Wave Computation on the Hyperbolic Double Doughnut
This work provides a numerical method for wave computation on hyperbolic surfaces, which is of interest to mathematicians and physicists studying spectral geometry and quantum chaos.
The authors compute wave propagation on a compact surface of constant negative curvature and genus 2 using finite elements, implementing Fuchsian group boundary conditions. They compute the first eigenvalues of the Laplace-Beltrami operator and test exponential decay and ergodicity of geodesic flow.
We compute the waves propagating on the compact surface of constant negative curvature and genus 2. We adopt a variational approach using finite elements. We have to implement the action of the fuchsian group by suitable boundary conditions of periodic type. A spectral analysis of the wave allows to compute the first eigenvalues of the Laplace-Beltrami operator. We test the exponential decay due to a localized dumping and the ergodicity of the geodesic flow.