# Boundary rigidity and filling volume minimality of metrics close to a flat one | Annals of Mathematics

Abstract We say that a Riemannian manifold (M,g) with a non-empty boundary ∂M is a minimal orientable filling if, for every compact orientable (M˜,g~) with ∂M˜=∂M, the inequality dg~(x,y)≥dg(x,y) for all x,y∈∂M implies vol(M˜,g~)≥vol(M,g). We show that if a metric g on a region M⊂Rn with a connected boundary is sufficiently C2-close to a Euclidean one, then it is a minimal filling. By studying the equality case vol(M˜,g~)=vol(M,g) we show that if dg~(x,y)=dg(x,y) for all x,y∈∂M then (M,g) is isometric to (M˜,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.

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### 数学年刊（Annals of Mathematics）

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