# Visualizing hyperbolic metric of punctured sphere

Uniformization of the 3-punctured sphere generates a "pants" configuration with three legs narrowing down to cusps. This is supposed to have a metric of constant negative curvature, and I can see this in the cusps, and also in the saddle regions where they join, but I am at a loss to understand how the curvature can remain negative in the middle region. Simplistic depictions (e.g. on the Wikipedia pair-of-pants page) clearly do NOT show negative curvature throughout. As another example, Hubbard Fig. 3.5.1 (ref below) shows a surface which does have negative curvature throughout, and has two cusps coming in; however, it has a finite-size hole going out, and that hole is clearly going to have to grow, rather than shrink, to maintain negative curvature.

In short, I can't understand how a surface can start at zero radius, then traverse a section where size is growing, and then contract back to zero size, without encountering a section where curvature is positive. I can calculate that it is happens using the uniformizing metric, but I would really appreciate some guidance in how to visualize or understand it property. Even if I think of embeddings in 3 space with self-intersections, I can't seem to construct a visualization that works.

ref: Hubbard, John H., "Teichmuller Theory", volume 1, Matrix Editions, 2006.