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Updated: Feb 19, 2026

Preparation of Free-Surface Hyperbolic Water Vortices
Published on: July 28, 2023
The motion of a vortex on a closed surface of constant negative curvature
1Instituto de Matemática e Estatística, Universidade de São Paulo, Sao Paulo, Brazil.
Abstract:
The purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one. The algorithm is based on a relation between the Laplace-Beltrami Green function and the heat kernel. The algorithm is used to compute the motion of a vortex on the Bolza surface. This is the first determination of the orbits of a vortex on a closed surface of genus greater than one. The numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the Bolza surface. Some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc. The following theorem is proved: 'a Weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium'.
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