Poincare inequality on balls to arbitrary open subset of manifolds

Question

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.

Answer 1

You can't.

Consider a function $f$ that is similar to the function $x^{-s}$ in $\mathbb{R}^n$, near a point and bounded from below on the rest of the manifold. If we pick $s>0$ small enough, both the function and its gradient will be integrable near the singularity, so it belongs to $W^{1,2}(M)$.

Now, subtracting a positive constant $f_C:=f - C$ we can make the $L^2$-norm of $f_C$ as big as we wish without changing the $L^2$-norm of the gradient. The support of this function is all of $M$, so is still not a counterexample. But notice that for $C$ big, you can truncate at $0$ without changing the norms to much, i.e. $$F_C=\min(f_C,0)$$ and $F_C$ is zero in a set of positive measure.

If what you want is to control the $L^2$ norm of the gradient from below but you don't have an estimate for the average of the function, sometimes you can use other arguments. For example, lets say you have an estimate for how big the sets $\{f\leq 0\}$ and $\{f\geq 1\}$ are. Then, you can use the isoperimetric profile of $M$, and the coarea formula, to control the $L^2$-norm of the gradient in the set $\{0\leq f \leq 1\}$. This is a kind of Sobolev isoperimetric inequality, an idea that goes back to De Giorgi.