Handle body of 3-manifold with boundary We know from Morse theory that smooth manifold(with or without boundary) is a handlebody.
However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, N. G.(2-AOS2); Burago, Yu. D.(2-AOS2); Zalgaller, V. A.(2-AOS2). They proved

Every 3-dimensional compact Riemannian manifold M (∂M≠∅) with nonnegative (at all points and in all directions) Ricci curvature (Ric≥0) and nonnegative mean curvature of the boundary (H≥0), which is positive at least at one point of ∂M, is diffeomorphic to a handle-body.

Why this need a proof? I am pretty sure that I missed something, but what is it? 
One thing I know is they proved the boundary is connected, which can be viewed as a splitting theorem.
 A: 3-dimensional handlebodies are obtained by gluing 1-handles to a 3-ball. It is true, that every compact, orientable 3-manifold has a handle decomposition, but this needs not just 1-handles but also 2- and (in the closed case) 3-handles. 
In fact, what you get from a handle decomposition (with w.l.o.g. just one 0- and one 3-handle) is the so-called Heegaard splitting, that is a decomposition of a closed, orientable 3-manifold into two handlebodies: the union of a 3-ball with the 1-handles is one of them, the other comes from the 2- and 3-handles.
For 3-manifolds with boundary, the handle decomposition provides you with a Heegaard splitting into one handlebody and one compression body. Here, the compression body is constructed by attaching 1-handles to $\partial M\times I$.
The relevant Wikipedia article is https://en.wikipedia.org/wiki/Heegaard_splitting
A: There are (at least) two different senses of the term "handlebody".
When speaking of manifolds of general dimension ("n-dimensional"), "handlebody" usually means "has a handle decomposition".  So, for example, every smooth manifold has the structure of a handlebody in this sense.
For 3-manifolds, "handlebody" usually means "built out of a 0-handle and some 1-handles".
The Wikipedia article on handlebodies discusses both of these senses.
(Ryan Budney's first comment on the original question says more or less the same thing, but I thought it was worth putting in an answer.)
