TITLE: Smoothing of the distance function on a Riemannian manifold
QUESTION [11 upvotes]: Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let $\epsilon>0$ be an arbitrary positive number. Is there a smooth function $\tilde{d}_p(x)$ on $M$, such that 
$$ | d_p(x)-\tilde{d}_p(x) | < \epsilon$$
$$ |\textrm{grad}(\tilde{d}_p)(x)|<2$$
for $\forall x \in M$ ?
Functions satisfying the first condition can be constructed easily by partition of unity and the standard technique of mollifiers. However, I can't see how to control the gradient of the approximate function.
I need this result when I'm trying to follow a proof. The existence of such a function is taken for granted in that proof, and is called the "regularization" of the distance function. This question also seems to be interesting by itself. 
I'm not sure if this question is too easy for math overflow. I put it on stack exchange last week but nobody answered it. Could you please help me? Thanks a lot.

REPLY [5 votes]: This is an additional comment about higher derivative bounds. Consider the
class of pointed complete noncompact Riemannian manifolds $(M^{n},g,O)$ with
$\left\vert \operatorname{sect}\left(  g\right)  \right\vert \leq K$. For the
distance-like function constructed from mollification by Greene and Wu, one
also has a uniform upper bound for the Hessian of $u$. To obtain a two-sided
bound for the Hessian, one can smooth $u$ by the heat equation. In a paper
titled "Construction of an exhaustion function on complete manifolds",
Luen-Fai Tam proves that there exists a $C^{\infty}$ function $f:M\rightarrow
\mathbb{R}$ with $d\left(  \cdot,O\right)  +1\leq f\leq d\left(
\cdot,O\right)  +C$, $\left\vert \nabla f\right\vert \leq C$, and $\left\vert
\nabla\nabla f\right\vert \leq C$, where $C$ depends only on $n$ and $K$.
Techniques that Tam uses include heat kernel estimates and weighted $L^{2}$
estimates. An exposition of Tam's result is given in Section 4 of Chapter 26
in the book "The Ricci Flow: Techniques and Applications: Part III:
Geometric-Analytic Aspects"; see therein for references to previous related work.