## A Willmore type inequality

We will show that a certain Willmore-type functional for closed submanifolds in ${\mathbb R^n}$ 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 ${\gamma}$ is a simple closed curve in ${\mathbb R^{3}}$ and ${k}$ is its curvature, then $\displaystyle \int_\gamma |k|\ge 2 \pi.$ The equality holds if and only if ${\gamma}$ is a plane convex curve.

This is closely related to the following

 Theorem 2 (Fary-Milnor ) [doCarmo p. 402] If ${\gamma}$ is a knotted simple closed curve in ${\mathbb R^3}$ with curvature ${k}$, then $\displaystyle \int_\gamma |k| \ge 4 \pi.$

Intuitively, ${\gamma}$ is knotted means that we cannot deform ${\gamma}$ continuously in ${\mathbb R^3}$ to the standard planar circle without crossing itself.

On the other hand, we have

 Proposition 3 For a closed surface ${\Sigma}$ in ${\mathbb R^3}$, if ${H}$ is is (un-normalized) mean curvature, then $\displaystyle \int_\Sigma H^2 \ge 16\pi,$ with the equality holds if and only if it is a round sphere.

Proof: Let ${\lbrace \lambda_i\rbrace_{i=1}^2}$ be the principal curvatures. Then

$\displaystyle H^2 - 4K=(\lambda_1+\lambda_2)^2-4\lambda_1\lambda_2=(\lambda_1-\lambda_2)^2\ge 0.$

Let ${M_+=\lbrace K> 0\rbrace}$. Then

$\displaystyle \int_MH^2\ge \int_{M_+}H^2 \ge 4\int_{M_+}K\ge 16\pi.$

The last inequality is true because the Gauss map ${\nu:M\rightarrow \mathbb S^2}$ is surjective onto the sphere which has area ${4\pi}$ and its Jacobian is exactly ${K}$. If the equality holds, then $M$ is totally umbilical and therefore is a sphere. $\Box$

Furthermore, recently Marques and Neves proved the following Willmore conjecture:

 Theorem 4 (Marques, Neves [Marques-Neves]) For any immersed torus ${\Sigma}$ in ${\mathbb R^3}$, $\displaystyle \int_\Sigma H^2 \ge 8 \pi ^2.$ The equality holds if and only if ${\Sigma}$ is the Clifford torus up to a conformal transformation in $S^3$.

It is interesting to ask if a similar inequality holds for higher dimensional submanifold (and with higher co-dimension) in ${\mathbb R^n}$. It turns out that this is true:

 Theorem 5 Let ${m\ge 2}$ be an integer. Then there exists a constant ${C}$ depending on ${m}$ only such that for any closed oriented smooth submanifold ${M^m}$ immersed in ${\mathbb R^n}$, we have $\displaystyle \int_M|H|^m \ge C,$ where ${H}$ is the mean curvature vector of ${M}$.

To prove theorem 5, we need the following

 Theorem 6 (Michael-Simon ) } Let ${m\ge 2}$ be an integer. Then there exists a constant ${C_m}$ depending on ${m}$ only such that for any ${m}$-dimensional oriented submanifold ${\Sigma}$ immersed in ${\mathbb R^n}$, if ${f\in W^{1,1}(M)}$ and ${H}$ is the mean curvature vector of ${M}$, then $\displaystyle C_m \|f\|_{\frac{m}{m-1}}\le (\|\nabla f\|_1+ \|fH\|_1).$ Here ${\|\cdot\|_p}$ is the ${L^p}$-norm on ${M}$.

The proof of Theorem 5 is now very straightforward.
Proof: } This follows immediately by choosing ${f=1}$ in Theorem 6:

$\displaystyle C_m\mathrm{Area}(M)^\frac{m-1}{m} \le \|H\|_1 \le \|H\|_m \mathrm{Area}(M)^\frac{m-1}{m}.$

$\Box$

 Remark 1 It is easy to see that the equality in Theorem 5 holds only if ${H}$ is of constant length. When ${M}$ is an embedded hypersurface, this happens only when ${\Sigma}$ is a distance sphere. In this case ${\int_M |H|^m = m \omega_m}$ where ${\omega_m}$ is the area of the standard ${m}$-dimensional sphere. However, as it is not easy to determine the optimal constant ${C_m}$ for the Michael-Simon Sobolev inequality, we do not know whether ${C_m}$ in Theorem 6 gives an optimal constant for Theorem 5. (For a choice of ${C_2}$ 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 ${M}$.
 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.

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### 2 Responses to A Willmore type inequality

1. waynelam says:

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?

2. KKK says:

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) $L^2$ norm of the derivatives of some functions or some kind of curvature R (which scales like $1/\mathrm{length}^2$)… roughly this corresponds to the sum of “kinetic energy” and “potential energy”.
For higher dimension, it seems that $\int |H|^m$ 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 $\mathbb{R}^n$, the best constant is exactly achieved by the spheres. cf. this paper of Chen: