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Multiplicity one for equivariant min-max theory in prescribed homology classes

Tongrui Wang

TL;DR

The article advances equivariant min-max theory by proving a generic multiplicity-one result for $G$-invariant minimal hypersurfaces in prescribed $G$-homology classes and by establishing an equivariant min-max framework for PMC hypersurfaces with $G$-index bounds. It introduces lifting techniques and $G_ ext{±}$-signed symmetries to cover all $G$-invariant cycles, develops a robust PMC min-max theory under symmetric prescriptions, and proves compactness and index bounds for min-max $G$-hypersurfaces. These results yield an infinite collection of $G$-invariant minimal hypersurfaces in a given class and define an equivariant volume spectrum, with multiplicity-one realizations and explicit growth rates tied to the action's cohomogeneity. The findings extend the Morse-theoretic perspective on the area functional to the equivariant setting, enabling a structured equivariant Morse theory for $G$-invariant hypersurfaces and informing potential applications to Yau-type conjectures in symmetry-constrained contexts.

Abstract

For a closed Riemannian manifold $M$ with a compact Lie group $G$ acting by isometries, we show a generic multiplicity one theorem in equivariant min-max theory, and show in generic sense that there are infinitely many $G$-invariant minimal hypersurfaces in a fixed $G$-homology class. We also establish an equivariant min-max theory for $G$-invariant hypersurfaces of prescribed mean curvature with $G$-index upper bounds.

Multiplicity one for equivariant min-max theory in prescribed homology classes

TL;DR

The article advances equivariant min-max theory by proving a generic multiplicity-one result for -invariant minimal hypersurfaces in prescribed -homology classes and by establishing an equivariant min-max framework for PMC hypersurfaces with -index bounds. It introduces lifting techniques and -signed symmetries to cover all -invariant cycles, develops a robust PMC min-max theory under symmetric prescriptions, and proves compactness and index bounds for min-max -hypersurfaces. These results yield an infinite collection of -invariant minimal hypersurfaces in a given class and define an equivariant volume spectrum, with multiplicity-one realizations and explicit growth rates tied to the action's cohomogeneity. The findings extend the Morse-theoretic perspective on the area functional to the equivariant setting, enabling a structured equivariant Morse theory for -invariant hypersurfaces and informing potential applications to Yau-type conjectures in symmetry-constrained contexts.

Abstract

For a closed Riemannian manifold with a compact Lie group acting by isometries, we show a generic multiplicity one theorem in equivariant min-max theory, and show in generic sense that there are infinitely many -invariant minimal hypersurfaces in a fixed -homology class. We also establish an equivariant min-max theory for -invariant hypersurfaces of prescribed mean curvature with -index upper bounds.
Paper Structure (27 sections, 34 theorems, 76 equations)

This paper contains 27 sections, 34 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. Main results
  4. Lifting constructions and $G_\pm$-signed symmetries
  5. Min-max theory for PMC hypersurfaces and related results
  6. Outline
  7. Preliminary
  8. Notations for manifolds and group actions
  9. Notations for geometric measure theory
  10. Lift to be null-homologous with $G_\pm$-signed symmetries
  11. Weighted area functional $\mathcal{A}^h$
  12. Prescription functions and generic good pairs
  13. Compactness of stable $(G,h)$-hypersurfaces
  14. Relative equivariant min-max theory for prescribing mean curvature hypersurfaces
  15. Equivariant min-max constructions for $(X,Z)$-homotopy class
  16. ...and 12 more sections

Key Result

Theorem 1.4

Let $M$ be a closed manifold, $G$ be a compact Lie group acting by diffeomorphisms on $M$ with Eq: dimension assuption satisfied, and $\Sigma_0\subset M$ be a $G$-invariant hypersurface of locally $G$-boundary-type.

Theorems & Definitions (86)

  • Conjecture 1.1: S.-T. Yau yau1982problem
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • ...and 76 more