Abstract
We consider a homotopic evolution in the space of smooth shapes startingf rom 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 tangent 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 makingu se of the canonical Poisson structure.We study an embeddingo f the Löwner-Kufarev trajectories into the Segal-Wilson Grassmannian. This gives a way to construct the \( \tau \) -function, the Baker-Akhiezer function, and finally, to give a class of solutions to the KP equation.
Mathematics Subject Classification (2010). Primary 81R10, 17B68, 30C35; Secondary 70H06.
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Markina, I., Vasil’ev, A. (2013). Löwner-Kufarev Evolution in the Segal-Wilson Grassmannian. In: Kielanowski, P., Ali, S., Odzijewicz, A., Schlichenmaier, M., Voronov, T. (eds) Geometric Methods in Physics. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0448-6_33
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