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Zoll Manifolds of Type $\mathbb{CP}^n$ with Entire Grauert Tubes

Chi Li, Kyobeom Song

TL;DR

This work proves a rigidity result for Zoll manifolds of type $\mathbb{CP}^n$ under the entire Grauert tube condition. By constructing the complexified tangent bundle $X=TM\cup D$ and analyzing its cohomology, the authors show $X$ is a $2n$-dimensional Fano manifold with index $n+1$, and then apply Wiśniewski's classification to identify $X\cong \mathbb{CP}^n\times\mathbb{CP}^n$. The algebro-geometric identification of the divisor $D$ and the action of the anti-holomorphic involution $N_{-1}$ lead to a rigidity that forces $M$ to be $\mathbb{CP}^n$ with the Fubini–Study metric, up to scaling. The result extends previous uniqueness findings for Zoll spheres to the CP^n type, consolidating the role of complexification and Fano geometry in metric rigidity for Zoll manifolds.

Abstract

We show that a Zoll manifold of type $\mathbb{CP}^n$ with an entire Grauert tube is isometric to $\mathbb{CP}^n$ with the canonical Fubini-Study metric, up to constant multiplication.

Zoll Manifolds of Type $\mathbb{CP}^n$ with Entire Grauert Tubes

TL;DR

This work proves a rigidity result for Zoll manifolds of type under the entire Grauert tube condition. By constructing the complexified tangent bundle and analyzing its cohomology, the authors show is a -dimensional Fano manifold with index , and then apply Wiśniewski's classification to identify . The algebro-geometric identification of the divisor and the action of the anti-holomorphic involution lead to a rigidity that forces to be with the Fubini–Study metric, up to scaling. The result extends previous uniqueness findings for Zoll spheres to the CP^n type, consolidating the role of complexification and Fano geometry in metric rigidity for Zoll manifolds.

Abstract

We show that a Zoll manifold of type with an entire Grauert tube is isometric to with the canonical Fubini-Study metric, up to constant multiplication.
Paper Structure (10 sections, 7 theorems, 63 equations)

This paper contains 10 sections, 7 theorems, 63 equations.

Table of Contents

  1. Introduction and main results
  2. Complexification and Compactification of Tangent Bundle
  3. Homogeneous Complex Monge-Ampère Equation on the Tangent Bundle
  4. Compactification of the Tangent Bundle of Zoll Manifolds
  5. Concrete Description on the Model Case
  6. Cohomology of $X$ and $D$
  7. Computation of Cohomology
  8. Behavior of Cocycles
  9. Evaluation of Fano Index
  10. Characterization of the Complex Structure and Metric

Key Result

Theorem 1.1

Let $(M,g)$ be a Zoll manifold of type $\mathbb{CP}^n$. Assume that $(M, g)$ has the entire Grauert tube condition. Then, $M$ is isometric to $\mathbb{CP}^n$ with its Fubini-Study Metric, up to constant multiplication.

Theorems & Definitions (11)

  • Theorem 1.1
  • Remark 2.1: Manifold structure on $D$
  • Proposition 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • Lemma 3.2
  • Lemma 4.1
  • Lemma 4.2
  • ...and 1 more