Non-differentiability of the Teichmüller cometric in infinite-dimensional Teichmüller spaces (Q1913916)

Let \(Y\) be a Riemann surface and \(T(Y)\) be the Teichmüller space of \(Y\). For \([X, f]\in T(Y)\), let \(Q_X\) be the set of holomorphic quadratic differentials \(\Phi= \Phi(z)dz^2\) on the Riemann surface \(X\) such that \(G(X, \Phi)= |\Phi|= \int_X |\Phi||dz^2|< + \infty\). It is well-known that the above norm is the infinitesimal form of the Teichmüller cometric, which is a function defined on the cotangent bundle \(Q_X\) of the Teichmüller space \(T(Y)\). In this paper, the author proves that for an infinite-dimensional Teichmüller space \(T(Y)\), the Teichmüller cometric is not of class \(C^1\). Actually, for any \([X, f]\in T(Y)\), the author constructs a point \((X, \Psi)\in Q_X\), such that \(G(X, \Phi)\) is not Fréchet differentiable at \((X, \Psi)\).