A problem on the T0 Teichmüller space

Let T(Δ) be the universal Teichmüller space and let T0(Δ) be the subspace of T(Δ) consisting of elements [μ]∈T(Δ) with boundary dilatation one. Our main result is that, if μ is an extremal Beltrami differential on Δ with ‖μ‖∞=k≠0 and [μ]∈T0(Δ), then [tμ]∈T0(Δ) for any t∈[0,1/k). This result answers a problem proposed by Earle, Gardiner and Lakic. Moreover, it is proved that the subspace T00(Δ) of T0(Δ) consisting of [μ]∈T0(Δ) with μ|U=0 where U is a neighborhood of ∂Δ is dense in T0(Δ). With this theorem, we provide a short proof of our main result.