Abstract
We consider subordination chains of simply connected domains with smooth boundaries in the complex plane. Such chains admit Hamiltonian and Lagrangian interpretations through the Löwner–Kufarev evolution equations. The action functional is constructed and its time variation is obtained. It represents the infinitesimal version of the action of the Virasoro–Bott group over the space of analytic univalent functions.
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Airault, H. and Malliavin, P., Unitarizing probability measures for representations of Virasoro algebra, J. Math. Pures Appl. 80 (2001), 627–667.
Airault, H., Malliavin, P. and Thalmaier, A., Support of Virasoro unitarizing measures, C. R. Acad. Sci. Paris 335 (2002), 621–626.
Bieberbach, L., Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln, Sitzungsber. Preuss. Akad. Wiss. (1916), 940–955.
Blumenfeld, R., Formulating a first-principles statistical theory of growing surfaces in two-dimensional Laplacian fields, Phys. Rev. E 50 (1994), 2952–2962.
de Branges, L., A proof of the Bieberbach conjecture, Acta Math. 154 (1985), 137–152.
Goddard, P. and Olive, D., Kac–Moody and Virasoro algebras in relation to quantum physics, Internat. J. Modern Phys. A 1 (1986), 303–414.
Gromova, L. and Vasil’ev, A., On the estimate of the fourth-order homogeneous coefficient functional for univalent functions, Ann. Polon. Math. 63 (1996), 7–12.
Gustafsson, B. and Vasil’ev, A., Conformal and Potential Analysis in Hele–Shaw Cells, Birkhäuser, Basel, 2006.
Howison, S. D., Complex variable methods in Hele–Shaw moving boundary problems, European J. Appl. Math. 3 (1992), 209–224.
Kirillov, A. A., Kähler structure on the K-orbits of a group of diffeomorphisms of the circle, Funktsional Anal. i Prilozhen 21:2 (1987), 42–45 (Russian). English transl.: Functional Anal. Appl. 21 (1987), 122–125.
Kirillov, A. A., Geometric approach to discrete series of unirreps for Vir, J. Math. Pures Appl. 77 (1998), 735–746.
Kirillov, A. A. and Yuriev, D. V., Kähler geometry of the infinite-dimensional homogeneous space M=Diff+(S1)/Rot(S1), Funktsional Anal. i Prilozhen 21:4 (1987), 35–46 (Russian). English transl.: Functional Anal. Appl. 21 (1987), 284–294.
Kirillov, A. A. and Yuriev, D. V., Representations of the Virasoro algebra by the orbit method, J. Geom. Phys. 5 (1988), 351–363.
Kufarev, P. P., On one-parameter families of analytic functions, Rec. Math. [Mat. Sbornik] 13(55) (1943), 87–118.
Lehto, P., On fourth-order homogeneous functionals in the class of bounded univalent functions, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 48 (1984).
Löwner, K., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, Math. Ann. 89 (1923), 103–121.
Neretin, Yu. A., Representations of Virasoro and affine Lie algebras, in Representation Theory and Noncommutative Harmonic Analysis, Encyclopedia Math. Sci. 22, pp. 157–225, Springer, Berlin, 1994.
Polyakov, A. M., Quantum geometry of bosonic strings, Phys. Lett. B 103 (1981), 207–210.
Pommerenke, C., Über die Subordination analytischer Funktionen, J. Reine Angew. Math. 218 (1965), 159–173.
Pommerenke, C., Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.
Prokhorov, D. and Vasil’ev, A., Univalent functions and integrable systems, Comm. Math. Phys. 262 (2006), 393–410.
Richardson, S., Hele–Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech. 56 (1972), 609–618.
Schiffer, M. and Hawley, N. S., Connections and conformal mapping, Acta Math. 107 (1962), 175–274.
Takhtajan, L. A., Liouville theory: quantum geometry of Riemann surfaces, Modern Phys. Lett. A 8 (1993), 3529–3535.
Tammi, O., Extremum Problems for Bounded Univalent Functions, Lecture Notes in Math. 646, Springer, Berlin, 1978.
Tammi, O., Extremum Problems for Bounded Univalent Functions II, Lecture Notes in Math. 913, Springer, Berlin, 1982.
Vasil’ev, A., Univalent functions in two-dimensional free boundary problems, Acta Appl. Math. 79 (2003), 249–280.
Vasil’ev, A., Evolution of conformal maps with quasiconformal extensions, Bull. Sci. Math. 129 (2005), 831–859.
Zograf, P. G. and Takhtajan, L. A., On the Liouville equation, accessory parameters and the geometry of Teichmüller space for Riemann surfaces of genus 0, Mat. Sb. 132(174):2 (1987), 147–166 (Russian). English transl.: Math. USSR-Sb. 60:1 (1988), 143–161.
Zograf, P. and Takhtajan, L., Hyperbolic 2-spheres with conical singularities, accessory parameters and Kähler metrics on M 0,n , Trans. Amer. Math. Soc. 355 (2003), 1857–1867.
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Vasil’ev, A. Energy characteristics of subordination chains. Ark Mat 45, 141–156 (2007). https://doi.org/10.1007/s11512-006-0031-8
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DOI: https://doi.org/10.1007/s11512-006-0031-8