I would like to know which explicit metrics on non-compact Calabi-Yau (CY) threefolds are known.

For instance, an important class of such spaces can be constructed algebraically, including local $\mathbb{CP}^1$ (a.k.a. the resolved conifold), local $\mathbb{CP}^2$, local $\mathbb{CP}^1 \times \mathbb{CP}^1$, and the deformed conifold. However, as far as I have searched the math and physics literature, I have found explicit CY metrics only in the case of the resolved and deformed conifold.

Is the CY metric for, e.g., local $\mathbb{CP}^2$ known? What about other cases?

(I have followed terminology from this paper).

explicitRG flow from the UV to the deep IR has/can be described? (At least in some cases such as the conifold?). $\endgroup$