# Definition of Willmore energy in higher dimensions

The Willmore energy is a well-known and studied quantity typically defined for surfaces immersed in a 3-dimensional manifold as the integral of the square of the mean curvature (see https://en.wikipedia.org/wiki/Willmore_energy).

I was wondering if there is any standard notion of Willmore energy for a $n$-dimensional submanifold of a $(n+1)$-dimensional Riemannian manifold. I couldn't find much material online, so I thought that this was the right place where to ask.

Any help will be very appreciated! Thanks a lot!

• This is a perfectly valid question with a rich history and a lot of amazing results to mention. I am really disappointed with the poor judgment of the members who put it on hold without even making comments or trying to improve it. This is not in the spirit of this community to repel the users by pushing them off the topics that you don't care about. Very disappointed... Jun 23, 2018 at 4:52
• I started writing a little survey on this question here. Provided there is an interest to this topic, I am happy to post updates and answer further questions. Jun 23, 2018 at 6:53
• @YuriVyatkin I am also surprised! I don't really understand why the question was put on hold.. I couldn't find much material online about a possible definition of the Willmore energy in higher dimension and I thought that this was the natural place where to ask. Thanks a lot for caring about my question and thanks for writing that nice note! :) Jun 23, 2018 at 9:18
• You are welcome. Anyway, the community here decides what is worthy, so I would suggest you to follow the guidelines above and edit your question, so that the folks change their mind and add more votes for reopening it. Add your definition of the Willmore energy and some motivation for your question. Jun 23, 2018 at 9:36
• Anyway, you are right, I made a mistake in the note. Please see the updated version there. Sorry, I have not been thinking on this topic for quite a while and need to recapitulate it for myself. Jun 23, 2018 at 12:16

Let us consider an embedding $\Sigma \to M$ where $\dim \Sigma = n$ and $M$ is a Riemannian manifold, with all the additional assumptions made when needed in order to the following could make sense.

Already Willmore  mentions (p.282) the integrals (for the case $M = \mathbb{R}^{n+1}$)

$$W(\Sigma) = \int_{\Sigma} |H|^n \mathrm{d}\Sigma \tag{1}$$

and the related work of B.-Y. Chen on computing their Euler-Lagrange equations. He notices that these integrals are not conformally invariant except when $n = 2$. Here $\mathrm{d}\Sigma$ is the Riemannian volume form induced on $\Sigma$.

In  Guo constructs a series of functionals $W_r, 2 \le r \le n,$ where the integrands are certain powers of polynomials of the principal curvatures of $\Sigma$. The functionals $W_r$ generalize the Willmore functional to higher dimensions ($W = C\cdot W_2$, where $C$ is a constant). Moreover, $W_r$ are conformally invariant (with respect to rescalings of the metric in $M$). Explicitly,

$$W_2(\Sigma) = \int_M (\sigma^2_1 - \sigma_2)^{n/2} \mathrm{d} \Sigma \tag{2}$$

The integrand in (2) is $n$-th power of the length of the trace-free part of the second fundamental form (the quantity, that I prefer to call the umbilicity tensor, for the sake of brevity).

The umbilicity tensor is a classical example of a hypersurface conformal invariant (these are the central topic of my PhD thesis, by the way). Then notation for this tensor is not widely adopted, so instead of writing something like $\overset{\circ}{L}$ (that I usually do), let me just denote it here by $U$, so that we can consider

$$\mathcal{W} = \int_{\Sigma} |U|^2 \mathrm{d}\Sigma \tag{3}$$ for the case $n = 2$. This is obviously a global conformal invariant of $\Sigma$ (Global conformal invariants in general have been studied in detail by Spyros Alexakis, but this one is rather special one, intimately related to the hypersurface structure).

The integrand in (3) is a nice Riemannian quantity: it arises as a polynomial expression in partial derivatives of the metric and transforms naturally under coordinate changes (in other words, this is a natural Riemannian ir curvature invariant of $\Sigma$).

If we require that the Willmore energy generalizes to the case of a Riemannian background and higher dimensions so that this generalizations satisfies the following conditions:

• the integrand is a natural Riemannian invariant (possibly, of a higher order, that is we can consider covariant derivatives of the curvature and the second fundamental form);
• the integral is a global conformal invariant of the hypersurface $\Sigma$;

then the problem of generalizing the Willmore energy becomes very non-trivial.

J.Guven in  found a $4$-dimensional analogue of $W$ in a Euclidean background:

$$H_2 = \frac{1}{2} \int \Big( (\nabla K)^2 - \frac{7}{8} K^4 + 2 K^2 \mathrm{tr} K^2 \Big) \mathrm{d}\Sigma$$ where by $K$ he denoted the shape tensor (essentially, the second fundamental form).

During my PhD study under supervision by A.Rod Gover, we discovered a quantity closely related to the so called $Q$-curvature, which also generalizes the Willmore functional to the case of $4$-dimensional hypersurfaces. Actually, I only scratched the surface gently...

The problem as stated above was completely solved by A.Rod Gover and Andrew Waldron using a set of powerful techniques, including so-called boundary calculus.

Some additional remarks and curious links you can find in my post here.

References:

1. T.J. Willmore, Riemannian Geometry (OUP, 1993)
2. Zhen Guo, Generalized Willmore functionals and related variational problems, Diff.Geom.and its App. 25 (2007) 543-551
3. Jemal Guven, Conformally invariant bending energy for hypersurfaces, J. Phys. A: Math. Gen. (2005), Vol.38, Num.37.
4. A. Rod Gover, Andrew Waldron, Generalising the Willmore equation: submanifold conformal invariants from a boundary Yamabe problem, (2014) arXiv:1407.6742 [hep-th]