Compact hyperbolic surface

Definition

A compact hyperbolic surface is a compact connected smooth surface $M$ with a Riemannian metric of constant sectional curvature $-1$ . Equivalently,

M\cong\Gamma\backslash\mathbb H^2,

where $\Gamma$ is a torsion-free cocompact discrete group of orientation- preserving isometries of the hyperbolic plane.

Disk model

The hyperbolic plane may be represented by the Poincaré disk $\mathbb D=\{z\in\mathbb C:|z|<1\}$ . Its ideal boundary is $S^1$ , and each complete geodesic has two endpoints there.

Dynamical role

The geodesic flow on the unit tangent bundle of $M$ is Anosov. Its stable and unstable directions create porous sets of boundary endpoints associated with geodesics avoiding an open observation region, which is how fractal uncertainty enters quantum chaos.

The horocycle flow supplies the recurrence used to prove porosity in the surface application.

References

David Borthwick, Spectral Theory of Infinite-Area Hyperbolic Surfaces, 2nd ed., Birkhäuser, 2016. DOI record.