## 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 (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 C^{2}-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.

Original language | English (US) |
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Pages (from-to) | 1183-1211 |

Number of pages | 29 |

Journal | Annals of Mathematics |

Volume | 171 |

Issue number | 2 |

DOIs | |

State | Published - 2010 |

## All Science Journal Classification (ASJC) codes

- Mathematics (miscellaneous)