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

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 d g (x, y) ≥ d g (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 C 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, g) = vol(M, g) we show that if d g (x, y) = d g (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.