g .x; y/ D d g .x; y/ for all x; y 2 @M then .M; g/ is isometric to . f M ; Q g/. 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. 1184