The p-widths of a surface
The p-widths of a surface
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DOI：
10.1007/s10240-023-00141-7
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发表时间：
2021-07
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影响因子：
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通讯作者：
Otis Chodosh;Christos Mantoulidis
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文献类型：
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作者：
Otis Chodosh;Christos Mantoulidis
闭黎曼流形的宽度是其Laplace-Beltrami算子的谱的非线性模拟，它对应于可能奇异的极小子流形的某个极大极小序列的面积。我们证明了任何闭黎曼双流形的-宽度对应于闭浸入测地线的并，而不是简单的测地线网.然后证明了由齐次多项式的零点集构造的圆双球面的扫掠的最优性，表明圆球面的-宽度是由大圆得到的.结果，我们找到了Liokumovich-Marques-Neves-Weyl定律中曲面的普适常数，在计算圆双球面的宽度的过程中，我们证明了两个新的结果：固定边长的静态测地线网的凹凸度量定理，以及一般地，具有有界质量和有界奇异集的静态测地线网具有Lusternik-Schnirelmann范畴零。
The-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which corresponds to areas of a certain min-max sequence of possibly singular minimal submanifolds. We show that the-widths of any closed Riemannian two-manifold correspond to a union of closed immersed geodesics, rather than simply geodesic nets.We then prove optimality of the sweepouts of the round two-sphere constructed from the zero set of homogeneous polynomials, showing that the-widths of the round sphere are attained bygreat circles. As a result, we find the universal constant in the Liokumovich–Marques–Neves–Weyl law for surfaces to be.En route to calculating the-widths of the round two-sphere, we prove two additional new results: a bumpy metrics theorem for stationary geodesic nets with fixed edge lengths, and that, generically, stationary geodesic nets with bounded mass and bounded singular set have Lusternik–Schnirelmann category zero.