Maximal cusps, collars, and systoles in hyperbolic surfaces.

*(English)*Zbl 0912.53026It is proved that a maximal cusp in any orientable hyperbolic surface has area at least 4, with the lower bound of 4 realized only for a cusp in the thrice-punctured sphere. For a maximal cusp in any other hyperbolic surfaces with \(p\)-punctures, it is shown that it has area at most \(6|\chi(S)|- (p-1)\), and that there is a metric that realizes this upper bound. Moreover, over all possible hyperbolic metrics, the area \(A\) of the maximal cusp takes on all values such that \(4< A\leq 6| \chi(S)|- (p-1)\).

If \(S\) is a punctured orientable hyperbolic surface other than the thrice-punctured sphere, and if \(S\) is endowed with any complete hyperbolic metric, then it is proved that there exists a choice of a maximal set of cusps in the hyperbolic surface with total area greater than \(5p/2\) if \(p\) is even and \(5p/2+3/2\) if \(p\) is odd. These universal lower bounds are the best possible.

Applications of these results to new lower bound on collar areas and upper bounds on systole lengths are included.

If \(S\) is a punctured orientable hyperbolic surface other than the thrice-punctured sphere, and if \(S\) is endowed with any complete hyperbolic metric, then it is proved that there exists a choice of a maximal set of cusps in the hyperbolic surface with total area greater than \(5p/2\) if \(p\) is even and \(5p/2+3/2\) if \(p\) is odd. These universal lower bounds are the best possible.

Applications of these results to new lower bound on collar areas and upper bounds on systole lengths are included.

Reviewer: Colin Adams (Williamstown/MA)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |