Weil Petersson Geodesics Research Articles

A new uniform lower bound on Weil–Petersson distance

In this paper we study the injectivity radius based at a fixed point along Weil-Petersson geodesics. We show that the square root of the injectivity radius based at a fixed point is $ 0.3884$-Lipschitz on Teichm\"uller space endowed with the Weil-Petersson metric. As an application we reprove that the square root of the systole function is uniformly Lipschitz on Teichm\"uller space endowed with the Weil-Petersson metric, where the Lipschitz constant can be chosen to be $0.5492$. Applications to asymptotic geometry of moduli space of Riemann surfaces for large genus will also be discussed.

Cusp excursions of random geodesics in Weil–Petersson type metrics

We analyse cusp excursions of random geodesics for Weil–Petersson type incomplete metrics on orientable surfaces of finite type: in particular, we give bounds for maximal excursions. We also give similar bounds for cusp excursions of random Weil–Petersson geodesics on non-exceptional moduli spaces of Riemann surfaces conditional on the assumption that the Weil–Petersson flow is polynomially mixing. Moreover, we explain how our methods can be adapted to understand almost greasing collisions of typical trajectories in certain slowly mixing billiards.

Bottlenecks for Weil-Petersson geodesics

We introduce a method for constructing Weil-Petersson (WP) geodesics with certain behavior in the Teichmüller space. This allows us to study the itinerary of geodesics among the strata of the WP completion and its relation to subsurface projection coefficients of their end invariants. As an application we demonstrate the disparity between short curves in the universal curve over a WP geodesic and those of the associated hyperbolic 3–manifold.

Growth of the Weil–Petersson inradius of moduli space

In this paper we study the systole function along Weil-Petersson geodesics. We show that the square root of the systole function is uniformly Lipschitz on Teichm\"uller space endowed with the Weil-Petersson metric. As an application, we study the growth of the Weil-Petersson inradius of moduli space of Riemann surfaces of genus $g$ with $n$ punctures as a function of $g$ and $n$. We show that the Weil-Petersson inradius is comparable to $\sqrt{\ln{g}}$ with respect to $g$, and is comparable to $1$ with respect to $n$. Moreover, we also study the asymptotic behavior, as $g$ goes to infinity, of the Weil-Petersson volumes of geodesic balls of finite radii in Teichm\"uller space. We show that they behave like $o((\frac{1}{g})^{(3-\epsilon)g})$ as $g\to \infty$, where $\epsilon>0$ is arbitrary.

Limit sets of Weil–Petersson geodesics with nonminimal ending laminations

In this paper, we construct examples of Weil–Petersson geodesics with nonminimal ending laminations which have [Formula: see text]-dimensional limit sets in the Thurston compactification of Teichmüller space.

Limit Sets of Weil–Petersson Geodesics

Abstract In this paper we prove that the limit set of any Weil–Petersson geodesic ray with uniquely ergodic ending lamination is a single point in the Thurston compactification of Teichmüller space. On the other hand, we construct examples of Weil–Petersson geodesics with minimal non-uniquely ergodic ending laminations and limit set a circle in the Thurston compactification.

Variation of Extremal Length Functions on Teichmüller Space

Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to $\mathbb{R}$-trees, we study the second variation of extremal length functions along Weil-Petersson geodesics. We show that the extremal length of any measured foliation is a pluri-subharmonic function on Teichmuller space.

Asymptotics of a class of Weil–Petersson geodesics and divergence of Weil–Petersson geodesics

We show that the strong asymptotic class of Weil-Petersson (WP) geodesics with narrow end invariant and bounded annular coefficients is determined by the forward ending lamination. This generalizes the Recurrent Ending Lamination Theorem of Brock-Masur-Minsky. As an application we provide a symbolic condition for divergence of WP geodesic rays in the moduli space.

Recurrent Weil–Petersson geodesic rays with non-uniquely ergodic ending laminations

We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichm\"uller geodesics does not hold for WP geodesics.

Asymptotics of Weil–Petersson Geodesics II: Bounded Geometry and Unbounded Entropy

We use ending laminations for Weil–Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for Weil–Petersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil–Petersson geodesics. As an application, we show theWeil–Petersson geodesic flow has compact invariant subsets with arbitrarily large topological entropy.

Asymptotics of Weil–Petersson Geodesics I: Ending Laminations, Recurrence, and Flows

We define an ending lamination for a Weil–Petersson geodesic ray. Despite the lack of a natural visual boundary for the Weil–Petersson metric [Bro2], these ending laminations provide an effective boundary theory that encodes much of its asymptotic CAT(0) geometry. In particular, we prove an ending lamination theorem (Theorem 1.1) for the full-measure set of rays that recur to the thick part, and we show that the association of an ending lamination embeds asymptote classes of recurrent rays into the Gromov-boundary of the curve complex $${\mathcal{C}(S)}$$ . As an application, we establish fundamentals of the topological dynamics of the Weil–Petersson geodesic flow, showing density of closed orbits and topological transitivity.

Extension of the Weil-Petersson connection

Convexity properties of Weil-Petersson (WP) geodesics on the Teichmuller space of punctured Riemann surfaces are investigated. A normal form is presented for the Weil-Petersson–Levi-Civita connection for pinched hyperbolic metrics. The normal form is used to establish approximation of geodesics in boundary spaces. Considerations are combined to establish convexity along Weil-Petersson geodesics of the functions, the distance between horocycles for a hyperbolic metric