The Hopf differential and harmonic maps between branched hyperbolic structures
Doctor of Philosophy
Given a surface of genus g with fundamental group π, a representation of π into PSL(2,R) is a homomorphism that assigns to each generator of π an element of P SL(2, R). The group P SL(2, R) acts on Hom(π, P SL(2, R)) by conjugation. Define therepresentationspaceRg tobethequotientspaceHom(π,PSL(2,R))\PSL(2,R). Associated to each representation ρ is a number e(ρ) called its Euler class. Goldman showed that the space Rg has components that can be indexed by Euler classes of rep- resentations, and that there is one component for each integer e satisfying |e| ≤ 2g−2. The two maximal components correspond to Teichmu ̈ller space, the space of isotopy classes of hyperbolic structures on a surface. Teichmu ̈ller space is known to be homeomorphic to a ball of dimension 6g − 6. The other components of Rg are not as well understood. The theory of harmonic maps between non-positively curved manifolds has been used to study Teichmu ̈ller space. Given a harmonic map between hyperbolic surfaces, there is an associated quadratic differential on the domain surface called the Hopf differential. Wolf, following Sampson, proved that via the Hopf differential, harmonic maps parametrize Teichmu ̈ller space. This thesis extends his work to the case of branched hyperbolic structures, which correspond to certain elements in non- maximal components of representation space. More precisely, a branched hyperbolic structure is a pair (M, σ|dz|2) where M is a compact surface of genus g and σ|dz|2 is a hyperbolic metric with integral order cone singularities at a finite number of points expressed in terms of a conformal parameter. Fix a base surface (M, σ|dz|2). For each target surface (M, ρ|dw|2) with the same number and orders of cone points as (M,σ|dz|2), there is a unique harmonic map w : (M,σ|dz|2) → (M,ρ|dw|2) homotopic to the identity that fixes the cone points of M pointwise. Thus we may define another map from the space of branched hyperbolic structures with the same number and orders of cone points to the space of meromorphic quadratic differentials on the base surface M. This map, Φ, takes the harmonic map w associated with a metric ρ|dw|2 to the Hopf differential of w. This thesis shows that the map Φ is injective.
Teichmuller theory; Harmonic maps