A problem of Goldberg and a Teichmüller differential (Q1906604)

Let \(R\) be a Riemann surface, and denote by \(Q(R)\) the family of holomorphic quadratic differentials \(\phi\) on \(R\) with \(|\phi|= \int_R |\phi|< + \infty\), and by \(S(R)\) the unit sphere of \(Q(R)\). For \(\psi\in Q(R)\), if the limit \(\lim_{t\to 0} {|\psi+ t\phi|- |\psi |\over t}\) exists uniformly for all \(\phi\in S(R)\), we call the \(L^1\)-norm on \(Q(R)\) is Fréchet differentiable at \(\psi\). In this paper, the author proves that the \(L^1\)-norm on \(Q(\Delta)\) is not Fréchet differentiable at \(\psi= dz^2/\pi\), where \(\Delta\) is the unit disk. This result is an answer to an open problem proposed by \textit{L. R. Goldberg} [Proc. Am. Math. Soc. 118, No. 4, 1179-1185 (1993; Zbl 0787.30029)]. The author also constructs a Teichmüller differential \(\mu= k\overline\psi/|\psi|\) with \(|\psi|< + \infty\) on an arbitrary Riemann surface \(R\) of conformally infinite type, wlhich has a degenerate Hamilton sequence. This example has applications in the theory of Teichmüller spaces.