Report on the teichmüller metric

Area of Science:

• Complex analysis
• Differential geometry
• Topology

Background:

• Teichmüller space T(g) represents conformal structures on Riemann surfaces.
• The Teichmüller modular group acts on T(g) via biholomorphic maps.

Purpose of the Study:

• To prove that every biholomorphic map of T(g) onto itself is induced by the Teichmüller modular group.
• To establish a converse to the known action of the modular group on Teichmüller space.

Main Methods:

• Demonstrating that isometries of T(g) with the Teichmüller metric originate from the Teichmüller modular group.
• Establishing the Teichmüller metric as the Kobayashi metric for T(g).

Main Results:

• All isometries of T(g) under the Teichmüller metric are induced by elements of the Teichmüller modular group.
• The Teichmüller metric is equivalent to the Kobayashi metric on T(g).

Conclusions:

• Every biholomorphic automorphism of T(g) is induced by an element of the Teichmüller modular group.
• This result provides a complete understanding of the automorphism group of Teichmüller space.