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Here, I try to give a exemple for each of the three curves with a convex polyhedron where I just draw the net. The geodesic are the dots lines (which are staight on the net). But I don't know any theorem that gives a general answer for your question. My only gess is that if you don't have a trivial estimate that says that on a subset $B$, $\int_B K \geq 4\pi $, then we should be able to construct a exemple.