A pair of pants is a surface homeorphic to the $3$-punctured sphere. It is a well known fact that a pair of pants admits a metric of constant curvature $-1$. Usually one is interested in pairs of pants with geodesic boundaries, and wants to know the length of the boundaries. I believe that this data defines a pair of pants of constant curvature $-1$ up to isometry.
I'm interested instead in pair of pants with just a negative upper bound to the curvature, and with prescribed lengths of the legs. More precisely, consider a tripod, that is is a metric tree with four vertices, one of which has degree $3$. Any tripod can be approximated as a metric space with a pair of pants. Formally, given a tripod $T$, for every $\epsilon$ there exist a pair of pants $P$ such that $$d_{GH}(T,P)<\epsilon$$ where $d_{GH}$ is the Gromov-Hausdorf distance. As a rough intuition one can think that there exist an isometric immersion $\phi:T\to P$ such that $N_{2\epsilon}(\phi(T))=P$, i.e. any point $p\in P$ is at distance less than $2\epsilon$ from $\phi(T)$.
Then my question is the following: given $T$ and $\epsilon>0$, does it exist a pair of pants $P$ such that $d_{GH}(T,P)<\epsilon$ and the curvature of $P$ is everywhere negative? Furthermore, can we give an estimate of the lower bound $K_{min}$ of the curvature of $P$?
My intuition tells me that the answers should be, yes and yes, with $K_{min}\to\infty$ as $\epsilon\to0$, but I wouldn't know how to prove it.