We will show that a certain Willmore-type functional for closed submanifolds in has a lower bound which depends on the dimension of the submanifold only.
We first recall some results which give lower bounds for certain energy functionals.
Theorem 1 (Fenchel ) [doCarmo, p. 399] Suppose is a simple closed curve in and is its curvature, then
The equality holds if and only if is a plane convex curve.
This is closely related to the following
Theorem 2 (Fary-Milnor ) [doCarmo p. 402] If is a knotted simple closed curve in with curvature , then
Intuitively, is knotted means that we cannot deform continuously in to the standard planar circle without crossing itself.
On the other hand, we have
Proposition 3 For a closed surface in , if is is (un-normalized) mean curvature, then
with the equality holds if and only if it is a round sphere.
Proof: Let be the principal curvatures. Then
Let . Then
The last inequality is true because the Gauss map is surjective onto the sphere which has area and its Jacobian is exactly . If the equality holds, then is totally umbilical and therefore is a sphere.
Furthermore, recently Marques and Neves proved the following Willmore conjecture:
Theorem 4 (Marques, Neves [Marques-Neves]) For any immersed torus in ,
The equality holds if and only if is the Clifford torus up to a conformal transformation in .
It is interesting to ask if a similar inequality holds for higher dimensional submanifold (and with higher co-dimension) in . It turns out that this is true:
Theorem 5 Let be an integer. Then there exists a constant depending on only such that for any closed oriented smooth submanifold immersed in , we have
where is the mean curvature vector of .
To prove theorem 5, we need the following
Theorem 6 (Michael-Simon ) } Let be an integer. Then there exists a constant depending on only such that for any -dimensional oriented submanifold immersed in , if and is the mean curvature vector of , then
Here is the -norm on .
- It is easy to see that the equality in Theorem 5 holds only if is of constant length. When is an embedded hypersurface, this happens only when is a distance sphere. In this case where is the area of the standard -dimensional sphere.
However, as it is not easy to determine the optimal constant for the Michael-Simon Sobolev inequality, we do not know whether in Theorem 6 gives an optimal constant for Theorem 5. (For a choice of which is however non-optimal, see [Topping]. )
- It is also interesting to ask if the constant in Theorem 5 can be improved under various topological conditions on .
Remark 2 There is a generalization of Michael-Simon-Sobolev inequality by Hoffman and Spruck [Hoffman-Spruck] which allows more general ambient space. Theorem 5 can be generalized using their result.