Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension
Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension
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黎曼流形中最小子流形的等周不等式：高维余维中的反例
DOI：
10.1007/s00526-011-0466-z
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发表时间：
2012
影响因子：
2.1
通讯作者：
R öttgen
中科院分区：
文献类型：
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作者：
Bangert;R öttgen
对于具有凸边界的紧致黎曼流形，B.白色证明了以下的替代：要么有一个等周不等式极小超曲面或存在一个封闭的极小超曲面，可能与一个小的奇异集。有一个自然的问题，如果类似的结果是正确的子流形更高的余维。具体来说，就是B。白色问，如果不存在的等周不等式fork-varifolds意味着存在一个非零的，固定的，integralk-varifold。我们提出的例子表明，这是不正确的余维大于2。其关键是在闭四维球B4上构造一个黎曼度量，它具有以下性质：（i）B4具有严格凸边界. (ii)存在一条完备的非常量测地线。(iii)B4中不存在闭测地线。
For compact Riemannian manifolds with convex boundary, B. White proved the following alternative: either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B. White asked if the non-existence of an isoperimetric inequality fork-varifolds implies the existence of a nonzero, stationary, integralk-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four–dimensional ballB4with the following properties: (i)B4has strictly convex boundary. (ii) There exists a complete nonconstant geodesic. (iii) There does not exist a closed geodesic inB4.
乔治·D·伯克霍夫 (GEORGE D. BIRKHOFF)，博士，理学士。
DOI：
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发表时间：
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期刊：
影响因子：
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作者：
Mises
通讯作者：
Mises