Let $X$ be a complete non compact manifold.

Q: Does the Poincare inequality also hold, i.e. $$\|u\|_{L^2}\leq C \|\nabla u\|_{L^2},$$ for any $u\in C^1_c(X)$.

If it holds, how to determine the Constant $C$?(what will it be related to ?)

Let $(X,g)$ be a non compact Riemannian manifold, such that its closure $\bar X=X\cup Y$ is a compact manifold with boundary $Y$.

Q: For the Poincare inequality $$\|u\|_{L^2}\leq C \|\nabla u\|_{L^2},$$ for any $u\in C^1_c(X)$, how to determine the Constant $C$  ?(what will it be related to ? e.g. dimension, diameter or volume?)

This is a revised version.

Thanks for the Arun Debray's comments.