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