Gromov's generalized Weyl law conjecture for widths of cycles

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Let MnM^n be a closed nn-dimensional manifold with n≥2n\geq 2, and let ωp1(Mn,g)\omega_p^1(M^n,g) denote its 11-dimensional pp-width with respect to a Riemannian metric gg. The volume of MnM^n is denoted by Vol⁡(Mn,g)\operatorname{Vol}(M^n,g). Gromov's generalized Weyl law conjecture. There exists a constant α(n,1)>0\alpha(n,1)>0 such that

lim⁡p→∞ωp1(Mn,g)p−n−1n=α(n,1)Vol⁡(Mn,g)1n.\lim_{p\to\infty}\omega_p^1(M^n,g)p^{-\frac{n-1}{n}}=\alpha(n,1)\operatorname{Vol}(M^n,g)^{\frac{1}{n}}.

This is the expected Weyl law for 11-dimensional cycles, extending the known volume-spectrum Weyl law to other dimensions and codimensions. The supplied text does not state whether this conjecture has been resolved.

References

Primary source

Xinze Li and Bruno Staffa, “On the equidistribution of closed geodesics and geodesic nets”, arXiv:2205.13694 (2023).

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