Hausdorff dimension and the Weil–Petersson extension to quasifuchsian space

Abstract

We consider a natural nonnegative two-form G on quasifuchsian space that extends the Weil-Petersson metric on Teichmüller space. We describe completely the positive definite locus of G , showing that it is a positive definite metric off the fuchsian diagonal of quasifuchsian space and is only zero on the "pure-bending" tangent vectors to the fuchsian diagonal. We show that G is equal to the pullback of the pressure metric from dynamics. We use the properties of G to prove that at any critical point of the Hausdorff dimension function on quasifuchsian space the Hessian of the Hausdorff dimension function must be positive definite on at least a halfdimensional subspace of the tangent space. In particular this implies that Hausdorff dimension has no local maxima on quasifuchsian space. 30F60, 30F40, 37D35 Statement of resultsLet S be a closed hyperbolic surface and T .S / be the associated Teichmüller space. Then the Weil-Petersson metric w is a Riemannian metric on T .S/. For simplicity, we normalize the Weil-Petersson metric to define the normalized Weil-Petersson metricIf QF.S/ is the quasifuchsian space of S , then by Bers simultaneous uniformization, QF.S/ ' T .S/ T . x S/ where x S has opposite orientation to S . This gives the natural diagonal embedding W T .S/ ! T .S/ T . x S/ ' QF.S/ given by .X / D .X; x X /. We let F.S/ D .T .S// the diagonal in QF.S /. Then F.S / corresponds to the subspace of fuchsian elements of QF.S/ and is called the fuchsian subspace. It is a smooth submanifold of QF.S/ and we have the natural identification T .S / ' F.S/ via .Quasifuchsian space QF.S / has a complex structure arising out of identifying the isometry group Isom C .H 3 / with PSL.2; C/ (see Marden [9]). This complex structure is given by a bundle map J W T .QF.S// ! T .QF.S// with J a lift of the identity map We define hW QF.S/ ! R by setting h.X / equal to the Hausdorff dimension of the limit set of X . Then by Ruelle [14], h is real-analytic. Also, associated with each X 2 QF.S / is the real-analytic length function L X W QF.S/ ! R of the unit Patterson-Sullivan geodesic current X of X .In [6], the author and Taylor showed that the function .hL X / on QF.S / is minimum at X . Using this we defined a natural nonnegative two-form G on QF.S/ given by taking the Hessian of .hL X / at X . ThusWe showed that G extends the normalized Weil-Petersson metric g on F.S/. Specifically:In this paper we answer the question of whether G is positive-definite on QF.S/. The answer is that G is "almost" a metric, in particular it is a metric off the fuchsian locus F.S/. The complete description of the positive-definite locus of G is given by the following Theorem.Although we will not discuss this aspect further, the Main Theorem has a simple description in terms of the geometry of the associated deformations. If X 2 F.S/ Â QF.S/ then the tangent space at X decomposes into T X .QF.S// D T X .F.S//J .T X .F.S/// (see Bonahon [2]). If w 2 T X .F.S// then w corresponds to deforming X inside the fuchsian subspace F.S/. By the Earthquake Th...

An introduction to pressure metrics for higher Teichmüller spaces

“…(Pollicott and Sharp [69] previously proved that the entropy associated to ℓ H varies analytically over H d (S).) However, this pressure form is degenerate in ways which are completely analogous to the degeneracy of the pressure metric on quasifuchsian space discovered by Bridgeman [14].…”

“…Our basic strategy is inspired by McMullen's [62] re-interpretation of Thurston's Riemannian metric on Teichmüller space and its generalization to quasifuchsian space by Bridgeman [14].…”

“…For example, Bridgeman's pressure metric on quasifuchsian space [14] is degenerate exactly on the set of pure bending vectors on the Fuchsian locus. However, Bridgeman's pressure metric still gives rise to a path metric.…”

“…McMullen [62] then introduced the use of the techniques of Thermodynamic Formalism to interpret both of these metrics as pullbacks of the pressure metric on the space of suspension flows on the shift space associated to the Bowen-Series coding. Bridgeman [14] then showed that the resulting pressure form on quasifuchsian space is degenerate exactly on the set of pure bending vectors on the Fuchsian locus.…”

“…Bridgeman [14] had previously defined and studied the pressure metric on quasifuchsian space QF (S) = CC(π 1 (S), PSL 2 (C)). He showed that the degenerate vectors in this case correspond exactly to pure bending vectors on the Fuchsian locus.…”