The Short Curve conjecture for Weil–Petersson geodesics

Given a Weil–Petersson geodesic $g:(a,b) \to \operatorname{Teich}(S)$, let its forward and backward end invariants be $(\nu^{-},\nu^{+})$. For each subsurface $Z\subseteq S$ that is not a three-holed sphere, let $d_Z(\nu^{-},\nu^{+})$ denote the distance in the curve complex of $Z$ between the projections of the end invariants. For a simple closed curve $\alpha$, write $\ell_\alpha(g(t))$ for its length along $g$.

Short Curve conjecture. If $g$ is a Weil–Petersson geodesic with end invariant $(\nu^{-},\nu^{+})$, then:

1. For every $\epsilon>0$ there is an $A>0$ such that, if $d_Z(\nu^{-},\nu^{+})>A$, then

$$\inf_t\ell_\alpha(g(t))\leq\epsilon$$

for every $\alpha\in\partial Z$. 2. For every $A>0$ there is an $\epsilon>0$ such that, if

$$\inf_t\ell_\alpha(g(t))\leq\epsilon,$$

then there is a subsurface $Z\subsetneq S$ such that $\alpha\in\partial Z$ and $d_Z(\nu^{-},\nu^{+})>A$.

The conjecture asserts that large subsurface coefficients exactly detect the appearance of short boundary curves along Weil–Petersson geodesics. It gives a subsurface-projection coding of geodesic behavior in the moduli space, generalizing the role of continued fractions for geodesics on the modular surface.