Differential operators associated with holomorphic mappings

Univalence of holomorphic mappings

“…If f has constant Jacobian in Ω, then an easy computation [4] shows that L (Y, Y ) ≡ 0. Hence condition (1) of the theorem is satisfied for all C > 0 and the theorem says that f is univalent on Ω ∩ B(R) where R is arbitrarily large.…”

“…In an earlier paper [4] we presented several equivalent ways of associating two differential operators with a holomorphic mapping, and we showed how these operators were analogous to the classical Schwarzian derivative. The classical result of Nehari [6] gives a sufficient condition for a holomorphic function defined on the disk to be univalent; the condition is a bound on the Schwarzian derivative.…”

“…These operators were studied in detail in our earlier paper [4], and play a role analogous to the Schwarzian derivative. They will be used to study injectivity criteria.…”

The Schwarzian derivative for maps between manifolds with complex projective connections

“…From the first definition, given in Section 6, it is easy to see that when the projective connections on M and N are flat, then Σ φ generalizes the operator Σ described in [13]. The second definition is considerably more complicated, but it gives a valuable geometric interpretation of Σ φ .…”

“…In our paper [13], we defined and studied two operators, Σ and L, associated to a locally biholomorphic mapping from a domain in C n to CP n . These two operators were shown to play a role analogous to the Schwarzian derivative…”