In LM, we proved a family version of the famous Witten rigidity theorems and several family vanishing theorems for elliptic genera. In this paper, we gerenalize our theorems LM in two directions. First we establish a family rigidity theorem for the Dirac operator on loop space twisted by general positive energy loop group representations. Second we prove a family rigidity theorem for spin^ c-manifolds. Several vanishing theorems on both cases are also obtained.

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.

We prove that the GromovHausdorff compactification of the moduli space of KhlerEinstein Del Pezzo surfaces in each degree agrees with certain algebro-geometric compactification. In particular, this recovers Tians theorem on the existence of KhlerEinstein metrics on smooth Del Pezzo surfaces and classifies all the degenerations of such metrics. The proof is based on a combination of both algebraic and differential geometric techniques.

In this paper we investigate the convergence for the mean curvature flow of closed submanifolds with arbitrary codimension in space forms. Particularly, we prove that the mean curvature flow deforms a closed submanifold satisfying a pinching condition in a hyperbolic space form to a round point in finite time.

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.