## The Teichmüller theory of harmonic maps |

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### Contents

The Topology of the Space of Hopf Differentials | 15 |

The WeilPetersson Geometry of the Space | 45 |

Conclusion | 59 |

### Common terms and phrases

affine map Ahlfors asymptotics ataatP base point Beltrami differentials BQD(a compactification compute conformal map conformal structures conformally equivalent converges Corollary 2.5 critical point curvature curves define derivatives Diff0 diffeomorphism dimensional Doctor of Philosophy dx pt dzdz embedding energy density Euler-Lagrange equation flat metric function theory genus g harmonic maps Hess F Hessian holomorphic quadratic differentials homeomorphism homotopy class HOPF DIFFERENTIALS horizontal arcs hyperbolic geometry hyperbolic metric infimum ion of Tg lattice Lemma M.cO mapping class group maps between surfaces moduli space oosh P(R+ parameter parameterization Petersson metric point in Tg projective measured foliations Proof Proposition 2.3 prove pt-length QD(a QD(cO QD(o Royden Sampson 25 Schoen-Yau 26 second variation show that H singular space of holomorphic Suppose surfaces of genus tangent target TeichmOller's Teichmuller space tensor Theorem 2.1 Thurston topology total energy transverse measure Tromba variational formulas vector vertical foliation Weil-Petersson geodesic Weil-Petersson metric Wolpert zero