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Balanced metrics, Zoll deformations and isosystolic inequalities in $\mathbb{C}P^n$

Luciano L. Junior

TL;DR

The paper addresses the local extremality of the $2$-systole and the $(2n-2)$-systole on $\mathbb{C}P^n$ under distinguished metric classes, revealing that homogeneous metrics maximize the normalized systole within their conformal class and that Kähler metrics locally minimize the codimension-two systole within balanced metrics. It develops a unified framework around the class $\mathcal{W}_k$ of almost Hermitian structures and a Zoll-like deformation class $\mathcal{Z}$, proving sharp integral-geometric and variational results. The authors obtain explicit formulas for the $2$- and $4n$-systoles along the homogeneous metric family $\{g_t\}$ and show systolic freedom in that class, while establishing local minimality of $\mathrm{Sys}^{\mathrm{nor}}_{2n-2}$ among balanced metrics and a rigidity statement for deformations in $\mathcal{Z}$. Collectively, these results illuminate how calibrated minimal submanifolds, balance conditions, and Zoll-type deformations govern the strand of extremal systolic geometry in complex projective spaces.

Abstract

The k-systole of a Riemannian manifold is the infimum of the volume over all homologically non-trivial k-cycles. In this paper we discuss the behavior of the dimension two and co-dimension two systole of the complex projective space for distinguished classes of metrics, namely the homogeneous metrics and the balanced metrics. In particular, we argue that every homogeneous metric maximizes the systole in its volume-normalized conformal class, as well as that each Kähler metric locally minimizes the systole on the set of volume-normalized balanced metrics. The proof demands the implementation of integral geometric techniques, and a careful analysis of the second variation of the systole functional. As an application, we characterize the systolic behavior of almost-Hermitian 1-parameter Zoll-like deformations of the Fubini-Study metric.

Balanced metrics, Zoll deformations and isosystolic inequalities in $\mathbb{C}P^n$

TL;DR

The paper addresses the local extremality of the -systole and the -systole on under distinguished metric classes, revealing that homogeneous metrics maximize the normalized systole within their conformal class and that Kähler metrics locally minimize the codimension-two systole within balanced metrics. It develops a unified framework around the class of almost Hermitian structures and a Zoll-like deformation class , proving sharp integral-geometric and variational results. The authors obtain explicit formulas for the - and -systoles along the homogeneous metric family and show systolic freedom in that class, while establishing local minimality of among balanced metrics and a rigidity statement for deformations in . Collectively, these results illuminate how calibrated minimal submanifolds, balance conditions, and Zoll-type deformations govern the strand of extremal systolic geometry in complex projective spaces.

Abstract

The k-systole of a Riemannian manifold is the infimum of the volume over all homologically non-trivial k-cycles. In this paper we discuss the behavior of the dimension two and co-dimension two systole of the complex projective space for distinguished classes of metrics, namely the homogeneous metrics and the balanced metrics. In particular, we argue that every homogeneous metric maximizes the systole in its volume-normalized conformal class, as well as that each Kähler metric locally minimizes the systole on the set of volume-normalized balanced metrics. The proof demands the implementation of integral geometric techniques, and a careful analysis of the second variation of the systole functional. As an application, we characterize the systolic behavior of almost-Hermitian 1-parameter Zoll-like deformations of the Fubini-Study metric.
Paper Structure (20 sections, 56 theorems, 142 equations)

This paper contains 20 sections, 56 theorems, 142 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. The Class $\mathcal{W}_k$
  4. Systole of Homogeneous Metrics
  5. Systole of Balanced Metrics
  6. Deformations in $\mathcal{Z}$
  7. The Class Wk
  8. Preliminaries
  9. The Classification Theorem
  10. Systole of Homogeneous Metrics
  11. Construction of Homogeneous Metrics
  12. 2-Systole
  13. 4n-Systole
  14. Systole of Balanced Metrics
  15. Manifold Structure of the space of Balanced Metrics
  16. ...and 5 more sections

Key Result

Theorem A

There exist an open neighborhood $g_{FS} \in \mathcal{U}$ of the Fubini-Study metric in the space $(\mathrm{Riem}(\mathbb{C}P^2), C^{\infty})$ of Riemannian metrics, such that: for every metric $g \in \mathcal{U}$. Moreover, the equality holds if and only if there is an almost complex structure $J$ such that $(\mathbb{C}P^2,J,g)$ is almost Kähler.

Theorems & Definitions (108)

  • Theorem A: Gromov-Berger, cf. gromov_pshol, section $0.2.$B
  • Definition B
  • Definition C
  • Theorem D
  • Corollary E: Integrable Case
  • Corollary F
  • Proposition G
  • Theorem H
  • Theorem I
  • Theorem J
  • ...and 98 more