• Subhashis Nag

      Articles written in Proceedings – Mathematical Sciences

    • Schiffer variation of complex structure and coordinates for Teichmüller spaces

      Subhashis Nag

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      Schiffer variation of complex structure on a Riemann surfaceX0 is achieved by punching out a parametric disc$$\bar D$$ fromX0 and replacing it by another Jordan domain whose boundary curve is a holomorphic image of$$\partial \bar D$$. This change of structure depends on a complex parameter ε which determines the holomorphic mapping function around$$\partial \bar D$$.

      It is very natural to look for conditions under which these ε-parameters provide local coordinates for Teichmüller spaceT(X0), (or reduced Teichmüller spaceT#(X0)). For compactX0 this problem was first solved by Patt [8] using a complicated analysis of periods and Ahlfors' [2] τ-coordinates.

      Using Gardiner's [6], [7] technique, (independently discovered by the present author), of interpreting Schiffer variation as a quasi conformal deformation of structure, we greatly simplify and generalize Patt's result. Theorems 1 and 2 below take care of all the finitedimensional Teichmüller spaces. In Theorem 3 we are able to analyse the situation for infinite dimensionalT(X0) also. Variational formulae for the dependence of classical moduli parameters on the ε's follow painlessly.

    • Canonical measures on the moduli spaces of compact Riemann surfaces

      Subhashis Nag

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      We study some explicit relations between the canonical line bundle and the Hodge bundle over moduli spaces for low genus. This leads to a natural measure on the moduli space of every genus which is related to the Siegel symplectic metric on Siegel upper half-space as well as to the Hodge metric on the Hodge bundle.

    • Self-dual connections, hyperbolic metrics and harmonic mappings on Riemann surfaces

      Subhashis Nag

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      The Sampson-Wolf model of Teichmüller space (using harmonic mappings) is shown to be exactly the same as the more recent Hitchin model (utilizing self-dual connections). Indeed, it is noted how the self-duality equations become the harmonicity equations. An interpretation of the modular group action in this model is mentioned.

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