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Evolution of Smooth Shapes and Integrable Systems

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Abstract

We consider a homotopic evolution in the space of smooth shapes starting from the unit circle. Based on the Löwner–Kufarev equation, we give a Hamiltonian formulation of this evolution and provide conservation laws. The symmetries of the evolution are given by the Virasoro algebra. The ‘positive’ Virasoro generators span the holomorphic part of the complexified vector bundle over the space of conformal embeddings of the unit disk into the complex plane and smooth on the boundary. In the covariant formulation, they are conserved along the Hamiltonian flow. The ‘negative’ Virasoro generators can be recovered by an iterative method making use of the canonical Poisson structure. We study an embedding of the Löwner–Kufarev trajectories into the Segal–Wilson Grassmannian, construct the \(\tau \)-function, and the Baker–Akhiezer function which are related to a class of solutions to the KP hierarchy.

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Acknowledgments

The authors are thankful for the support of NILS mobility project and Professor Fernando Pérez-González (Universidad de La Laguna, España), the University of Chicago and Professor Paul Wiegmann for support and helpful discussions. The authors acknowledge also helpful discussions with Professor Roland Friedrich during his visit to the University of Bergen and valuable remarks made by an anonymous referee.

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  1. Department of Mathematics, University of Bergen, P.O. Box 7800, 5020, Bergen, Norway

    Irina Markina & Alexander Vasil’ev

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  1. Irina Markina
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  2. Alexander Vasil’ev
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Correspondence to Alexander Vasil’ev.

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Communicated by Filippo Bracci.

The authors have been supported by the grants of the Norwegian Research Council #239033/F20, #213440/BG; and EU FP7 IRSES program STREVCOMS, Grant No. PIRSES-GA-2013-612669. This work was supported by the Erwin Schrödinger Institute in Vienna.

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Markina, I., Vasil’ev, A. Evolution of Smooth Shapes and Integrable Systems. Comput. Methods Funct. Theory 16, 203–229 (2016). https://doi.org/10.1007/s40315-015-0133-z

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