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Maximal cusps, collars, and systoles in hyperbolic surfaces. (English) Zbl 0912.53026
It 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.

53C20 Global Riemannian geometry, including pinching
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