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 3For 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 5Let 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 .

The proof of Theorem 5 is now very straightforward.

*Proof:* } This follows immediately by choosing in Theorem 6:

Remark 1

- 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 2There 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.

I know that the case m=2 for a closed surface in R^3 is particularly interesting since it is a Möbius invariant and it is regarded as bending energy. People are interested to find the least bended surfaces in R^3.

Would there be similar motivation for the optimal constant in the case m>2?

I don’t exactly know, but I don’t really think the integrals are related to some kind of energy functional, as these functionals are usually (perhaps weighted) norm of the derivatives of some functions or some kind of curvature R (which scales like )… roughly this corresponds to the sum of “kinetic energy” and “potential energy”.

For higher dimension, it seems that lacks this kind of physical interpretation. On the other hand, the best constant C should be closedly related to the isoperimetric property for the ambient space (see e.g. here )

Indeed, in , the best constant is exactly achieved by the spheres. cf. this paper of Chen:

http://link.springer.com/article/10.1007%2FBF01351818