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Grafting of real projective surfaces with Hitchin holonomy

Toshiki Fujii

Abstract

We define graftable curves on real projective surfaces. In particular, we construct graftable ones in Hitchin case and show that real projective structures with the same Hitchin holonomy, carrying the same weight type, are related to each other via multi-graftings.

Grafting of real projective surfaces with Hitchin holonomy

Abstract

We define graftable curves on real projective surfaces. In particular, we construct graftable ones in Hitchin case and show that real projective structures with the same Hitchin holonomy, carrying the same weight type, are related to each other via multi-graftings.
Paper Structure (19 sections, 13 theorems, 17 equations, 23 figures)

This paper contains 19 sections, 13 theorems, 17 equations, 23 figures.

Table of Contents

  1. Introduction
  2. Graftable curves
  3. Special $\mathbb{RP}^2$-annuli and tori
  4. Grafting torus
  5. Graftable curves
  6. Grafting along closed curves
  7. Isotopy of graftable curves
  8. Construction of graftable curves
  9. Convex geometry
  10. Real projective strctures with Hitchin holonomy
  11. Construction of graftable multi-curves
  12. Grafting along the curves
  13. The operation on grafting data
  14. Grafting along $\beta_R$ and $\beta_L$
  15. The action of mapping class group on grafting data
  16. ...and 4 more sections

Key Result

Theorem A

Let $\sigma_1$ and $\sigma_2$ be two real projective structures on $\Sigma$ with the same Hitchin holonomy. Suppose in addition that they correspond to the grafting data with the same weight type. Then, $\sigma_2$ can be obtained from $\sigma_1$ by a composition of at most $6g$ multi-graftings.

Figures (23)

  • Figure 1: The triangular domains and lines preserved by a hyperbolic transformation $A$.
  • Figure 2: $\beta_R$ in the grafting annuli (red lines). Blue points are fixed points of $\rho(\beta_j)$.
  • Figure 3: $\beta_L$ in the grafting annuli (red lines). Blue points are fixed points of $\rho(\beta_j)$.
  • Figure 4: A weighting scheme along $\beta_R$.
  • Figure 5: The developing image of $\beta_R$ in the grafting annuli.
  • ...and 18 more figures

Theorems & Definitions (29)

  • Theorem A
  • Theorem B
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 19 more