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do someone have an idea, how to prove that :

For every polyhedral metric $(S,d)$ of curvature $\leq -1$, (S=surface), there exists a sequence of smooth metrics, converging uniformly to $(S,d)$.

And why the metric in the neigbourhood of any singularity will be of the shape $dr^2+f^2(r)d\phi^2$ with $f$ a suitable function.

Thnak you for any answer

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