Proofpsurfs

Proofpsurfs, also known as proof surfaces, are mathematical objects used in the study of hyperbolic geometry and complex analysis. They were introduced by Henri Poincaré in the late 19th century as a tool for visualizing and understanding the geometry of hyperbolic spaces. A proofpsurf is essentially a two-dimensional surface that is embedded in a three-dimensional hyperbolic space, and it has the property that any geodesic (shortest path) on the surface is also a geodesic in the ambient hyperbolic space.

The study of proofpsurfs is closely related to the theory of Kleinian groups, which are discrete subgroups of the group of isometries of hyperbolic space. These groups act on the proofpsurf, and the quotient space of the proofpsurf by the action of the group is a hyperbolic manifold. The study of these manifolds is an active area of research in geometry and topology.

Proofpsurfs have also been used in the study of complex analysis, where they are used to construct examples of Riemann surfaces with interesting geometric properties. For example, a proofpsurf can be used to construct a Riemann surface that is homeomorphic to a punctured torus, but has a hyperbolic metric.

In recent years, there has been renewed interest in the study of proofpsurfs, due in part to their connections to other areas of mathematics, such as Teichmüller theory and the theory of hyperbolic 3-manifolds. The study of proofpsurfs continues to be an active area of research, and new results and techniques are being developed all the time.