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We present a brief but nearly self-contained proof of a formula for the Weil-Petersson Hessian of the geodesic length of a closed curve (either simple or not simple) on a hyperbolic surface. The formula is the sum of the integrals of two naturally defined positive functions over the geodesic, proving convexity of this functional over Teichmuller space (due to Wolpert (1987)). We then estimate this Hessian from below in terms of local quantities and distance along the geodesic. The formula extends to proper arcs on punctured hyperbolic surfaces, and the estimate to laminations. Wolpert’s result that the Thurston metric is a multiple of the Weil-Petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves. We give further applications to upper bounds of the Hessian, especially near pinching loci, recover through a geometric argumentWolpert’s result on the convexity of length to the half-power, and give a lower bound for growth of length in terms of twist.

Given an open book decomposition (Σ,τ) of a three manifold Y , Thurston and Winkelnkemper [TW] construct a specific contact form a on Y . Given a spin-c Dirac operator D on Y , the contact form naturally associates a one parameter family of Dirac operators Dr = D− ir 2 cl(a) for r ≥ 0. When r >> 1, we prove that the spectrum of Dr = D0 − ir 2 cl(a) within [−1 2r 1 2 , 1 2r 1 2 ] are almost uniformly distributed. With the result in Part I [Ts1], it implies that the subleading order term of the spectral flow from D0 to Dr is of order r(logr) 9 2 . Besides the interests of the spectral flow, the method of this paper provide a tool to analyze the Dirac operator on an open book decomposition.

We give a proof of the Hard Lefschetz Theorem for orbifolds that does not involve intersection homology. We use a foliated version of the Hard Lefschetz Theorem due to El Kacimi.

In this article we show how to compute the semiclassical spectral measure associated with the Schr¨odinger operator on Rn, and, by examining the first few terms in the asymptotic expansion of this measure, obtain inverse spectral results in one and two dimensions. (In particular we show that for the Schr¨odinger operator on R2 with a radially symmetric electric potential, V , and magnetic potential, B, both V and B are spectrally determined.) We also show that in one dimension there is a very simple explicit identity relating the spectral measure of the Schr¨odinger operator with its Birkhoff canonical form.