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Index gap of the systole function

Changjie Chen

TL;DR

The paper proves a universal index-gap for Morse-type systole function critical points on the moduli space $\\mathcal{M}_{g,n}$, showing that the Morse index grows at least as $C\log\log(g+n)$, with finitely many exceptions. It leverages a rank-augmentation mechanism for Weil-Petersson gradients of geodesic-length functions, encoded via $j$-systems and essential subsurfaces, to bound indices and relate low-index critical points to the Deligne-Mumford boundary. The authors also provide a quantitative bound on the growth of the index gap and classify low-index critical points, establishing that low-degree (co)homology in the Deligne-Mumford compactification is governed by boundary strata. This work connects hyperbolic geometry, Teichmüller theory, and Morse theory to illuminate the topological structure of moduli spaces.

Abstract

It is known that the systole function is topologically Morse on the moduli space $\mathcal M_{g,n}$ and the $\text{sys}_T$ functions are $C^2$-Morse on the Deligne-Mumford compactification $\overline{\mathcal M}_{g,n}$. In this paper, We show that these Morse functions admit an index gap on $\mathcal M_{g,n}$. Specifically, there exists a universal constant $C>0$ such that any critical point in $\mathcal M_{g,n}$ has Morse index at least $C\log\log(g+n)$. This implies by Morse theory that the low degree homology of the Deligne-Mumford compactification $\overline{\mathcal M}_{g,n}$ comes from the boundary $\partial\mathcal M_{g,n}$.

Index gap of the systole function

TL;DR

The paper proves a universal index-gap for Morse-type systole function critical points on the moduli space , showing that the Morse index grows at least as , with finitely many exceptions. It leverages a rank-augmentation mechanism for Weil-Petersson gradients of geodesic-length functions, encoded via -systems and essential subsurfaces, to bound indices and relate low-index critical points to the Deligne-Mumford boundary. The authors also provide a quantitative bound on the growth of the index gap and classify low-index critical points, establishing that low-degree (co)homology in the Deligne-Mumford compactification is governed by boundary strata. This work connects hyperbolic geometry, Teichmüller theory, and Morse theory to illuminate the topological structure of moduli spaces.

Abstract

It is known that the systole function is topologically Morse on the moduli space and the functions are -Morse on the Deligne-Mumford compactification . In this paper, We show that these Morse functions admit an index gap on . Specifically, there exists a universal constant such that any critical point in has Morse index at least . This implies by Morse theory that the low degree homology of the Deligne-Mumford compactification comes from the boundary .
Paper Structure (7 sections, 32 theorems, 61 equations, 10 figures)

This paper contains 7 sections, 32 theorems, 61 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Morse functions
  3. Curves on subsurfaces
  4. Essentiality of subsurfaces
  5. Critical points of the Morse functions
  6. Bounding the index gap
  7. Classification of Low Index Critical Points

Key Result

Theorem 1.1

For any $k\ge0$, with finitely many exceptions on $(g,n)$, all critical points have Morse index greater than $k$ in $\mathcal{M}_{g,n}$, for the systole function or the $\mathop{\mathrm{sys_T}}\nolimits$ functions.

Figures (10)

  • Figure 1: The right subsurface is non-essential; $\alpha$ and $\lambda$ are not necessarily disjoint
  • Figure 2: New endpoints are to the left to the old ones, meaning that $\overline\delta(t)$ rotates counterclockwise
  • Figure 3: $(g,n)=(0,3)$
  • Figure 4: $(g,n)=(1,1),\#S(X)=2$, $\frac{\pi}{2}$-intersection
  • Figure 5: $(g,n)=(0,4),\#S(X)=2$, $\frac{\pi}{2}$-intersection
  • ...and 5 more figures

Theorems & Definitions (66)

  • Theorem 1.1: = Theorem \ref{['main theorem down']}
  • Theorem 1.2: = Theorem \ref{['kissingnumbersubsurface']}
  • Theorem 1.3: = Theorem \ref{['induction']}
  • Theorem 1.4: = Theorem \ref{['maxm']} and \ref{['maxcap']}
  • Definition 2.1
  • Definition 2.2: Eutacticity
  • Definition 2.3: Topological Morse function, cf. morse1959topologically
  • Theorem 2.4: akrout2003singularites
  • Theorem 2.5: chen2023c
  • Remark 2.6
  • ...and 56 more