The Weil-Petersson geometry on the thick part of the moduli space of Riemann surfaces

Abstract

Abstract. In the thick part of the moduli space of Riemann surfaces, we show that the sectional curvature of the Weil-Petersson metric is bounded independently of the genus of the surface.

The Weil–Petersson geometry of the moduli space of Riemann surfaces

“…Proof. The proof follows closely the proofs to obtain lower and upper bounds given in [16,4]. For completeness, we repeat it here.…”

“…Therefore, attention must be shifted to find lower bounds of the sectional curvature on compact subsets of the moduli space M g . This problem was studied by Huang in [4]. To describe his result in more detail, we need to introduce some notation first.…”

“…Given a positive constant r 0 , the thick part of the moduli space M g (with respect to r 0 ) is defined as the subset of M g consisting of those points where the injectivity radius of the corresponding Riemann surfaces is greater than r 0 . In [4], Huang showed that on the thick part of the moduli space M g , the holomorphic sectional and sectional curvatures of the WP metric are both bounded below by negative constants −C 1 and −C 2 depending on r 0 , but independent of the genus g. As a result, the Ricci and scalar curvatures are bounded below by −C 3 g and −C 4 g 2 respectively, where C 3 and C 4 are two positive constants depending on r 0 , but independent of g. The main tool used by Huang is the analysis of harmonic maps between hyperbolic surfaces. In the present article, we are going to give a different proof of Huang's result without using harmonic maps.…”

“…To obtain the lower bounds of curvatures, Huang [4] used harmonic maps to show that if a harmonic Beltrami differential has unit Weil-Petersson norm, then its sup-norm is bounded above by a constant depending on the injectivity radius of the underlying Riemann surface. Here we re-prove this result without resorting to harmonic maps, which better reveals its elementary nature and also allows us to generalize this result to the universal Teichmüller space later.…”

Harmonic mappings and moduli spaces of Riemann surfaces

“…The curvature of the Weil-Petersson metric is not bounded negatively from above, as was first shown by Huang [27] who found that the curvature goes to 0 in certain directions when approaching the boundary of moduli space. This issue has been further investigated in [29,74]. Also, in [56] it was shown that the Weil-Petersson metric is not Gromov hyperbolic.…”