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Compactness Theorems for The Spaces of Distance Measure Spaces and Riemann Surface Laminations

Electronic Theses of Indian Institute of Science

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Title Compactness Theorems for The Spaces of Distance Measure Spaces and Riemann Surface Laminations
 
Creator Divakaran, D
 
Subject Compactness Theorems
Riemannian Geometry
Metric Geometry
Riemann Surfaces
Hyperbolic Geometry
Gromov's Compactness Theorem
Distance Measure Spaces
Deligne-Mumford Compactification
Metric Spaces
Riemann Surface Laminations
Bers’ Compactness Theorem
Mathematics
 
Description 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 theorem with many applications. In this thesis, we give a generalisation of this landmark result, more precisely, we give 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 compactification is the completion of the moduli space of Riemann surfaces under the 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 compactness theorem for J-holomorphic curves in symplectic manifolds (in particular curves in an algebraic variety) are important results for many areas of mathematics.
While Gromov compactness theorem for J-holomorphic curves in symplectic manifolds, is an important tool in symplectic topology, its applicability is limited by the lack of general methods to construct pseudo-holomorphic curves. One hopes that considering a more general class of objects in place of pseudo-holomorphic curves will be useful. Generalising the domain of pseudo-holomorphic curves from Riemann surfaces to Riemann surface laminations is a natural choice. Theorems such as the uniformisation theorem for surface laminations by 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 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.
 
Contributor Gadgil, Siddhartha
 
Date 2018-02-17T20:59:36Z
2018-02-17T20:59:36Z
2018-02-18
2014
 
Type Thesis
 
Identifier http://hdl.handle.net/2005/3131
http://etd.ncsi.iisc.ernet.in/abstracts/3988/G26382-Abs.pdf
 
Language en_US
 
Relation G26382