Teichmüller space

Definition.

Let $S_{0}$ be a Riemann surface. Consider all pairs $(S,f)$ where $S$ is a Riemann surface and $f$ is a sense-preserving quasiconformalmapping of $S_{0}$ onto $S$ . We say $(S_{1},f_{1})\\sim(S_{2},f_{2})$ if $f_{2}\\circ f_{1}^{-1}$ is homotopic to a conformal mapping of $S_{1}$ onto $S_{2}$ . In this case we say that $(S_{1},f_{1})$ and $(S_{2},f_{2})$ are Teichmüller equivalent. The space of equivalence classes under this relation is called the Teichmüller space $T(S_{0})$ and $(S_{0},I)$ is called the initialpoint of $T(S_{0})$ . The equivalence relation is called Teichmüller equivalence.

Definition.

There exists a natural Teichmüller metric on $T(S_{0})$ , where the distancebetween $(S_{1},f_{1})$ and $(S_{2},f_{2})$ is $\\log K$ where $K$ is the smallest maximal dilatation of a mapping homotopic to $f_{2}\\circ f_{1}^{-1}$ .

There is also a natural isometry between $T(S_{0})$ and $T(S_{1})$ defined bya quasiconformal mapping of $S_{0}$ onto $S_{1}$ . The mapping $(S,f)\\mapsto(S,f\\circ g)$ induces an isometric mapping of $T(S_{1})$ onto $T(S_{0})$ . So we could think of $T(\\cdot)$ as a contravariant functor fromthe category of Riemann surfaces with quasiconformal maps to the category ofTeichmüller spaces (as a subcategory of metric spaces).

References

1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966

单词	TeichmullerSpace
释义	Teichmüller space Definition. Let $S_{0}$ be a Riemann surface. Consider all pairs $(S,f)$ where $S$ is a Riemann surface and $f$ is a sense-preserving quasiconformalmapping of $S_{0}$ onto $S$ . We say $(S_{1},f_{1})\\sim(S_{2},f_{2})$ if $f_{2}\\circ f_{1}^{-1}$ is homotopic to a conformal mapping of $S_{1}$ onto $S_{2}$ . In this case we say that $(S_{1},f_{1})$ and $(S_{2},f_{2})$ are Teichmüller equivalent. The space of equivalence classes under this relation is called the Teichmüller space $T(S_{0})$ and $(S_{0},I)$ is called the initialpoint of $T(S_{0})$ . The equivalence relation is called Teichmüller equivalence. Definition. There exists a natural Teichmüller metric on $T(S_{0})$ , where the distancebetween $(S_{1},f_{1})$ and $(S_{2},f_{2})$ is $\\log K$ where $K$ is the smallest maximal dilatation of a mapping homotopic to $f_{2}\\circ f_{1}^{-1}$ . There is also a natural isometry between $T(S_{0})$ and $T(S_{1})$ defined bya quasiconformal mapping of $S_{0}$ onto $S_{1}$ . The mapping $(S,f)\\mapsto(S,f\\circ g)$ induces an isometric mapping of $T(S_{1})$ onto $T(S_{0})$ . So we could think of $T(\\cdot)$ as a contravariant functor fromthe category of Riemann surfaces with quasiconformal maps to the category ofTeichmüller spaces (as a subcategory of metric spaces). References 1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966
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