Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform

Atia, Mohamed Jalel; Thabet, Faouzi

doi:10.1007/s10587-016-0260-3

Geodesic

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Quadratic differentials

(A (z - a) (z - b) / (z - c)^{2}) d z^{2}

and algebraic Cauchy transform

Atia, Mohamed Jalel ; Thabet, Faouzi

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 351-363.

Voir la notice de l'article dans Czech Digital Mathematics Library

Résumé

We discuss the representability almost everywhere (a.e.) in

C

of an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. This brings us to the study of trajectories of the particular family of quadratic differentials

A (z - a) (z - b) \* (z - c)^{- 2} d z^{2}

. More precisely, we give a necessary and sufficient condition on the complex numbers

a

and

b

for these quadratic differentials to have finite critical trajectories. We also discuss all possible configurations of critical graphs.

MR Zbl

DOI : 10.1007/s10587-016-0260-3

Classification : 28A99, 30L05
Mots-clés : algebraic equation; Cauchy transform; quadratic differential

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     title = {Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic {Cauchy} transform},
     journal = {Czechoslovak Mathematical Journal},
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Atia, Mohamed Jalel; Thabet, Faouzi. Quadratic differentials $(A(z-a)(z-b)/(z-c)^{2}) {\rm d} z^{2}$ and algebraic Cauchy transform. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 351-363. doi : 10.1007/s10587-016-0260-3. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-016-0260-3/

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