By the Teichmuller uniqueness theorem, given a homeomorphism $f:X \rightarrow X$ where $X$ is the $n$-punctured sphere, there is a unique quasiconformal homeomorphism $g$ fixing $0$, $1$, and $\infty$ (in the same homotopy class as $f$) whose Beltrami coefficient $\mu$ has the smallest $L^{\infty}$ norm $x$. 


My question is : If $g\_n$ are q.c homeomorphisms fixing $0$, $1$, and $\infty$ in the same homotopy class as $f$ such that the $L^{\infty}$ norms $x\_n$ of their Beltrami coefficients $\mu\_n$ decrease to a limit $x$ (but we don't know whether the $\mu\_n$ converge to anything), then do the $g\_n$ converge to $g$ in some space (ex: Sobolev space)?



A continuous version of Teichmuller uniqueness
Here are some details. First, in your setting, all the maps $g\_n$ are K-quasiconfirmal for a certain K. Thus, by the convergence property for qc maps, the sequence $(g\_n)$ has a convergent subsequence and the limit is qc. Clearly, the limit is homotopic to the maps $g\_n$. It suffices to show then that if $(g\_n)$ itself converges to some $h$ then this limit is the extremal map. By the same convergence property, since norms of Beltrami differentials are assumed to converge to the norm of the external map $g$, we get
$$
K(h)=K(g).
$$
Hence, by uniqueness of the extremal map, $h=g$. qed


