I have a soft question regarding the main theorem in the Cheeger--Gromov paper Chopping Riemannian Manifolds.
Notation: Let $M^n$ denote a complete Riemannian manifold and let $X \subset M^n$ be a subset of $M$ (not necessarily smooth). We denote by $T_r(X)$ the set of points at distance $\leq r$ from $X$. Moreover, for any smooth submanifold $Y \subset M^n$, we denote by $\Pi_Y$ the second fundamental form of $Y$.
The main theorem is the following:
Theorem 0.1. Let $M^n$ have bounded sectional curvature $| K | \leq 1$. Given $X \subset M^n$, and $0 < r \leq 1$, there exists a smooth submanifold $U^n \subset M^n$ with smooth boundary $\partial U^n$ such that $$X \subset U \subset T_r(X),$$ $$\text{Vol}(\partial U) \ \leq \ c(n) \text{Vol} \left(T_r(X) - T_{r/2}(X) \right) r^{-1},$$ and $$\left \| \Pi_{\partial U} \right \| \leq c(n) r^{-1}.$$
Essentially, the theorem asserts that for any set $X$ in $M^n$, regardless of how `bad' $X$ is, we can approximate $X$ by a smooth submanifold with a smooth boundary. Moreover, we can do this approximation with a top-dimensional smooth submanifold. It is also interesting that we can control the second fundamental form of the boundary of $U$.
Question: What is the geometric significance of having a second fundamental form bounded in such a way? Or for that matter, what is the significance of having a bounded second fundamental form?
I admit the question is rather vague, but are there some particularly nice results that hold (curvature control for example) if the second fundamental form is bounded?