Two univalent functions are equivalent, f∼g, if they have the same Schwarzian derivative. The equivalence relation ∼ being defined up to an homographic transformation, it gives an isomorphism between the manifold of Jordan curves and the quotient manifold . It permits to obtain vector fields on and on . The action of these vector fields on the Neretin polynomials is explicited. The existence of a unitarizing measure on the quotient manifold is discussed and for such a measure, orthogonality relations for the Neretin polynomials are obtained. This work is a concrete realization on the complex space of the abstract quotient considered in Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626.
Deux fonctions univalentes sont équivalentes, f∼g, si elles ont même dérivée Schwarzienne. La relation d'équivalence ∼ étant définie à une transformation homographique près, on obtient un isomorphisme entre la variété des courbes de Jordan et la variété quotient . Cela permet de déduire des champs de vecteurs sur et sur . On explicite l'action de ces champs de vecteurs sur les polynômes de Neretin. On étudie l'existence de mesures unitarisantes sur le quotient de l'ensemble des fonctions univalentes par cette relation d'équivalence et pour une telle mesure, on établit des relations d'orthogonalité entre les polynômes de Neretin. Ce travail est une réalisation concrète du quotient de Airault et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 621–626 sur l'espace complexe produit d'une infinité dénombrable de .
Accepted:
Published online:
Helene Airault 1, 2; Vladimir Bogachev 3
@article{CRMATH_2003__336_5_429_0, author = {Helene Airault and Vladimir Bogachev}, title = {Realization of {Virasoro} unitarizing measures on the set of {Jordan} curves}, journal = {Comptes Rendus. Math\'ematique}, pages = {429--434}, publisher = {Elsevier}, volume = {336}, number = {5}, year = {2003}, doi = {10.1016/S1631-073X(03)00085-2}, language = {en}, }
TY - JOUR AU - Helene Airault AU - Vladimir Bogachev TI - Realization of Virasoro unitarizing measures on the set of Jordan curves JO - Comptes Rendus. Mathématique PY - 2003 SP - 429 EP - 434 VL - 336 IS - 5 PB - Elsevier DO - 10.1016/S1631-073X(03)00085-2 LA - en ID - CRMATH_2003__336_5_429_0 ER -
Helene Airault; Vladimir Bogachev. Realization of Virasoro unitarizing measures on the set of Jordan curves. Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 429-434. doi : 10.1016/S1631-073X(03)00085-2. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00085-2/
[1] Mesure unitarisante; algèbre de Heisenberg, algèbre de Virasoro, C. R. Acad. Sci. Paris, Ser. I, Volume 334 (2002), pp. 787-792
[2] Unitarizing probability measures for representations of Virasoro algebra, J. Math. Pures Appl., Volume 80 (2001) no. 6, pp. 627-667
[3] Support of Virasoro unitarizing measures, C. R. Acad. Sci. Paris, Ser. I, Volume 335 (2002), pp. 621-626
[4] An algebra of differential operators and generating functions on the set of univalent functions, Bull. Sci. Math. (2002)
[5] Cours à l'Université de Pise. Differentiable measures and the Malliavin calculus (Mai 1995), Scuola Normale Superiore, Pisa, J. Math. Sci., Volume 87 (1997) no. 4, pp. 3577-3731
[6] Geometric approach to discrete series of unireps for Virasoro, J. Math. Pures Appl., Volume 77 (1998), pp. 735-746
[7] Univalent Functions and Teichmüller Spaces, Graduate Texts in Math., 109, Springer-Verlag, 1987
[8] Holomorphic extensions of representations of the group of diffeomorphisms of the circle, Math. USSR-Sb., Volume 67 (1990) no. 1, pp. 75-96
Cited by Sources:
Comments - Policy