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Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces

Toru Kajigaya, Keita Kunikawa

TL;DR

This work advances index estimates for minimal hypersurfaces in compact semi-simple Riemannian symmetric spaces by introducing trace-formula methods that leverage harmonic 1-forms and wedge^2 g structures. By embedding the ambient manifold into a compact semi-simple RSS and constructing test variations linked to the ambient Killing fields, the authors derive explicit trace identities that bound the Morse index in terms of the first Betti number b1(Σ). The main result yields a universal linear bound index(Σ) ≥ 2/(d(d−1)+2(2n−3)) · b1(Σ) for unstable Σ, with corollaries for two-sided and simply-connected M and applications to Berger spheres and K-projective geodesic spheres. The framework generalizes and unifies prior approaches (Savo, ACS, MR) and provides concrete geometric criteria to certify index bounds in families of symmetric spaces, contributing toward the Marques–Neves–Schoen conjecture in this broader setting.

Abstract

We show that the Morse index of unstable closed minimal hypersurface $Σ$ in a compact semi-simple Riemannian symmetric space $M=G/K$ is bounded from below by constant times the first Betti number of $Σ$. Our proof is based on a natural extension of the previous method and this also provides a novel approach for the index estimate.

Index estimate by first Betti number of minimal hypersurfaces in compact symmetric spaces

TL;DR

This work advances index estimates for minimal hypersurfaces in compact semi-simple Riemannian symmetric spaces by introducing trace-formula methods that leverage harmonic 1-forms and wedge^2 g structures. By embedding the ambient manifold into a compact semi-simple RSS and constructing test variations linked to the ambient Killing fields, the authors derive explicit trace identities that bound the Morse index in terms of the first Betti number b1(Σ). The main result yields a universal linear bound index(Σ) ≥ 2/(d(d−1)+2(2n−3)) · b1(Σ) for unstable Σ, with corollaries for two-sided and simply-connected M and applications to Berger spheres and K-projective geodesic spheres. The framework generalizes and unifies prior approaches (Savo, ACS, MR) and provides concrete geometric criteria to certify index bounds in families of symmetric spaces, contributing toward the Marques–Neves–Schoen conjecture in this broader setting.

Abstract

We show that the Morse index of unstable closed minimal hypersurface in a compact semi-simple Riemannian symmetric space is bounded from below by constant times the first Betti number of . Our proof is based on a natural extension of the previous method and this also provides a novel approach for the index estimate.

Paper Structure

This paper contains 16 sections, 27 theorems, 172 equations, 1 table.

Table of Contents

  1. Introduction
  2. Index and the method of estimate
  3. Second variation of minimal hypersurfaces
  4. Method of index estimate
  5. Preliminaries on symmetric spaces
  6. Trace formula I
  7. Trace formula
  8. An application
  9. Trace formula II
  10. Trace formula
  11. Applications
  12. Proof of Theorem \ref{['thm:main0']}
  13. Relation to previous methods
  14. The case of $\mathbb{R}^{n+k}$
  15. The method of Mendes-Radeschi
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $M$ be a compact semi-simple Riemannian symmetric space and $\Sigma$ be a closed minimal hypersurface embedded in $M$. If $\Sigma$ is unstable, then it holds that where $d$ is the dimension of the isometry group of $M$ and $n={\rm dim} M$.

Theorems & Definitions (50)

  • Theorem 1.1: Theorem \ref{['thm:main']}
  • Corollary 1.2
  • Corollary 1.3
  • Remark 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 40 more