[Paper Review] Polynomial cubic differentials and convex polygons in the projective plane

David Dumas, Michael Wolf|arXiv (Cornell University)|Jul 30, 2014

Advanced Differential Equations and Dynamical Systems32 references4 citations

TL;DR

This paper establishes a homeomorphism between the moduli space of polynomial cubic differentials of degree $d$ on $\mathbb{C}$ and the space of projective equivalence classes of oriented convex $(d+3)$-gons in $\mathbb{RP}^2$. Using the Cheng-Yau theorem and Wang's method, it constructs complete hyperbolic affine spheres from polynomial cubic differentials, showing that each such differential arises as the Pick differential of an affine sphere asymptotic to a convex polygon, with symmetry properties preserved under the correspondence.

ABSTRACT

We construct and study a natural homeomorphism between the moduli space of polynomial cubic differentials of degree d on the complex plane and the space of projective equivalence classes of oriented convex polygons with d+3 vertices. This map arises from the construction of a complete hyperbolic affine sphere with prescribed Pick differential, and can be seen as an analogue of the Labourie-Loftin parameterization of convex RP^2 structures on a compact surface by the bundle of holomorphic cubic differentials over Teichmuller space.

Motivation & Objective

• To characterize polynomial cubic differentials on $\mathbb{C}$ that arise as Pick differentials of complete hyperbolic affine spheres asymptotic to convex polygons in $\mathbb{RP}^2$.
• To extend the Labourie-Loftin parameterization from compact surfaces to the non-compact case of convex polygons in $\mathbb{RP}^2$.
• To construct a natural homeomorphism between the moduli space of degree-$d$ polynomial cubic differentials and the moduli space of projective equivalence classes of convex $(d+3)$-gons.
• To prove the existence of complete hyperbolic affine spheres with prescribed polynomial Pick differential using a method of super- and sub-solutions for quasilinear PDEs.
• To establish a correspondence that preserves geometric and dynamical symmetries, such as rotational symmetry in the pentagon case corresponding to invariance under $z \mapsto e^{2\pi i/5}z$.

Proposed method

• Constructs a complete hyperbolic affine sphere over $\mathbb{R}^3$ asymptotic to the cone over a convex polygon in $\mathbb{RP}^2$ using the Cheng-Yau theorem on affine spheres.
• Applies Wang's method to reduce the existence of the affine sphere to solving a quasilinear PDE for the Blaschke metric, given a holomorphic cubic differential.
• Uses the method of super- and sub-solutions to prove existence of solutions to the quasilinear PDE for the Blaschke metric associated with a polynomial cubic differential.
• Lifts the construction to a universal cover to define a map between orbifold moduli spaces, then descends to the quotient under $\mathrm{Aut}(\mathbb{C})$ and $\mathrm{SL}_3\mathbb{R}$ actions.
• Establishes continuity of the forward and inverse maps between the moduli space of polynomial cubic differentials and the moduli space of convex polygons, proving the homeomorphism.
• Employs ODE asymptotics and matrix exponential estimates to control the behavior of solutions to the associated connection equations in the limit.

Experimental results

Research questions

• RQ1 Which polynomial cubic differentials on $\mathbb{C}$ arise as the Pick differential of a complete hyperbolic affine sphere asymptotic to a convex polygon in $\mathbb{RP}^2$?
• RQ2 Can the Labourie-Loftin parameterization of $\mathbb{RP}^2$ structures on compact surfaces be generalized to non-compact convex domains with polygonal boundary at infinity?
• RQ3 How does the symmetry of a convex polygon in $\mathbb{RP}^2$ correspond to the automorphism group of its associated polynomial cubic differential?
• RQ4 What is the precise relationship between the degree of a polynomial cubic differential and the number of vertices of the corresponding convex polygon?
• RQ5 Is the map from polynomial cubic differentials to convex polygons smooth, and what is its regularity on the orbifold moduli spaces?

Key findings

• The affine sphere construction yields a well-defined homeomorphism $\alpha: \mathcal{MC}_d \to \mathcal{MP}_{d+3}$, where $\mathcal{MC}_d$ is the moduli space of $\mathrm{Aut}(\mathbb{C})$-equivalence classes of degree-$d$ polynomial cubic differentials, and $\mathcal{MP}_{d+3}$ is the moduli space of $\mathrm{SL}_3\mathbb{R}$-equivalence classes of convex $(d+3)$-gons.
• For the regular pentagon, the associated cubic differential is $z^2 dz^3$, and its invariance under $z \mapsto e^{2\pi i/5}z$ reflects the 5-fold rotational symmetry of the polygon.
• The construction of the Blaschke metric via super- and sub-solutions ensures existence of a complete hyperbolic affine sphere for any polynomial cubic differential on $\mathbb{C}$.
• The method of proof relies on controlling the growth of the $|\phi|$-metric in a conformal coordinate system, ensuring the existence of a zero-free disk of linearly growing radius near the boundary of the domain.
• The asymptotic behavior of solutions to the associated ODEs is controlled via matrix exponential estimates, with bounds depending on the $L^1$-norm of the coefficient and small perturbations.
• The map $\alpha$ is continuous in both directions, and the spaces $\mathcal{MC}_d$ and $\mathcal{MP}_n$ are shown to be smooth orbifolds, suggesting the possibility of smoothness of $\alpha$ beyond continuity.