Lecture 14: Deforming Strictly Convex Projective Manifolds

Let $M$ be a closed, connected, smooth $n$ -manifold.

Our goal is to describe all strictly convex projective structures on $M$ . To this end, we seek a strictly convex domain $\Omega \subseteq \mathbb R \mathrm P^n$ such that $M$ is homeomorphic to the orbit space $\Omega / \Gamma$ for some subgroup $\Gamma$ of $\mathrm{PGL}(\Omega)$ . Recall that $\Omega$ is properly convex if its closure lies in an affine patch, and that $\Omega$ is strictly convex if it is properly convex and if $\partial \overline \Omega$ contains no line segment of positive length.

A marking $h : M \to \Omega / \Gamma$ gives a holonomy representation

$\rho = h_* : \pi_1 M \stackrel{\cong}{\longrightarrow} \Gamma.$

Proposition. $\Gamma$ determines $\Omega$ ; hence, $\rho$ determines $\Omega$ .

Proof. This follows from the fact that $\partial \overline \Omega$ is the limit set of $\Gamma$ . $\square$

Thus describing strictly convex projective structures on $M$ amounts to describing holonomy representations $\pi_1 M \to \Gamma \leq \mathrm{PGL}(n + 1)$ .

Let $SC(M)$ denote the set of all holonomy representations $\pi_1 M \to \mathrm{PGL}(n + 1)$ of strictly convex projective structures on $M$ . We have

$SC(M) \leq \mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1)) \cong \mathrm{Hom}(\pi_1 M, \mathrm{SL}_\pm (n + 1)).$

Theorem (Benoist 2000). $SC(M)$ is both open and closed in $\mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1))$ .

We will prove the “open” part of this theorem using a general result of Thurston and Ehresmann.

$(G, X)$ -MANIFOLDS, THE HOLONOMY THEOREM

Let $X$ be a manifold, and let $G$ be a Lie group that acts continuously (and hence analytically) and transitively on $X$ by diffeomorphisms.

Recall that a manifold $M$ is a $(G, X)$ -manifold if there exists an atlas $\{\varphi_i : U_i \subseteq M \to \varphi_i(U_i) \subseteq X\}$ of $M$ such that the transition map

$g_{ij} = \varphi_j \circ \varphi_i^{-1} : \varphi_i^{-1}(U_i \cap U_j) \longrightarrow \varphi_j^{-1}(U_i \cap U_j)$

is the restriction of an element of $G$ whenever $U_i \cap U_j$ is nonempty, and such that the following cocycle condition is satisfied:

$g_{ik} = g_{jk} \circ g_{ij} \quad\text{whenever}\quad U_i \cap U_j \cap U_k \neq \emptyset.$

Alternatively, $M$ is a $(G, X)$ -manifold if and only if it is homeomorphic to a quotient $X / \Gamma$ for some discrete torsion-free subgroup of $G$ . Then the action of $\Gamma$ on $X$ is properly discontinuous and free, and the diagram

$\begin{matrix} \pi_1 M & \smash{\stackrel{\mathrm{hol}}{\longrightarrow}} & \Gamma \\ \curvearrowright & & \curvearrowright \\ \widetilde M & \smash{\stackrel{\mathrm{dev}}{\longrightarrow}} & X \\ \end{matrix}$

commutes. Here $\pi_1 M$ acts on the universal cover $\widetilde M$ of $M$ by deck transformations, $\mathrm{dev}$ is the developing map, and $\mathrm{hol}$ is the holonomy of $M$ .

Theorem (Thurston–Ehresmann). Let $M$ be a closed, connected, smooth $(G, X)$ -manifold. Then the set of all holonomies $\pi_1 M \to G$ of $(G, X)$ -structures on $M$ is open in $\mathrm{Hom}(\pi_1 M, G)$ .

Sketch of proof (Thurston 1978). Let $\pi : \widetilde M \to M$ be the universal cover, and let $D \subseteq \widetilde M$ be a fundamental domain. There exist finitely many charts fo $\varphi_: U_i \to X$ covering $D$ . For each index $i$ , we choose $U_i$ small enough that $\pi$ is one-to-one on $U_i$ , and we use $\pi$ to identify $U_i$ with a subset of $M$ . We have

$M = \bigsqcup U_i \Big / \sim,$

where the equivalence relation $\sim$ is given by

$x_i \sim x_j \Longleftrightarrow g_{ij} \circ \varphi_i(x_i) = \varphi_j(x_j).$

Here is the idea of the proof: If $\rho'$ is close to a holonomy representation $\rho : \pi_1 M \to G$ , then there are maps $g'_{ij}$ close to the maps $g_{ij}$ satisfying the cocycle condition. Then define

$M' = \bigsqcup U_i \Big / \sim',$

where the relation $\sim'$ is defined by replacing $g_{ij}$ with $g'_{ij}$ above.

Why can we do this? Take the nerve of the cover, i.e., a graph $H$ with one vertex for each set $U_i$ and one edge between corresponding vertices whenever $U_i \cap U_j$ is nonempty. We may assume without loss of generality that the sets $U_i$ are round balls, so that $H$ is connected. Now let $T$ be a maximal tree with a chosen basepoint. Define $g'_{ij}$ to be $g_{ij}$ if the edge for $g_{ij}$ is in $T$ ; otherwise, choose $g'_{ij}$ so that the based loop $\alpha_{ij}$ in $H$ containing the edge for $g_{ij}$ is $\rho(\alpha_{ij})$ , a product of maps $g_{kl}$ around the loop. $\square$

Theorem (Kozul 1962). $SC(M)$ is open in $\mathrm{Hom}(\pi_1 M, \mathrm{PGL}(n + 1))$ .

Proof. Suppose we are given a strictly convex projective structure, and hence a holonomy representation $\rho : \pi_1 M \to \mathrm{PGL}(n + 1)$ by the Proposition, and a nearby representation $\rho'$ . By the holonomy theorem, there exists a nearby projective structure, and this structure is a strictly convex structure (why?). $\square$

HESSIAN METRICS

Let $U \subseteq \mathbb R^n$ be open, and let $c : U \to \mathbb R^n$ be a smooth function that is strictly convex (i.e., whose Hessian matrix is positive definite). The map $c$ induces a Riemannian metric on $U$ given by

$\langle u, v \rangle_c = u^\top A v \quad\text{for all }v, w \in T_x U \text{ and all } x \in U,$

where

$A = D^2 c = {\left(\displaystyle{\frac{\partial^2 c}{\partial x_i \partial x_j}}\right)}$

is the Hessian matrix of $c$ .

For example, the map $c : \mathbb R^2 \to \mathbb R$ given by $c(x_1, x_2) = x_1^2 + x_2^2$ induces the Euclidean metric on $\mathbb R^2$ .

Recall that a nonzero tangent vector $v \in T_x U$ is specified by a smooth curve $\gamma : (-\epsilon, \epsilon) \to U$ with $\gamma(0) = x$ and $\gamma'(0) = v$ , where $x \in U$ and $\epsilon > 0$ . Then $\|v\|_c^2 = F''(0) > 0$ by strict convexity, where $F = c \circ \gamma$ .

Let $M$ be an affine $n$ -manifold. Then we have charts $\varphi_i : U_i \subseteq M \to \mathbb R^n$ and transition maps in the affine group $\mathrm{Aff}(n)$ .

Let $c : M \to \mathbb R$ be smooth. Then

$\begin{align*} c \text{ is strictly convex} &\Longleftrightarrow {} \text{all maps } c \circ \varphi_i^{-1} \text{ are strictly convex} \\ &\Longleftrightarrow {} (c \circ \gamma)''(t) > 0 \text{ for all } t \in (-\epsilon, \epsilon) \text{ for all curves } \gamma \text{ defined as above.} \end{align*}$

Thus $c$ determines a Hessian metric on $M$ .

Remark. The Vinberg character functions on cones (April 14) are examples of this phenomenon.

Theorem. Suppose $M$ is a simply connected boundaryless affine $n$ -manifold with a complete Hessian metric $d$ . Then the developing map $M \to \mathbb R^n$ is injective and has image a convex set.

Proof. Since $M$ is simply connected, the developing map is a diffeomorphism, so we identify $M$ with its image in $\mathbb R^n$ .

Let $a, b \in M$ be distinct; we must show that the line segment joining them is contained in $M$ . Consider the triangle formed by $a$ , $b$ , and an arbitrary point $p \in M$ . Let $\alpha, \beta : [0, 1] \to M$ be the straight-line paths paths from $p$ to $a$ and from $p$ to $b$ , respectively. For each $t \in [0, 1]$ , let $\gamma_t : [0, 1] \to M$ be the straight-line path from $\alpha(t)$ to $\beta(t)$ . We claim that

$\mathrm{length}(\gamma_t) \leq K < \infty\quad \text{for some } K > 0 \text{ for all } t.$

Then $d(x, p) < L$ for some $L > 0$ for all $x \in M$ , and we are done by completeness.

Let $F_t = c \circ \gamma_t$ for each $t$ . We have

$\mathrm{length}(\gamma_t) = \int^1_0 \sqrt{F''_t(s)}\,ds \leq {\left(\int^1_0 F''_t(s)\,ds\right)}^{1/2} {\left(\int^1_0 ds\right)}^{1/2} \leq (F'_t(1) + F'_t(0))^{1/2} \leq K.$

The first inequality follows from the Cauchy–Schwarz ineqality, and the last inequality (for some $K > 0$ ) follows from compactness. This proves the claim. $\square$

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