Teichmüllerrum

Teichmüllerrum is not a standard term in mathematics; it appears to be a misspelling or conflation of Teichmüllerraum (Teichmüller space) or Teichmüller theory.

Teichmüller space T(S) of a surface S is the parameter space of complex structures on S up to isotopy, equivalently the space of marked conformal structures or marked hyperbolic metrics modulo isotopy. For a closed oriented surface of genus g ≥ 2, T(S) has complex dimension 3g−3 (real dimension 6g−6). It can be realized as the quotient of the space of marked conformal structures by the action of the identity component of the diffeomorphism group, or via a quasiconformal model: two marked structures are equivalent if there is a conformal map isotopic to the mark linking them.

The Teichmüller metric, defined by extremal quasiconformal mappings, turns T(S) into a complete Finsler metric space; the corresponding Teichmüller map realizes the distance between points. There are additional geometric structures, notably the Weil–Petersson metric (incomplete but Kähler), and various complex-analytic descriptions such as the Bers embedding that places T(S) into a Banach space of holomorphic quadratic differentials.

The moduli space M_g, parametrizing Riemann surfaces up to biholomorphism, is the quotient of Teichmüller space by the mapping class group. Teichmüller theory studies the interaction of these spaces with deformation theory, hyperbolic geometry, and algebraic geometry.

If you meant Teichmüller space specifically, the rest of this article applies; otherwise Teichmüllerrum is not established as a standard term in mathematical literature.