Cross-ratios and Schwarzian Derivatives in Rn

1988 ◽  
pp. 1-15 ◽  
Author(s):  
Lars V. Ahlfors
Keyword(s):  
Schwarzian Derivatives

scholarly journals PROJECTIVE AND CONFORMAL SCHWARZIAN DERIVATIVES AND COHOMOLOGY OF LIE ALGEBRAS VECTOR FIELDS RELATED TO DIFFERENTIAL OPERATORS

2006 ◽  
Vol 03 (04) ◽  
pp. 667-696 ◽  
Author(s):  
SOFIANE BOUARROUDJ
Keyword(s):  
Lie Algebras ◽  
Cohomology Group ◽  
Vector Fields ◽  
Differential Operators ◽  
Projective Manifold ◽  
Conformal Class ◽  
Tensor Fields ◽  
Relative Cohomology ◽  
Schwarzian Derivatives ◽  
Cohomology Of Lie Algebras

Let M be either a projective manifold (M, Π) or a pseudo-Riemannian manifold (M, g). We extend, intrinsically, the projective/conformal Schwarzian derivatives we have introduced recently, to the space of differential operators acting on symmetric contravariant tensor fields of any degree on M. As operators, we show that the projective/conformal Schwarzian derivatives depend only on the projective connection Π and the conformal class of the metric [g], respectively. Furthermore, we compute the first cohomology group of Vect(M) with coefficients in the space of symmetric contravariant tensor fields valued in the space of δ-densities, and we compute the corresponding sl(n + 1, ℝ)-relative cohomology group.

scholarly journals Schwarzian derivatives of convex mappings

2011 ◽  
Vol 36 ◽  
pp. 449-460 ◽  
Author(s):  
Martin Chuaqui ◽  
Peter Duren ◽  
Brad Osgood
Keyword(s):  
Convex Mappings ◽  
Schwarzian Derivatives ◽  
Derivatives Of

scholarly journals Carleson Measure of Harmonic Schwarzian Derivatives Associated with a Finitely Generated Fuchsian Group of the Second Kind

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Guangming Hu ◽  
Yutong Liu ◽  
Yu Sun ◽  
Xinjie Qian
Keyword(s):  
Harmonic Function ◽  
Unit Disk ◽  
Fuchsian Group ◽  
Fundamental Domain ◽  
Carleson Measure ◽  
Schwarzian Derivative ◽  
Finitely Generated ◽  
Schwarzian Derivatives ◽  
Carleson Condition

Let S H f be the Schwarzian derivative of a univalent harmonic function f in the unit disk D , compatible with a finitely generated Fuchsian group G of the second kind. We show that if S H f 2 1 − z 2 3 d x d y satisfies the Carleson condition on the infinite boundary of the Dirichlet fundamental domain F of G , then S H f 2 1 − z 2 3 d x d y is a Carleson measure in D .

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