# volume of a submanifold implies bounds on curvature

I would like to ask the following question: Suppose an m-dimensional manifold in an n-dimensional euclidean space, choose some point on this manifold and take an n-dimensional ball of radius R centred at this point. If the volume of the part of the manifold "enclosed" in this ball is V, what can we tell about the bounds for the different types of curvatures (scalar, mean, sectional curvatures) for the manifold inside this ball? Thanks very much in advance!

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By taking a whirling $n$-manifold, possibly connecting flat pieces if $V$ is large, one should be able to achieve any $V$ while curvature are large at least at some points. So you should not expect any uniform bound. I do not know if integral bounds can be expected. You should be more precise in what you want (upper or lower bound? would $L^p$ bound be useful to you?), and explain what you have tried for simple cases (e.g. curves in the plane). – Benoît Kloeckner Aug 20 '13 at 10:22
Yes, i forgot to mention another condition that i had in mind. The restriction here is the existance of certain constant $l$, representing the largest number allowing an open normal bundle around the manifold of radius $r<l$ to be embedded in $\mathbf{R}^n$. Thus, our manifold has certain smoothness restriction. Now if we establish some fixed sized grid in $\mathbf{R}^n$ and then register that in one of the cells of this grid the volume of the manifold is relatively higher compared to other cells, what can i tell about the bounds of different types of curvature in this particular cell? Thanks – bourbaquez Aug 20 '13 at 11:22
you should not change the question along the way, especially in a comment. If you want to ask another question than the one above, you should probably close it, then open a new question. Be careful to think your new question through before posting, so that it is complete and precise. Your comment seems rather fuzzy to me. – Benoît Kloeckner Aug 20 '13 at 11:53
OK, thanks, I'll open a new question. What is fuzzy about my new comment? – bourbaquez Aug 20 '13 at 11:58
The last sentence is quite imprecise, you do not tell which bounds you are talking about exactly, the relation between the grid and of $l$ is unclear... Overall, you seem to have tried to much to make it short, which may be suitable for a comment but not for a question. – Benoît Kloeckner Aug 20 '13 at 16:21

Not much if you do not restrict the relation between the dimensions $m$ and $n$. According to Nash' isometric embedding theorem (see his paper in Annals of Math. (2) 63, theorem 3), the isometric embedding can be obtained in an arbitrarily small ball in $\mathbb{R}^n$.