In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X , then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil–Petersson norm of μ over the square root of the systole of X up to a uniform positive constant multiplication. We apply the uniform bound above to show that the Weil–Petersson Ricci curvature, restricted at any hyperbolic surface of short systole in the moduli space, is uniformly bounded from below by the negative reciprocal of the systole up to a uniform positive constant multiplication. As an application, we show that the average total Weil–Petersson scalar curvature over the moduli space is uniformly comparable to -g as the genus g goes to infinity.

For fixed subgroups Fix(ϕ) of automorphisms ϕ on hyperbolic 3-manifold groups π1(M), we observed that rk(Fix(ϕ))<2rk(π1(M)) and the constant 2 in the inequality is sharp; we also classify all possible groups Fix(ϕ).

Dunfield-Garoufalidis and Boyer-Zhang proved that the A-polynomial of a nontrivial knot in S3 is nontrivial. In this paper, we use holonomy perturbations to prove the non-triviality of the A-polynomial for a nontrivial, null-homotopic knot in an irreducible 3-manifold. Also, we give a strong constraint on the A-polynomial of a knot in the 3-sphere.

Suppose that <i></i>₀ is an unknotted simple closed curve contained in the 3-sphere which happens to be invariant under a subgroup <i>G</i> of the Mbius group of <i>S³</i> = the group (generated by inversions in 2-spheres). It is shown that there is an equivariant isotopy <i></i>_<i>t</i>, 0 <i>t</i> 1, from <i></i>₀ to a round circle <i></i>₁.

Given a 3-holed sphere decomposition of an orientable closed surface, it is shown that each orientation preserving homeomorphism of the surface is isotopic to a composition AB where A is a product of positive Dehn twists and B is a product of negative Dehn twists on the decomposition curves.