The half-volume spectrum of a manifold The half-volume spectrum of a manifold

We define the half-volume spectrum { ω ~ p } p ∈ N of a closed manifold ( M n + 1 , g ) . This is analogous to the usual volume spectrum of M , except that we restrict to p -sweepouts whose slices each enclose half the volume of M . We prove that the Weyl law continues to hold for the half-volume sp...

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Bibliographic Details
Published inCalculus of variations and partial differential equations Vol. 64; no. 5
Main Authors Mazurowski, Liam, Zhou, Xin
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2025
Springer Nature B.V
Subjects
Analysis
Calculus of Variations and Optimal Control; Optimization
Control
Manifolds
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Minimal surfaces
Phase transitions
Systems Theory
Theoretical
53A10
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ISSN0944-2669
1432-0835
DOI10.1007/s00526-025-02949-z

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Summary:We define the half-volume spectrum { ω ~ p } p ∈ N of a closed manifold ( M n + 1 , g ) . This is analogous to the usual volume spectrum of M , except that we restrict to p -sweepouts whose slices each enclose half the volume of M . We prove that the Weyl law continues to hold for the half-volume spectrum. We define an analogous half-volume spectrum c ~ ( p ) in the phase transition setting. Moreover, for 3 ≤ n + 1 ≤ 7 , we use the Allen–Cahn min-max theory to show that each c ~ ( p ) is achieved by a constant mean curvature surface enclosing half the volume of M plus a (possibly empty) collection of minimal surfaces with even multiplicities.
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ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-025-02949-z