We give a complete proof of the Bers–Sullivan–Thurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds. The ending invariants are either Riemann surfaces or filling laminations supporting Masur domain measured laminations and satisfying some mild additional conditions. With any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence. We derive the necessary compactness theorem combining the Rips machine with non-existence results for certain small actions on real trees of free products of surface groups and free groups. We prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface. In order to be able to apply the ending lamination theorem, we have to prove that this algebraic limit has the desired topological type and that these non-realized laminations are ending laminations. That this is the case is the main novel technical result of this paper. Loosely speaking, we show that a filling Masur domain lamination which is not realized is an ending lamination.

Let$M$be a compact complex manifold. The corresponding Teichmüller space Teich is the space of all complex structures on$M$up to the action of the group $${{\rm Diff}_0(M)}$$ of isotopies. The mapping class group $${\Gamma:={\rm Diff}(M)/{{\rm Diff}_0(M)}}$$ acts on Teich in a natural way. An$ergodic complex structure$is a complex structure with a $${\Gamma}$$ -orbit dense in Teich. Let$M$be a complex torus of complex dimension $${\ge 2}$$ or a hyperkähler manifold with $${b_2 > 3}$$ . We prove that$M$is ergodic, unless$M$has maximal Picard rank (there are countably many such$M$). This is used to show that all hyperkähler manifolds are Kobayashi non-hyperbolic.

It is proved that if the twistor space of a self-dual four-manifold of positive scalar curvature contains a real effective divisor of degree two, then the four-manifold is diffeomorphic to the connected sum n\mathbb {C}{P^ 2} of n complex projective planes for some n. It follows that if the four-manifold is known to be homeomorphic to n\mathbb {C}{P^ 2} , then it is also diffeomorphic to n\mathbb {C}{P^ 2} .

In a recent series of articles, the authors have studied the transition behavior of partial Bergman kernels $\Pi_{k, [E_1, E_2]} (z,w)$ and the associated DOS (density of states) $\Pi_{k, [E_1, E_2]} (z)$ across the interface $\mathcal{C}$ between the allowed and forbidden regions. Partial Bergman kernels are Toeplitz Hamiltonians quantizing Morse functions $H : M \to \mathbb{R}$ on a Kähler manifold. The allowed region is $H^{-1} ([E_1, E_2])$ and the interface $\mathcal{C}$ is its boundary. In prior articles it was assumed that the endpoints $E_j$ were regular values of $H$. This article completes the series by giving parallel results when an endpoint is a critical value of $H$. In place of the Erf scaling asymptotics in a $k^{-\frac{1}{2}}$ tube around $\mathcal{C}$ for regular interfaces, one obtains $\delta$-asymptotics in $k^{-\frac{1}{4}}$ tubes around singular points of a critical interface. In $k^{-\frac{1}{2}}$ tubes, the transition law is given by the osculating metaplectic propagator.

It is shown that an HKT space with closed parallel potential 1-form has D (2, 1; 1) symmetry. Every locally conformally hyper-Khler manifold generates this type of geometry. The HKT spaces with closed parallel potential 1-form arising in this way are characterized by their symmetries and an inhomogeneous cubic condition on their torsion.