RUI: Complex Structures, Hyperbolic Invariants, Infinitesimal Currents and Intersection Numbers for Deformation Spaces

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The Principal Investigator studies Teichmuller spaces of surfaces, both closed and open. The Teichmuller space of a surface is the space of all possible shapes of the surface, where a shape of the surface is the hyperbolic metric on a surface up to isometries homotopic to the identity. Therefore the invariants of hyperbolic metrics on a surface are used in the study of Teichmuller spaces. The PI?s approach is to first consider the Teichmuller space of the hyperbolic plane called the universal Teichmuller space as this space contains all other Teichmuller spaces. An invariant called a shear associated to an ideal triangulation of the hyperbolic plane is used to parameterize the universal Teichmuller space. The PI intends to continue his study of the universal Teichmuller space and related Teichmuller spaces in terms of these invariants called shears. In particular, the PI intends to describe the Weil-Petersson metric on the Teichmuller space of a finitely punctured closed surface in terms of shears on ideal triangulations of the surface, where triangulations can be both locally finite and locally infinite. He also intends to implement these formulas on the computer with the help of some undergraduate and masters students from Queens College. Another direction in applying shear invariants to the Teichmuller spaces is to find a parameterization of Takhtajan-Teo Teichmuller space in terms of shears and to find a formula for the Weil-Petersson metric in this space. This direction has possible applications to Sharon-Mumford?s approach to two-dimensional shape analysis in Computer Vision. The Quasifuchsian space of a closed surface supports a Weil-Petersson metric as well which is defined by taking the second partial derivative of the product of the Hausdorff dimension of the limit quasicircle and the Sullivan-Paterson measure. The PI intends to investigate the infinitesimal Sullivan-Paterson measures and their intersection numbers to obtain another expression for the Weil-Petersson metric on the Quasifuchsian space similar to the situation of the Fuchsian (Teichmuller) space.Riemann surfaces are two-dimensional objects which locally look like open subsets of a plane and that have transition maps which preserve angles. Each Riemann surface supports a unique hyperbolic metric in its class of conformal metrics. The PI studies the variations of hyperbolic metrics on a surface thought of as a single space of metrics called the Teichmuller space. The Teichmuller space is of interest in complex analysis, low-dimensional topology, dynamics, differential geometry and physics. One aspect of the project is related to the Computer Vision given by the approach of Sharon-Mumford as well as to the mathematical physics in the approach of Nag-Sullivan and Takhtajan-Teo. The project is building tools for study of the universal Teichmuller space and it has a potential for applications to the above mentioned fields. Another part of the project involves undergraduate and masters students from Queens College. The students participating in the project will be exposed to an active research agenda thus contributing to the human resource development in the sciences and engineering.