Abstract
Let \(T(X)\) be the Teichmuller space of a closed surface \(X\) with genus \(g \ge 2\). For a given point \([\sigma ] \in T(X)\), let \(Q(\sigma )\) be the space of holomorphic quadratic differentials on \((X, \sigma )\). Denote by \(\Phi : T(X) \rightarrow Q(\sigma )\) the Hopf mapping which maps a point \([\rho ] \in T(X)\) to the Hopf differential of the unique harmonic map \(w: (X, \sigma ) \rightarrow (X, \rho )\) in the homotopy class of the identity \(id: (X, \sigma ) \rightarrow (X, \rho )\). In this paper, we give quantitative comparisons of both the energy and \(L_1\)-norm of the Hopf differential involved in \(\Phi \) with the length spectrum metric, the Teichmuller metric, and Thurston’s asymmetric metrics, respectively.
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