Let $M$ be an n-dim compact Riemannian manifold with $Ric \geqslant -(n-1)$, it's well known that the following Poincare inequality holds for any function $f\in W^{1,2}(M)$ $$ \frac{1}{m(B)}\int_B |f-\frac{1}{m(B)}\int_B f|\leqslant C r (\frac{1}{m(B)}\int_B |\nabla f|^2)^{\frac12} $$ For a function $f\in W^{1,2}(M)$ with $vol(M)>vol(Spt(f))$, where $Spt(f)$ is the support of $f$. Let $U$ be an open neighbourhood of $Spt(f)$, how can we get the following Poincare inequality for $U$? $$ \int_U f^2 \leqslant C \int_U |\nabla f|^2 $$ If we use partion of unity, let $\varphi_i$ be the functions supported on balls $B_i$, such that $U\subset \cup B_i$. Then $$ |\nabla (\varphi_i f)|^2 \leqslant 2(|\nabla \varphi|^2 f^2+\varphi^2|\nabla f|^2) $$

We still have $f^2$ on the right hand side. So partion of unity is not feasible.