TITLE: Does a small-area sphere in a 3-manifold bound a small ball?
QUESTION [16 upvotes]: Let $M$ be a closed riemannian 3-manifold. I think that the following fact should be true and should have a relatively simple proof, but I cannot figure it out.

For every $\varepsilon>0$ there is a $\delta>0$ such that every smooth 2-sphere in $M$ of area smaller than $\delta$ bounds a ball of volume smaller than $\varepsilon$.

Roughly, small-area spheres must bound small-volume balls. 
Note that: 


If $M\neq S^3$ then $M$ contains spheres that bound regions of arbitrarily small volume that are not balls (just take a spine of $M$ and small regular neighborhoods of it).  
 It suffices to prove that the 2-sphere is contained in a small-volume ball and invoke Alexander theorem.
 The same fact stated for 3-spheres in $S^4$ would imply the (open) Schoenflies problem (every 3-sphere bounds a 4-ball), since every 3-sphere in $S^4$ can be shrinked to have arbitrarily small area.
 It is not true in general that a torus of small area is contained in a ball (pick neighborhoods of a homotopically non-trivial knot).

REPLY [6 votes]: This is a direct corollary of Federer -Flemming deformation Lemma saying that vary small area sphere can be homotoped to 1-skeleton of the fixed triangulation of the manifold. The dimension assumption is irrelevant. Proof of this Lemma can be found somewhere in Federer's book (I will chase down the precise reference when I can).