TY - JOUR

T1 - Boundary rigidity and filling volume minimality of metrics close to a flat one

AU - Burago, Dmitri

AU - Ivanov, Sergei

PY - 2010/7/6

Y1 - 2010/7/6

N2 - 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 dg̃ (x, y) ≥ dg(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 ⊂ 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(M̃, g̃)= vol(M, g) we show that if dg̃ (x y) = dg(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.

AB - 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 dg̃ (x, y) ≥ dg(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 ⊂ 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(M̃, g̃)= vol(M, g) we show that if dg̃ (x y) = dg(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.

UR - http://www.scopus.com/inward/record.url?scp=77954169705&partnerID=8YFLogxK

U2 - 10.4007/annals.2010.171.1183

DO - 10.4007/annals.2010.171.1183

M3 - Article

AN - SCOPUS:77954169705

VL - 171

SP - 1183

EP - 1211

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 2

ER -