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Morse index of min-max stationary integral varifolds

Mitchell Gaudet, Talant Talipov

TL;DR

This paper proves a sharp Morse index upper bound for min-max stationary integral $d$-varifolds realizing the $d$-dimensional $p$-width $oldsymbol{oldsymbol{\omega}}_p^d(M,g)$ of a closed manifold, showing $ ext{index}(S)\le p(n-d)$ for a realizing $S$. The authors introduce a hierarchical deformation framework and a notion of $k$-unstability to systematically push min--max sequences away from $(p(n-d)+1)$-unstable varifolds while preserving the width, combining this with compactness arguments to secure the bound. In the 1-dimensional case, stationary geodesic nets provide a concrete realization with analogous index control, and the construction recovers corollaries about decompositions into disjoint embedded stationary geodesic nets with total index bounded by $p(n-1)$. The bound’s dependence on $n$ is shown to be essential and optimal on spheres, highlighting a fundamental codimension-sensitive aspect of higher-dimensional min--max theory with potential implications for the study of minimal submanifolds in higher codimension. Overall, the work extends Almgren--Pitts min--max theory to yield quantitative index control for width-realizing varifolds and clarifies the role of ambient dimension in Morse-theoretic bounds.

Abstract

We prove an upper bound for the Morse index of min-max stationary integral varifolds realizing the $d$-dimensional $p$-width of a closed Riemannian manifold.

Morse index of min-max stationary integral varifolds

TL;DR

This paper proves a sharp Morse index upper bound for min-max stationary integral -varifolds realizing the -dimensional -width of a closed manifold, showing for a realizing . The authors introduce a hierarchical deformation framework and a notion of -unstability to systematically push min--max sequences away from -unstable varifolds while preserving the width, combining this with compactness arguments to secure the bound. In the 1-dimensional case, stationary geodesic nets provide a concrete realization with analogous index control, and the construction recovers corollaries about decompositions into disjoint embedded stationary geodesic nets with total index bounded by . The bound’s dependence on is shown to be essential and optimal on spheres, highlighting a fundamental codimension-sensitive aspect of higher-dimensional min--max theory with potential implications for the study of minimal submanifolds in higher codimension. Overall, the work extends Almgren--Pitts min--max theory to yield quantitative index control for width-realizing varifolds and clarifies the role of ambient dimension in Morse-theoretic bounds.

Abstract

We prove an upper bound for the Morse index of min-max stationary integral varifolds realizing the -dimensional -width of a closed Riemannian manifold.
Paper Structure (9 sections, 17 theorems, 133 equations, 5 figures)

This paper contains 9 sections, 17 theorems, 133 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Almgren--Pitts min--max theory
  5. Index upper bound
  6. Unstable variations and index
  7. Compactness under index bounds
  8. Deformation theorem
  9. Optimality of the bound

Key Result

Theorem 1.1

Let $(M^2,g)$ be a closed Riemannian surface. For every $p\in\mathbb N$ there exist closed immersed geodesics $\{\sigma_{p,j}\}_{j=1}^{N(p)}$ and integers $m_{p,j}\ge1$ such that

Figures (5)

  • Figure 1: Stationary twisted figure-eight (left) and stationary eyeglass (right)
  • Figure 2: $A^S$ for stationary integral varifold $S$. Note $m(S)=(0,0)$.
  • Figure 3: $A^V$ for $V$ not stationary. Note $m(V)\neq(0,0)$.
  • Figure 4: Obtaining a homotopy on $F'_k$ from one on $Y$
  • Figure 5: Obtaining a homotopy on $F_k$ from one on $F'_k$

Theorems & Definitions (42)

  • Theorem 1.1: chodosh2023p
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2: Min--max theorem for $d$-dimensional widths
  • Theorem 2.3: Min--max theorem for $1$-dimensional widths
  • Definition 2.4
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 32 more