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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.

##### MSC:
 53C20 Global Riemannian geometry, including pinching
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