Abstract
We study one-parameter expanding evolution families of simply connected domains in the complex plane described by infinite systems of evolution parameters. These evolution parameters in some cases admit Hamiltonian formulation and lead to integrable systems. One example of such parameters is complex moments for the Laplacian growth that form a Whitham-Toda integrable hierarchy. Another example we deal with is related to expanding coefficient bodies for conformal maps given by Löwner subordination chains. The coefficients' bodies are proved to form a Liouville partially integrable Hamiltonian system for each fixed index and the first integrals are obtained. We also discuss the contact structure of this system.
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Agam, O., Bettelheim, E., Wiegmann, P., Zabrodin, A.: Viscous fingering and a shape of an electronic droplet in the Quantum Hall regime. Phys. Rev. Lett. 88, 236801 (2002)
Aleksandrov, I.A.: Parametric continuations in the theory of univalent functions. Moscow: Nauka, 1976 (in Russian)
Arnold, V.I.: Mathematical methods of classical mechanics. New York: Springer-Verlag, 1989
Babelon, O., Bernard, D., Talon, M.: Introduction to classical integrable systems. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press, 2003
Babenko, K.I.: The theory of extremal problems for univalent functions of class S. Proc. Steklov Inst. Math., No. 101 (1972). Transl. American Mathematical Society, Providence, R.I.: Amer. math. soc., 1975
Bambusi, D., Gaeta, G.: On persistence of invariant tori and a theorem by Nekhoroshev. Math. Phys. Electron. J. 8, Paper 1, 13 pp (2002)
Bolsinov, A.V., Fomenko, A.T.: Integrable Hamiltonian systems. Geometry, topology, classification. Boca Raton, FL: Chapman & Hall/CRC, 2004
Bieberbach, L.: Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln. S.-B. Preuss. Akad. Wiss. S.940–955 (1916)
de Branges, L.: A proof of the Bieberbach conjecture. Acta Math. 154 , no. 1–2, 137–152 (1985)
Fiorani, E., Giachetta, G., Sardanashvily, G.: The Liouville-Arnold-Nekhoroshev theorem for non-compact invariant manifolds. J. Phys. A 36, no. 7, L101–L107 (2003)
Gaeta, G.: The Poincaré-Lyapounov-Nekhoroshev theorem. Ann. Physics 297, no. 1, 157–173 (2002)
Goluzin, G.M.: Geometric theory of functions of a complex variable. Transl. Math. Monographs, Vol 26, Providence, RI: AMS, 1969
Hohlov, Yu.E., Howison, S.D., Huntingford, C., Ockendon, J.R., Lacey, A.A.: A model for non-smooth free boundaries in Hele-Shaw flows. Quart. J. Mech. Appl. Math. 47, 107–128 (1994)
Howison, S.D.: Complex variable methods in Hele-Shaw moving boundary problems. European J. Appl. Math. 3, no. 3, 209–224 (1992)
Kostov, I.K., Krichever, I., Mineev-Weinstein, M., Wiegmann, P.B., Zabrodin, A.: The τ-function for analytic curves. In: Random matrix models and their applications, Math. Sci. Res. Inst. Publ. 40, Cambridge: Cambridge Univ. Press, 2001, pp. 285–299
Kufarev, P.P.: On one-parameter families of analytic functions. Rec. Math. [Mat. Sbornik] N.S. 13(55), 87–118 (1943)
Löwner, K.: Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. Math. Ann. 89, 103–121 (1923)
Marshakov, A., Wiegmann, P., Zabrodin, A.: Integrable structure of the Dirichlet boundary problem in two dimensions. Commun. Math. Phys. 227, no. 1, 131–153 (2002)
Nekhoroshev, N.N.: The Poincaré-Lyapunov-Liouville-Arnol'd theorem. Funkt. Anal. i Pril. 28, no. 2, 67–69 (1994); translation in Funct. Anal. Appl. 28, no. 2, 128–129 (1994)
Pommerenke, Ch.: Über die Subordination analytischer Funktionen. J. Reine Angew. Math. 218, 159–173 (1965)
Pommerenke, Ch.: Univalent functions, with a chapter on quadratic differentials by G. Jensen. Göttingen: Vandenhoeck & Ruprecht, 1975
Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The mathematical theory of optimal processes. New York-London: Interscience Publishers/John Wiley & Sons, Inc., 1962
Prokhorov, D.: Sets of values of systems of functionals in classes of univalent functions. Mat. Sb. 181, no. 12, 1659–1677 (1990); translation in Math. USSR-Sb. 71, no. 2, 499–516 (1992)
Richardson, S.: Hele-Shaw flows with a free boundary produced by the injecton of fluid into a narrow channel. J. Fluid Mech., 56, no. 4, 609–618 (1972)
Schaeffer, A.C., Spencer, D.C.: Coefficient regions for schlicht functions (with a chapter on the region of the derivative of a schlicht function by Arthur Grad). American Mathematical Society Colloquium Publications, Vol. 35. New York: American Mathematical Society, 1950
Vasil'ev, A.: Mutual change of initial coefficients of univalent functions. Matemat. Zametki 38, no. 1, 56–85 (1985); translation in Math. Notes 38, no. 1–2, 543–548 (1985)
Wiegmann, P.B., Zabrodin, A.: Conformal maps and integrable hierarchies. Commun. Math. Phys. 213, no. 3, 523–538 (2000)
Zakharov, V.E.(ed.): What is integrability? Springer Series in Nonlinear Dynamics. Berlin: Springer-Verlag, 1991
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Communicated by L. Takhtajan
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Prokhorov, D., Vasil'ev, A. Univalent Functions and Integrable Systems. Commun. Math. Phys. 262, 393–410 (2006). https://doi.org/10.1007/s00220-005-1499-y
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DOI: https://doi.org/10.1007/s00220-005-1499-y