The Michael-Simon-Sobolev inequality states that
Theorem 1 Let () be a smooth immersed submanifold in and be a non-negative smooth function with compact support, then
Here is the norm of on , is the mean curvature vector of and depends on only.
In our convention, the mean curvature vector of is where is the position.
- Of course, by completion the above inequality also holds on (completion of w.r.t. norm).
- When is (or subset of it), then it becomes the basic ordinary Sobolev inequality
- The term can’t be dropped in general when has non-zero curvature. E.g. when is the sphere of radius in , choose , then LHS is of order , whereas . As the mean curvature of the sphere is of order , the term is also of order , so this inequality is quite “reasonable”.
The interesting thing about the inequality is that depends on only (not even depends on )! (Of course, the constant in the ordinary Sobolev inequality (2) also doesn’t depend on , even if we restrict our domain to , the reason is that such a function can always be extended (trivially) as a function compactly supported in . )
I haven’t really read the proof of the most general form (1) of this inequality, but I come across a cute proof of it when (i.e. is a surface), which I am going to give here. As a bonus, can be explicitly given in this case. (Actually Michael-Simon’s paper also gives an explicit constant. )
Proof: (Proof of Theorem 1 for )
For an -valued vector field which is defined on , we define
where is a local orthonormal frame on and is the connection on .
Suppose is a smooth submanifold in with boundary with being the unit outward normal of w.r.t. , then as (here for the proof),
Remark 2 In particular, if has no boundary, this becomes
This becomes zero if is a tangent vector field (as ), thus this formula generalizes the ordinary divergence theorem.
Now, assume contains for the time being and let .
Let to be the geodesic ball around of radius , and . Then applying (3) on ,
As , we have and , where is the area of the standard -dimensional sphere, so taking in the above, we have
Translating the origin to on , we have
By translating the origin from to , we have
So (6) becomes
In particular the constant can be taken to be .
Remark 3 The constant above is not optimal. Suppose not, by translating , we can assume contains , but then the proof shows that for all . So for all . However, the choice of the origin is arbitrary in the proof, i.e. for any points , the vector is tangential to both and . We then conclude that is a domain in a -plane . Also, for fixed , by tracing the proof, we see that is a multiple of for all . But this is impossible unless is zero, for if , for any which does not lie on , clearly and are not parallel.