We generalise work of Young-Eun Choi to the setting of ideal triangulations with vertex links of arbitrary genus, showing that the set of all (possibly incomplete) hyperbolic cone-manifold structures realised by positively oriented hyperbolic ideal tetrahedra on a given topological ideal triangulation and with prescribed cone angles at all edges is (if non-empty) a smooth complex manifold of dimension the sum of the genera of the vertex links. Moreover, we show that the complex lengths of a collection of peripheral elements give a local holomorphic parameterisation of this manifold.

Given a bounded Euclidean domain Ω, we consider the sequence of optimisers of the k-th Laplacian eigenvalue within the family consisting of all possible disjoint unions of scaled copies of Ω with fixed total volume. We show that this sequence encodes information yielding conditions for Ω to satisfy Pólya’s conjecture with either Dirichlet or Neumann boundary conditions. This is an extension of a result by Colbois and El Soufi which applies only to the case where the family of domains consists of all bounded domains. Furthermore, we fully classify the different possible behaviours for such sequences, depending on whether Pólya’s conjecture holds for a given specific domain or not. This approach allows us to recover a stronger version of Pólya’s original results for tiling domains satisfying some dynamical billiard conditions, and a strenghtening of Urakawa’s bound in terms of packing density.

We use the construction of unfolded Seiberg-Witten Floer spectra of general 3-manifolds defined in our previous paper to extend the notion of relative Bauer-Furuta invariants to general 4-manifolds with boundary. One of the main purposes of this paper is to give a detailed proof of the gluing theorem for the relative invariants.

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