The isopermetric problem is to identify the least perimeter enclosure of a given area. The authors study this problem on hyperbolic surfaces; it is assumed these are connected and geometrically finite. Such surfaces may be compact or have finitely many ends (cusps with exponentially shrinking thickness and finite area or flared ends which are asymptotic to the hyperbolic plane). The authors show that the least perimeter enclosures form four types: 1) circles 2) horocycles around cusps 3) boundaries of annular neighborhoods of geodesics 4) collections of neighboring curves of geodesics. If the surface has at least one puncture or cusp, types 1) and 3) do not occur and the least perimeter \(L\) satisfies \(L\leq A\); if \(A<\pi\), then a minimizer is of type 2).
The authors study various examples (once punctured torus, thrice punctured sphere, four punctured sphere, genus 2 surface without punctures, etc.). They also study higher dimensional phenomena.