Update: Announcing a New Spine for Teichmueller Space of Closed Hyperbolic Surfaces

This post is a summary of our preprint A New Spine for Teichmueller Space of Closed Hyperbolic Surfaces. We let \((S,g)\) be a closed hyperbolic surface with constant Gauss curvature \(K=-1\). Our preprint claims the following

Main Theorem. The subvariety \(W\) consisting of hyperbolic surfaces which are homologically filled by their shortest essential separating curves is a minimal dimension spine of Teichmueller space \(Teich(S)\).

A complete proof of this fact requires that we construct continuous equivariant deformation retract \(\text{Teich} \leadsto W\), and that we prove \(\dim W = \dim Teich - (2g-1)\). Our retraction is based on constructing harmonic one forms \(\phi\) on the surface which are “adapted” to the short curves on \(S\). The key fact is that we can simultaneously increase the lengths of these short curves in the “\(\phi\)-direction” by flowing along a Teichmueller deformation in the direction of the above harmonic one form.

Let \(C'\) be the set of short essential nonseparating curves on \((S,g)\). If \(C'\) does not homologically fill \(S\), then there exists a canonical harmonic one form \(\phi\) on \(S\) such that:

• 1. we have \(\phi^*=0\) along \(C' \subset S\) (abelian condition).
• 2. we have \(|\phi(\alpha'(s))|=1\) for all \(\alpha \in C'\) when \(\alpha=\alpha(s)\) is parameterized by arclength \(s\) (uniform circulation along geodesics \(\alpha \in C'\))
• 3. we have \(\phi\) is the minimal energy harmonic one form satisfying the above conditions.

The following Lemma is our main observation:

Belt Tightening Lemma: Let \((S,g)\) be hyperbolic surface with \(C\)-systoles \(C'\). There exists a one-parameter deformation \(\{g_t\}\) in \(Teich(S)\) such that:

• the metric \(g_t\) is hyperbolic for all \(t\geq 0\) and \(g_0=g\);

• the curve lengths \(\ell(\gamma, g_t)\) are simultaneously increasing for all \(t \geq 0\) and all \(\gamma \in C'\).

The proof of Belt Tightening is achieved by studying the initial value problem defined by: \[g'=Re(q)=(\phi \phi - \phi^* \phi^*), \quad g(0)=g.\]

The definitions imply that the variation of length along integral curves of the above IVP satisfies the following key identity valid for \(\alpha \in C'\) and \(\phi\) the canonical harmonic one form defined above:

\[\frac{d}{dt} \ell(\alpha, g_t) = \frac{1}{2} \int \phi(\alpha'(s))^2 ds=\frac{1}{2}\int 1 ds =\frac{1}{2} \ell(\alpha, g_t).\]

In the expression \(\alpha'=\alpha'(s)\) represents the unit speed parameterization of the geodesic \(\alpha\) with respect to the metric \(g_t\), and this implies e.g. $ g(‘(s), ’(s))^{1/2}=1$ as a function of arclength parameter \(s\). Here we are assuming the unit speed parameterization of the geodesic curve \(\alpha\). This computation establishes Belt Tightening Lemma. Thus we obtain a canonical flow by which we can simultaneously increase the lengths of the short curves \(C'\).

The next key idea is that we simultaneously increase the lengths of these short curves until a new homologically independant curve appears. The time when these homologically independant curves appear is a continuous function of the initial data. This is key observation necessary to construct the strong deformation retracts as formally defined in algebraic topology.

Definition: For every index \(1 \leq j \leq 2g\), let \(W_j\) be the subvariety of \(T\) consisting of hyperbolic metrics whose \(C\)-systoles satisfy \(\xi(C) \geq j\).

Theorem 1. For every index \(1 \leq j \leq 2g-1\), there exists a continuous equivariant deformation retract \(W_j \to W_{j+1}\). Moreover \(W_{j+1}\) has codimension one in \(W_j\).

The general retract \(W_{j} \to W_{j+1}\) is defined as follows: Let \((S,g)\) be a hyperbolic surface in \(W_j\) with \(\xi(C(g))=j < 2g\). Let \(\{g_t\}\) be the unique one-parameter deformation of hyperbolic metrics constructed in Belt Tightening Lemma which simultaneously increase the lengths of \(C'\). The theorem follows from the following Claims (i), (ii), (iii):

• Claim (i) There exists a minimal finite stopping time \(\tau=\tau(g)\) for \(g\in W_j\) which depends continuously on \(g\) such that \(g_\tau \in W_{j+1}\). Equivalently \(\tau\) is the unique minimal time such that a new homologically independant \(C\)-systole appears and which strictly increases the complexity \(\xi\).

• Claim (ii) The one-parameter deformation \(g_t\) defines a continuous strong deformation retraction \(g\mapsto g_\tau\) from \(W_j\) to \(W_{j+1}\).

• Claim (iii) The subvariety \(W_{j+1}\) is a codimension one subvariety of \(W_j\).

We argue that Claim (i) follows from the integral formula for variation of length along the Belt Tightening. For curves which are homologically independant of \(C'\), the negative term \(-\int \phi^* \phi^*\) dominates for large values of \(t\).

Proof of Main Theorem. The retract \(T \to W\) is defined as the composition of retracts \(W_1 \to W_2 \to \cdots \to W_{2g}\) constructed in Theorem 1. It follows that \(W_{2g}\) is a codimension \(2g-1\) subvariety of \(T\), and this is the minimal possible dimension according to Bieri-Eckmann homological duality.

Remark. Geometric minimality requires a further homological duality argument a la Souto-Pettet [@ps].

[Updated February 29 2024 following important comments by MFB.]

[-JHM]