DIVAKARAN DIVAKARAN

(under the supervision of Siddhartha Gadgil)

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Since the landmark work of Gromov, a very fruitful technique in Riemannian geometry is to consider limits of Riemannian manifolds that are of a more general nature, such as metric spaces. The limits are taken in the sense of the Gromov-Hausdorff distance, which can be defined for compact metric spaces or for metric spaces with given base points. As Riemannian manifolds are equipped with canonical measures, namely volumes, it has also turned out to be fruitful to consider limits of these as metric measure spaces. However, one case of fundamental importance in mathematics, the Deligne-Mumford compactification of the Moduli space of Riemann surfaces, or equivalently hyperbolic surfaces, cannot be naturally viewed (at least directly) in this setting. This is because the limit of hyperbolic surfaces is not compact, and indeed has in general several non-compact components. However, a limit using basepoints will only have one limiting non-compact component. Motivated by the desire to generalize convergence of metric measure spaces to include this case in a natural way, and furthermore allow us to study limits of surface laminations, we gave a generalization of Gromov's compactness theorem in my thesis. More precisely, we proved a compactness theorem for the space of distance measure spaces. As a consequence, the Deligne-Mumford compactification turns out to have a very natural description, namely, it is the completion of the space of hyperbolic metrics (with respect to our metric).

Further, along the same lines, we proved a compactness theorem for the space of Riemann surface laminations. This has scope for many applications. It is often useful to study an object as a blackbox - analyse something based on how it behaves. For topological spaces, this would mean analysing collection of all maps into the space or collection of maps from the space. Loops, maps from a circle to the space, and its higher dimensional analogues have proved very useful and forms a big part of the subject of Algebraic topology. J-holomorphic curves, maps from a Riemann surface to an almost complex manifold (satisfying some “nice” conditions) similarly proved useful in symplectic topology. For this reason, Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology. However, its applicability is limited by the lack of general methods to construct J-holomorphic curves. Now that we have compactness theorem for Reimann surface laminations analogous to the Deligne-Mumford compactification, we believe, J-holomorphic laminations, maps from Reimann surface laminations to an almost complex manifold can be a powerful tool in symplectic topology.