Boundary rigidity and filling volume minimality of metrics close to a flat one
Boundary rigidity and filling volume minimality of metrics close to a flat one
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边界刚度和填充体积指标接近于平坦指标的极小值
DOI：
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发表时间：
2010
期刊：
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通讯作者：
S. Ivanov
中科院分区：
文献类型：
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作者：
D. Burago;S. Ivanov
我们说一个边界为∂M的非空黎曼流形（M, g）是一个最小可定向填充，如果对于∂M =∂M的每一个紧致可定向（M, g），不等式dg (x, y)≥dg （x, y）对于所有x， y∈∂M意味着vol(M, g)≥vol（M, g）。我们证明，如果一个度规g在具有连通边界的区域M C R n上与欧几里得域足够C 2 -接近，那么它就是一个最小填充。通过研究等式vol(M, g) = vol（M, g）的情况，我们证明如果dg (x, y) = dg （x, y）对于所有x， y∈∂M，那么（M, g）与（M, g）是等距的。这给出了已知的第一个二维以上的边界刚性流形的开放类，并向证明米歇尔猜想迈出了一步。
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.