Gromov’s compactness theorem for metric spaces, a compactness theorem for the
space of compact metric spaces equipped with the Gromov-Hausdorff distance, is
a landmark theorem with many applications. We give a generalisation of this
result - more precisely, we prove a compactness theorem for the space of
distance measure spaces equipped with the generalised
Gromov-Hausdorff-Levi-Prokhorov distance. A distance measure space is a triple
`$(X, d, μ)$`

, where $(X, d)$ forms a distance space (a generalisation of a metric
space where, we allow the distance between two points to be infinity) and μ is
a finite Borel measure.

Using this result, we prove that the Deligne-Mumford compactifiaction is the completion of the moduli space of Riemann surfaces under generalised Gromov-Hausdorff-Levi-Prokhorov distance. The Deligne-Mumford compactification, a compactification of the moduli space of Riemann surfaces with explicit description of the limit points, and the closely related Gromov’s compactness theorem for pseudo-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.

While Gromov’s compactness theorem for pseudo-holomorphic curves is an important tool in symplectic topology, its applicability is limited due to the non-existence of a general method to construct pseudo-holomorphic curves. Considering a more general class of domains (in place of Riemann surfaces) is likely to be useful. Riemann surface laminations are a natural generalisation of Riemann surfaces. Theorems such as the uniformisation theorem for surface laminations due to Alberto Candel (which is a partial generalisation of the uniformisation theorem for surfaces), generalisations of the Gauss-Bonnet theorem proved for some special cases, and the topological classification of “almost all” leaves using harmonic measures reinforces the usefulness of this line on enquiry. Also, the success of essential laminations,as generalised incompressible surfaces, in the study of 3-manifolds suggests that a similar approach may be useful in symplectic topology. With this motivation we prove a compactness theorem analogous to the Deligne-Mumford compactification for the space of Riemann surface laminations.

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Last updated: 22 Jun 2024