论文标题
具有有界指数的最小超曲面的奇异集合的定量估计值
Quantitative Estimates on the Singular Set of Minimal Hypersurfaces with Bounded Index
论文作者
论文摘要
我们证明了一组奇异的固定积分$ n $ -varifolds $ v $在Riemannian歧管中的奇异集合的单数固定集成量的近距离范围的本地措施界限，这两者都具有：（i）在其平稳嵌入的部分上有限索引；和（ii）$ \ MATHCAL {H}^{n-1} $ - null Singular Set。这种结合的直接后果是在这种varifold的总质量和索引方面的奇异集的上部minkowski含量上的结合。这样的结果改善了已知界限，即对相应的绑定在$ \ MATHCAL {H}^{n-7} $ - A. Song（[10]）建立的单数集的测量值，以及A. Naber和D. Valtorta（[6]）建立的相同界限，用于编码iMimension-naber和D. valtorta（[6]），以最小化的一个面积最小化的通行率。我们的结果还提供了有关与有限索引（常规部件）的编纂One积分varifolds的单数集合的更多结构信息，并且没有N. wickramasekera（[11]）建立的经典奇异性。
We prove local measure bounds on the tubular neighbourhood of the singular set of codimension one stationary integral $n$-varifolds $V$ in Riemannian manifolds which have both: (i) finite index on their smoothly embedded part; and (ii) $\mathcal{H}^{n-1}$-null singular set. A direct consequence of such a bound is a bound on the upper Minkowski content of the singular set of such a varifold in terms of its total mass and its index. Such a result improves on known bounds, namely the corresponding bound on the $\mathcal{H}^{n-7}$-measure of the singular set established by A. Song ([10]), as well as the same bounds established by A. Naber and D. Valtorta ([6]) for codimension one area minimising currents. Our results also provide more structural information on the singular set for codimension one integral varifolds with finite index (on the regular part) and no classical singularities established by N. Wickramasekera ([11]).
