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The Space of Strictly-convex Real-projective structures on a closed manifold

Daryl Cooper, Stephan Tillmann

TL;DR

The paper investigates which holonomies arise from strictly convex real-projective structures on a closed manifold M by proving that the corresponding holonomy set Rep_S(M) is clopen in Rep(M). It develops a self-contained framework combining convex geometry in projective spaces with Vinberg’s convex-cone theory to control deformations, including a new box-estimate for degenerations. The Open Theorem is established by an explicit triangulation-based deformation that preserves radial convexity, while the Closed Theorem uses compactness arguments and conjugation tricks to rule out degenerations and preserve strict convexity in limits. Together, these results place Rep_S(M) as a union of connected components of Rep(M) and generalize known 2D/classical cases, enhancing the understanding of higher Teichmüller-type representations in $\operatorname{PGL}(n+1,\mathbb{R})$.

Abstract

This is an expository proof that, if $M$ is a compact $n$-manifold with no boundary, then the set of holonomies of strictly-convex real-projective structures on $M$ is a subset of $\operatorname{Hom}(π_1M,\operatorname{PGL}(n+1,\mathbb R))$ that is both open and closed.

The Space of Strictly-convex Real-projective structures on a closed manifold

TL;DR

The paper investigates which holonomies arise from strictly convex real-projective structures on a closed manifold M by proving that the corresponding holonomy set Rep_S(M) is clopen in Rep(M). It develops a self-contained framework combining convex geometry in projective spaces with Vinberg’s convex-cone theory to control deformations, including a new box-estimate for degenerations. The Open Theorem is established by an explicit triangulation-based deformation that preserves radial convexity, while the Closed Theorem uses compactness arguments and conjugation tricks to rule out degenerations and preserve strict convexity in limits. Together, these results place Rep_S(M) as a union of connected components of Rep(M) and generalize known 2D/classical cases, enhancing the understanding of higher Teichmüller-type representations in .

Abstract

This is an expository proof that, if is a compact -manifold with no boundary, then the set of holonomies of strictly-convex real-projective structures on is a subset of that is both open and closed.

Paper Structure

This paper contains 4 sections, 34 theorems, 43 equations, 8 figures.

Table of Contents

  1. Properly and strictly convex
  2. Convex Cones
  3. Open
  4. Closed

Key Result

Proposition 1.1

Every projective structure on $M$ lifts to a positive projective structure.

Figures (8)

  • Figure 1: Projecting onto a line
  • Figure 2: Distance between two lines near infinity
  • Figure 3: Quasi-geodesics
  • Figure 4: $T\cap \mathcal{D}$ is the centroid of $T\cap\mathcal{C}$
  • Figure 5: Volume forms on $\Omega\subset S^n$ and $\mathcal{C}_{\phi}$
  • ...and 3 more figures

Theorems & Definitions (69)

  • Proposition 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • Corollary 1.5
  • proof
  • Lemma 1.6
  • proof
  • ...and 59 more