scalar curvature bounds and volume bounds

-1

Hello. Could someone please indicate me how to resolve the following problem: I have an open submanifold in some Euclidean Space. Given a compact subset of this submanifold, there are bounds for the scalar curvature on this compact region of the submanifold. What can there be said about the bounds of the diameter of this compact subset ( in terms of the induced metric) and the dimension of the submanifold in terms of the bounds on the scalar curvature on this set? What is relationship between these concepts in this context?

On the other hand, if I take another compact subset on this submanifold containing the first subset, how can the information of the scalar curvature bounds and diameter of the first subset determine the correspondent concepts for this second, bigger subset?

Any suggestions, clues, references are welcome. Thanks very much

flag	asked Nov 15 at 17:09 Temniy 1

Why have you posted this question twice in the course of an hour? mathoverflow.net/questions/112498/… – Gerry Myerson Nov 15 at 22:08

Sorry, this is my first post, still don't know how to manage it – Temniy Nov 16 at 9:06

Nothing can be said (think about 1-dimensional curves in $\mathbb{R}^n$). – YangMills Nov 16 at 14:28

Ehm... what do you mean? – Temniy Nov 16 at 14:40

The scalar curvature of the induced Riemannian metric on a curve in $\mathbb{R}^n$ is always zero, yet the curve can have diameter (=volume in this case) arbitrarily large. – YangMills Nov 16 at 17:10