{
  "abstract": "Let $(M^{n+1},g)$ be a closed smooth Riemannian manifold, and let \\[ 0=λ_0\\leqλ_1\\leq\\ldots \\] be the spectrum of the Laplace-Beltrami operator, with corresponding real eigenfunctions $\\{ϕ_j\\}_{j\\geq0}$. Let $\\{ω_p(M,g)\\}_{p\\geq1}$ denote Gromov's volume spectrum. In 2014, Marques-Neves conjectured that the sweepout generated by the nodal sets of $ϕ_0,\\ldots,ϕ_p$ is asymptotically optimal for $ω_p(M,g)$ as $p\\to\\infty$. 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):=\\operatorname{span}\\{ϕ_0,\\ldots,ϕ_N\\}, \\qquad a_N\\in H^1\\bigl(\\mathbb P(E_N(g));\\mathbb Z_2\\bigr) \\] be the generator, and for $p\\geq1$ define \\[ \\mathcal N_p^N(g) := \\left\\{ Ψ\\mid X\\text{ is a finite complex},\\ Ψ\\in C\\bigl(X,\\mathbb P(E_N(g))\\bigr),\\ Ψ^*(a_N^p)\\neq0 \\right\\}. \\] We define the nodal $(p,N)$-width by \\[ ν_p^N(M,g) := \\inf_{Ψ\\in\\mathcal N_p^N(g)} \\sup_{x\\in\\operatorname{dmn}(Ψ)} \\mathcal H_g^{n}\\bigl(\\{Ψ_x=0\\}\\bigr). \\] We prove the following filtration formula \\[ ω_p(M,g) = \\lim_{N\\to\\infty}ν_p^N(M,g) \\] for every $p\\geq1$. We state several questions on nodal geometry and min-max theory.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37769",
  "author_total": 1,
  "authors": [
    {
      "name": "Talant Talipov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8hcqt5fgmtnv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37769",
  "fulltext_url": null,
  "id": "d8hcqt5fgmtnv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37769",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214885088",
  "pdf_url": "https://arxiv.org/pdf/2609.37769",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8hcqt5fgmtnv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37769",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37769"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On filtration formula and nodal sets in min-max theory",
  "title_text": "On filtration formula and nodal sets in min-max theory",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/d8hcqt5fgmtnv",
  "venue": "arXiv",
  "versions": []
}