The String Equation for Some Rational Functions

  • Björn GustafssonEmail author
Part of the Trends in Mathematics book series (TM)


For conformal maps defined in the unit disk one can define a certain Poisson bracket that involves the harmonic moments of the image domain. When this bracket is applied to the conformal map itself together with its conformally reflected map the result is identically one. This is called the string equation, and it is closely connected to the governing equation, the Polubarinova-Galin equation, for the evolution of a Hele-Shaw blob of a viscous fluid (or, by another name, Laplacian growth). For non-univalent analytic functions the Poisson bracket may become ambiguous, hence the string equation need not make sense. In the present paper we show that for a certain class of (non-univalent) rational functions related to quadrature Riemann surfaces, the string equation does make sense, and holds.


Polubarinova-Galin equation String equation Poisson bracket Harmonic moments Branch points Hele-Shaw flow Laplacian growth Quadrature Riemann surface 

Mathematics Subject Classification (2010)

Primary 30C55 Secondary 31A25 34M35 37K05 76D27 



The author is grateful to an anonymous referee for constructive comments on the paper.


  1. 1.
    P.J. Davis, The Schwarz Function and Its Applications. The Carus Mathematical Monographs, No. 17 (The Mathematical Association of America, Buffalo, 1974). MR 0407252 (53 #11031)Google Scholar
  2. 2.
    J. Escher, G. Simonett, Classical solutions of multidimensional Hele-Shaw models. SIAM J. Math. Anal. 28(5), 1028–1047 (1997). MR 1466667 (98i:35213)MathSciNetCrossRefGoogle Scholar
  3. 3.
    B. Gustafsson, The string equation for nonunivalent functions. arXiv:1803.02030 (2018)Google Scholar
  4. 4.
    B. Gustafsson, The string equation for polynomials. Anal. Math. Phys. 8(4), 637–653 (2018). MathSciNetCrossRefGoogle Scholar
  5. 5.
    B. Gustafsson, Y.-L. Lin, Non-univalent solutions of the Polubarinova-Galin equation. arXiv:1411.1909 (2014)Google Scholar
  6. 6.
    B. Gustafsson, V.G. Tkachev, The resultant on compact Riemann surfaces. Commun. Math. Phys. 286(1), 313–358 (2009). MR 2470933 (2009i:32015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    B. Gustafsson, V.G. Tkachev, On the exponential transform of multi-sheeted algebraic domains. Comput. Methods Funct. Theory 11(2), 591–615 (2011). MR 2858963MathSciNetCrossRefGoogle Scholar
  8. 8.
    B. Gustafsson, A. Vasilev, Conformal and Potential Analysis in Hele-Shaw Cells. Advances in Mathematical Fluid Mechanics (Birkhäuser Verlag, Basel, 2006). MR 2245542 (2008b:76055)Google Scholar
  9. 9.
    B. Gustafsson, R. Teoderscu, A. Vasilev, Classical and Stochastic Laplacian Growth. Advances in Mathematical Fluid Mechanics (Birkhäuser Verlag, Basel, 2014). MR 2245542 (2008b:76055)Google Scholar
  10. 10.
    H. Hedenmalm, A factorization theorem for square area-integrable analytic functions. J. Reine Angew. Math. 422, 45–68 (1991). MR 1133317 (93c:30053)Google Scholar
  11. 11.
    H. Hedenmalm, N. Makarov, Coulomb gas ensembles and Laplacian growth. Proc. Lond. Math. Soc. (3) 106(4), 859–907 (2013). MR 3056295MathSciNetCrossRefGoogle Scholar
  12. 12.
    H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman Spaces. Graduate Texts in Mathematics, vol. 199 (Springer, New York, 2000). MR 1758653 (2001c:46043)CrossRefGoogle Scholar
  13. 13.
    S. Kharchev, A. Marshakov, A. Mironov, A. Morozov, A. Zabrodin, Towards unified theory of 2 d gravity. Nuclear Phys. B 380(1–2), 181–240 (1992). MR 1186584MathSciNetCrossRefGoogle Scholar
  14. 14.
    I.K. Kostov, I. Krichever, M. Mineev-Weinstein, P.B. Wiegmann, A. Zabrodin, The τ-Function for Analytic Curves. Random Matrix Models and Their Applications, Math. Sci. Res. Inst. Publ., vol. 40 (Cambridge University Press, Cambridge, 2001), pp. 285–299. MR 1842792 (2002h:37145)Google Scholar
  15. 15.
    I. Krichever, A. Marshakov, A. Zabrodin, Integrable structure of the Dirichlet boundary problem in multiply-connected domains. Commun. Math. Phys. 259(1), 1–44 (2005). MR 2169966 (2006g:37109)MathSciNetCrossRefGoogle Scholar
  16. 16.
    O.S. Kuznetsova, V.G. Tkachev, Ullemar’s formula for the Jacobian of the complex moment mapping. Complex Var. Theory Appl. 49(1), 55–72 (2004). MR 2031026 (2004k:30085)MathSciNetCrossRefGoogle Scholar
  17. 17.
    H. Lamb, Hydrodynamics. Cambridge Mathematical Library, 6th edn. (Cambridge University Press, Cambridge, 1993). With a foreword by R.A. Caflisch [Russel E. Caflisch]. MR 1317348Google Scholar
  18. 18.
    Y.-L. Lin, Perturbation theorems for Hele-Shaw flows and their applications. Ark. Mat. 49(2), 357–382 (2011). MR 2826949MathSciNetCrossRefGoogle Scholar
  19. 19.
    A.V. Marshakov, Matrix models, complex geometry, and integrable systems. I. Teoret. Mat. Fiz. 147(2), 163–228 (2006). MR 2254744 (2007j:81081)Google Scholar
  20. 20.
    A.V. Marshakov, Matrix models, complex geometry, and integrable systems. II. Teoret. Mat. Fiz. 147(3), 399–449 (2006). MR 2254723 (2007j:81082)Google Scholar
  21. 21.
    A. Marshakov, P. Wiegmann, A. Zabrodin, Integrable structure of the Dirichlet boundary problem in two dimensions. Commun. Math. Phys. 227(1), 131–153 (2002). MR 1903842 (2004a:37092)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. Mineev-Weinstein, A. Zabrodin, Whitham-Toda hierarchy in the Laplacian growth problem. J. Nonlinear Math. Phys. 8(suppl), 212–218 (2001). Nonlinear evolution equations and dynamical systems (Kolimbary, 1999). MR 1821533Google Scholar
  23. 23.
    M. Reissig, L. von Wolfersdorf, A simplified proof for a moving boundary problem for Hele-Shaw flows in the plane. Ark. Mat. 31(1), 101–116 (1993). MR MR1230268 (94m:35250)MathSciNetCrossRefGoogle Scholar
  24. 24.
    S. Richardson, Hele-Shaw flows with a free boundary produced by the injection of fluid into a narrow channel. J. Fluid Mech. 56, 609–618 (1972)CrossRefGoogle Scholar
  25. 25.
    J. Ross, D. Witt Nyström, The Hele-Shaw flow and moduli of holomorphic discs. Compos. Math. 151(12), 2301–2328 (2015). MR 3433888MathSciNetCrossRefGoogle Scholar
  26. 26.
    M. Sakai, A moment problem on Jordan domains. Proc. Am. Math. Soc. 70(1), 35–38 (1978). MR 0470216 (57 #9974)MathSciNetCrossRefGoogle Scholar
  27. 27.
    M. Sakai, Domains having null complex moments. Complex Variables Theory Appl. 7(4), 313–319 (1987). MR 889117 (88e:31002)MathSciNetCrossRefGoogle Scholar
  28. 28.
    M. Sakai, Finiteness of the family of simply connected quadrature domains, in Potential Theory (Prague, 1987) (Plenum, New York, 1988), pp. 295–305. MR 986307 (90a:30114)CrossRefGoogle Scholar
  29. 29.
    H.S. Shapiro, The Schwarz Function and Its Generalization to Higher Dimensions. University of Arkansas, Lecture Notes in the Mathematical Sciences, 9 (Wiley, New York, 1992). A Wiley-Interscience Publication. MR 1160990 (93g:30059)Google Scholar
  30. 30.
    R. Teodorescu, E. Bettelheim, O. Agam, A. Zabrodin, P. Wiegmann, Normal random matrix ensemble as a growth problem. Nuclear Phys. B 704(3), 407–444 (2005). MR 2116267 (2006f:82039)MathSciNetCrossRefGoogle Scholar
  31. 31.
    F.R. Tian, A Cauchy integral approach to Hele-Shaw problems with a free boundary: the case of zero surface tension. Arch. Rational Mech. Anal. 135(2), 175–196 (1996). MR 1418464 (97j:35167)MathSciNetCrossRefGoogle Scholar
  32. 32.
    V.G. Tkachev, Ullemar’s formula for the moment map. II. Linear Algebra Appl. 404, 380–388 (2005). MR 2149671 (2006b:30064)Google Scholar
  33. 33.
    C. Ullemar, Uniqueness theorem for domains satisfying a quadrature identity for analytic functions. Research Bulletin TRITA-MAT-1980-37, Royal Institute of Technology, Department of Mathematics, Stockholm, 1980Google Scholar
  34. 34.
    B.L. van der Waerden, Moderne Algebra (Springer, Berlin, 1940). MR 0002841CrossRefGoogle Scholar
  35. 35.
    A.N. Varchenko, P.I. Etingof, Why the Boundary of a Round Drop Becomes a Curve of Order Four. University Lecture Series, 3rd edn. (American Mathematical Society, Providence, 1992). MR MR924157 (88k:00002)Google Scholar
  36. 36.
    A. Vasilev, From the Hele-Shaw experiment to integrable systems: a historical overview. Complex Anal. Oper. Theory 3(2), 551–585 (2009). MR 2504768 (2010f:76034)Google Scholar
  37. 37.
    P.B. Wiegmann, A. Zabrodin, Conformal maps and integrable hierarchies. Commun. Math. Phys. 213(3), 523–538 (2000). MR 1785428 (2002g:37105)MathSciNetCrossRefGoogle Scholar
  38. 38.
    A. Zabrodin, Matrix models and growth processes: from viscous flows to the quantum Hall effect, in Applications of Random Matrices in Physics. NATO Sci. Ser. II Math. Phys. Chem., vol. 221 (Springer, Dordrecht, 2006), pp. 261–318. MR 2232116 (2007d:82081)Google Scholar
  39. 39.
    A. Zabrodin, Random matrices and Laplacian growth. The Oxford Handbook of Random Matrix Theory (Oxford University Press, Oxford, 2011), pp. 802–823. MR 2932659Google Scholar
  40. 40.
    L. Zalcman, Some inverse problems of potential theory, in Integral Geometry (Brunswick, Maine, 1984). Contemp. Math., vol. 63 (American Mathematical Society, Providence, 1987), pp. 337–350. MR 876329 (88e:31012)Google Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsKTHStockholmSweden

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