On filtration formula and nodal sets in min-max theory
Abstract
Let (Mn+1,g) be a closed smooth Riemannian manifold, and let 0=λ0≤λ1≤… be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions {ϕj}j≥0. Let {ωp(M,g)}p≥1 denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured that the sweepout generated by the nodal sets of ϕ0,…,ϕp is asymptotically optimal for ωp(M,g) as p→∞. We show that this conjecture is false in every dimension, even for positively curved real-analytic metrics. Nevertheless, we show that a less rigid notion of nodal sweepouts recovers the volume spectrum. Let EN(M,g):=span{ϕ0,…,ϕN}, aN∈ H1(ℙ(EN(g));ℤ2) be the generator, and for p≥1 define 𝒩pN(g) := { Ψ| X is a finite complex, Ψ∈ C(X,ℙ(EN(g))), Ψ*(aNp)≠0 }. We define the nodal (p,N)-width by νpN(M,g) := infΨ∈𝒩pN(g) supx∈dmn(Ψ) ℋgn({Ψx=0}). We prove the following filtration formula ωp(M,g) = limN→∞νpN(M,g) for every p≥1. We state several questions on nodal geometry and min-max theory.
Related
- Sharp Bounds for Nodal Sets of Steklov Eigenfunctions on Smooth Surfaces
- Outer Lipschitz Geometry of Unions of Tangential Nodal Sets
- Sharp lower bounds for the volumes of nodal and positivity sets of harmonic functions
- Bourgain--Brezis--Mironescu formula for BV functions on arbitrary open sets and applications
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BibTeX
@misc{talipov2026filtration,
title = {{On filtration formula and nodal sets in min-max theory}},
author = {Talant Talipov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37769},
eprint = {2609.37769},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37769},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On filtration formula and nodal sets in min-max theory
AU - Talant Talipov
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37769
UR - https://arxiv.org/abs/2609.37769
AB - Let (M^n+1,g) be a closed smooth Riemannian manifold, and let 0=λ_0≤λ_1≤… be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions {ϕ_j}_j≥0. Let {ω_p(M,g)}_p≥1 denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured that the sweepout generated by the nodal sets of ϕ_0,…,ϕ_p is asymptotically optimal for ω_p(M,g) as p→∞. We show that this conjecture is false in every dimension, even for positively curved real-analytic metrics. Nevertheless, we show that a less rigid notion of nodal sweepouts recovers the volume spectrum. Let E_N(M,g):=span{ϕ_0,…,ϕ_N}, a_N∈ H^1(ℙ(E_N(g));ℤ_2) be the generator, and for p≥1 define 𝒩_p^N(g) := { Ψ| X is a finite complex, Ψ∈ C(X,ℙ(E_N(g))), Ψ^*(a_N^p)≠0 }. We define the nodal (p,N)-width by ν_p^N(M,g) := inf_Ψ∈𝒩_p^N(g) sup_x∈dmn(Ψ) ℋ_g^n({Ψ_x=0}). We prove the following filtration formula ω_p(M,g) = lim_N→∞ν_p^N(M,g) for every p≥1. We state several questions on nodal geometry and min-max theory.
ER -