The cut-off function on Riemannian manifold

Question

Let $M$ be a complete Riemannian manifold. Can we find a constant $C>0$ so that for any $p\in M$ and $R>0, 2R<\text{inj}(M)$, we can find a function $\varphi \in C^\infty _c(B_p(2R))$, such that $\varphi=1 $ on $B_p(R)$ and $|\nabla \varphi| \leq C/R$? ($B_p(R)$ is the geodesic ball with center $p$)

Answer 1

Let $g : \mathbb R \to \mathbb R$ be a function such that $g(x)=1$ when $x\leq r$, $g(x)=0$ when $x \geq 2r$ and $g'(x) \leq C/r$ for all $x$. such a function can be found by (e.g.) taking a convolution of a small smooth bump function with a piecewise linear function. We need $C>1$ or that $g$ does not exist.

Then consider $\phi : B_p(2r) \to \mathbb R$, $\phi (x) = g(d(x))$, where $d(x) = d(x, p)$. Note that $\phi$ is smooth in $B_p(r)$ as it is constant. Away from $p$, $d$ is smooth as $2r < Inj(M,p)$ and

$$\nabla \phi = g' \nabla d = g' \frac{\partial }{\partial r}\ .$$

Thus $|\nabla \phi| \leq C |\frac{\partial }{\partial r}| = C$ by Gauss lemma. As Jason DeVito mentioned, you need to have $\text{inj}(M)>0$ in order to choose an $r$ independent of $p$.