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

  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]. )

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

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