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edited Mar 30, 2012 at 12:20

Strebel discovered (in 1962) that Teichmüller's existence theorem fails for quasiconformal maps of the unit disk. Uniqueness fails too for points in $T(D^2)$ which do not have Teichmuller representatives. See Strebel's examples in the book by Gardiner and Lakic, "Quasiconformal Teichmüller Theory," p. 177-178.
On the other hand, the uniqueness theorem in certain sense does hold (Theorem 5, Chapter 4 of the same book). More More precisely, if a quasiconformal map $f: D^2\to D^2$ is a Teichmüller map, i.e., it has Beltrami differential of the Teichmüller form $t \bar{\phi}/|\phi|$ (where $\phi$ is holomorphic), then $f$ is the unique extremal quasiconformal map in its equivalence class $[f]\in T(D^2)$. In particular, $[f]$ contains no other Teichmüller maps.
On third hand, Strebel proved in 1976 ("On the existence of extremal Teichmueller mappings") that for $[f]\in T(D^2)$ given by, say, smooth boundary values, the Teichmüller existence theorem does hold.

Strebel discovered (in 1962) that Teichmüller's existence theorem fails for quasiconformal maps of the unit disk. See Strebel's examples in the book by Gardiner and Lakic, "Quasiconformal Teichmüller Theory," p. 177-178.
On the other hand, the uniqueness theorem does hold (Theorem 5, Chapter 4 of the same book). More precisely, if a quasiconformal map $f: D^2\to D^2$ is a Teichmüller map, i.e., it has Beltrami differential of the Teichmüller form $t \bar{\phi}/|\phi|$ (where $\phi$ is holomorphic), then $f$ is the unique extremal quasiconformal map in its equivalence class $[f]\in T(D^2)$. In particular, $[f]$ contains no other Teichmüller maps.
On third hand, Strebel proved in 1976 ("On the existence of extremal Teichmueller mappings") that for $[f]\in T(D^2)$ given by, say, smooth boundary values, the Teichmüller existence theorem does hold.

Strebel discovered (in 1962) that Teichmüller's existence theorem fails for quasiconformal maps of the unit disk. Uniqueness fails too for points in $T(D^2)$ which do not have Teichmuller representatives. See Strebel's examples in the book by Gardiner and Lakic, "Quasiconformal Teichmüller Theory," p. 177-178.
On the other hand, the uniqueness theorem in certain sense does hold (Theorem 5, Chapter 4 of the same book). More precisely, if a quasiconformal map $f: D^2\to D^2$ is a Teichmüller map, i.e., it has Beltrami differential of the Teichmüller form $t \bar{\phi}/|\phi|$ (where $\phi$ is holomorphic), then $f$ is the unique extremal quasiconformal map in its equivalence class $[f]\in T(D^2)$. In particular, $[f]$ contains no other Teichmüller maps.
On third hand, Strebel proved in 1976 ("On the existence of extremal Teichmueller mappings") that for $[f]\in T(D^2)$ given by, say, smooth boundary values, the Teichmüller existence theorem does hold.

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created Mar 30, 2012 at 5:33

Strebel discovered (in 1962) that Teichmüller's existence theorem fails for quasiconformal maps of the unit disk. See Strebel's examples in the book by Gardiner and Lakic, "Quasiconformal Teichmüller Theory," p. 177-178.
On the other hand, the uniqueness theorem does hold (Theorem 5, Chapter 4 of the same book). More precisely, if a quasiconformal map $f: D^2\to D^2$ is a Teichmüller map, i.e., it has Beltrami differential of the Teichmüller form $t \bar{\phi}/|\phi|$ (where $\phi$ is holomorphic), then $f$ is the unique extremal quasiconformal map in its equivalence class $[f]\in T(D^2)$. In particular, $[f]$ contains no other Teichmüller maps.
On third hand, Strebel proved in 1976 ("On the existence of extremal Teichmueller mappings") that for $[f]\in T(D^2)$ given by, say, smooth boundary values, the Teichmüller existence theorem does hold.