On infinitesimal Strebel points (Q1689862)

Let \(X\) be a Riemann surface of infinite analytic type, \(T(X)\) and \(B(X)\) be the Teichmüller space and the infinitesimal Teichmüller space over \(X\), respectively. The paper under review proves that, for any element \([\mu]_T\in T(X)\), there exists \(\mu_1\in [\mu]_T\) such that \([\mu_1]_B\in B(X)\) is an infinitesimal Strebel point. Since \([\mu]_T\) and \([\mu]_B\) shear many similar properties in the theory of extremal quasiconformal mappings, there was a question to ask whether \([\mu]_T\in T(X)\) is a Strebel point if and only if \([\mu]_B\in B(X)\) is a Strebel point. Under the condition that \(\|\mu\|_\infty\) is small, \textit{Y. Hu} and \textit{Y. Shen} [Sci. China, Ser. A 52, No. 9, 2019--2026 (2009; Zbl 1179.30045)] proved that the fact that \([\mu]_T\in T(X)\) is a Strebel point does not imply that \([\mu]_B\in B(X)\) is a Strebel point, and the fact that \([\mu]_B\in B(X)\) is a Strebel point does not imply that \([\mu]_T\in T(X)\) is a Strebel point. When \(X\) is the unit disk \(\mathbb D\), in one direction, \textit{H. Huang} and \textit{B. Long} [Comput. Methods Funct. Theory 17, No. 1, 121--127 (2017; Zbl 1362.30030)] obtained a general result that \([\mu]_B\in B(\mathbb D)\) is a Strebel point does not imply \([\mu]_T\in T(\mathbb D)\) is a Strebel point. In the paper under review, the author generalizes the result of Huang and Long for any Riemann surface of infinite analytic type.