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Tangent spaces to the Teichmüller space from the energy-conscious perspective
4 Going from the cohomological description to the analytic description
4.1 -invariant partition of unity on D
Contents
Introduction
1 Preliminaries
1.1 Some facts from hyperbolic geometry
1.2 The Teichmüller space, a kaleidoscopic view
1.2.1 Classical definition
1.2.2 T(g) as a representation variety
1.3 Tangent spaces to the Teichmüller space
1.3.1 Cohomological description
1.3.2 Analytic description: Holomorphic quadratic differentials
2 Explicit expressions of harmonic vector fields on H2
2.1 Harmonic maps
2.2 The notion of a harmonic vector field
2.2.1 Constructing harmonic vector fields on U H2
2.2.2 Extending harmonic vector fields on H2 to the boundary circle S1
3 Going from the analytic description to the cohomological description
3.1 Vector fields on D and S1
4 Going from the cohomological description to the analytic description
4.1 -invariant partition of unity on D
4.2 The Poisson map adapted to vector fields
4.2.1 Reincarnation of the Poisson integral formula
4.3 A detailed map from H1(; g) to HQD(D, )
4.4 Open Problems
A The genesis of the potential equation F=(z-)2(z)
A.1 A swift introduction to Beltrami differentials
A.2 Filling in the gap
Dissertation
Tangent spaces to the Teichmüller space from the energy-conscious perspective
Entstehung
2021
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