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Let $S$ be a genus $g$, $g > 1$ Riemann surface, and let $h \colon S \to S$ be a homeomorphism of $S$. We denote by $[h] \in \text{Map}(S)$ the corresponding element of the mapping class group of $S$. We can without loss of generality assume that $h$ is quasiconformal. By Teichmuller's uniqueness theorem, among all quasiconformal mappings which are isotopic to $h$, we may choose $h$ to be a unique quasiconformal mapping which has the smallest dilatation.

Now let us add some marked points.

Let $S$ be a surface with $k$ marked points $x_1,\ldots,x_k$, and let $\text{Map}(S,k)$ be the group of isotopy classes of mappings of $S$ that fix the set $x_1,\ldots,x_k$ (the pure mapping class group).

Let $[h] \in \text{Map}(S,k)$. My questions are:

Will it be possible to find a mapping $h$ such that it has the smallest dilatation among all quasiconformal mappings which correspond to the element $[h] \in \text{Map}(S,k)$? In other words, do we have extremal quasiconformal mappings among mappings fixing $k$ points?

Given some extremal mapping, is it unique? defined up to a biholomorphism preserving marked points?

In particular, I am interested in the case where $S$ is the Riemann sphere and $k > 3$.

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Yes. See, for example, W. Abikoff, Real analytic theory of Teichmuller space, Springer, 1980, Chap. II, section 1.5 Theorem 2.

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