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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.

Wehrl used Glauber coherent states to define a map from quantum density matrices to classical phase space densities and conjectured that for Glauber coherent states the mininimum classical entropy would occur for density matrices equal to projectors onto coherent states. This was proved by Lieb in 1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for every angular momentum$J$. This conjecture is proved here. We also recall our 1991 extension of the Wehrl map to a quantum channel from J to $${K=J+\frac{1}{2}, J+1, ... ,}$$ with $${K=\infty}$$ corresponding to the Wehrl map to classical densities. These channels were later recognized as the optimal quantum cloning channels. For each$J$and $${J < K \leqslant \infty}$$ we show that the minimal output entropy for the channels occurs for a J coherent state. We also show that coherent states both Glauber and Bloch minimize any concave functional, not just entropy.

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