The MAA has posted to its facebook page a link to an article about a recent proposed proof of what is called the Willmore conjecture, after Thomas Willmore.
Wikipedia's article titled Wilmore conjecture includes the following:
Let $v:M\to\mathbb{R}^3$ be a smooth immersion of a compact, orientable surface (of dimension two). Giving $M$ the Riemannian metric induced by $v$, let $H:M\to\mathbb{R}$ be the mean curvature (the arithmetic mean of the principal curvatures $\kappa_1$ and $\kappa_2$ at each point). Let $K$ be the Gaussian curvature. In this notation, the Willmore energy $W(M)$ of $M$ is given by $W(M) = \int_S H^2 \, dA - \int_S K \, dA.$ In the case of the torus, the second integral above is zero.
A little bit of this came from editing by me within the past hour.
Knowing very little of differential geometry, I hesitate to do much more with this paragraph before clarifying some things. It seems $M$ is a particular parametrization of the surface, but the integrals look like things that should not depend on which suitably well-behaved parametrization is chosen. Yet the definition seems to attribute the Willmore energy to the parametrization $M$, rather than to the surface, which might be parametrized in any of many different ways. Notice the use of the capital letter $S$ in the expression $\displaystyle\int_S$, when nothing called $S$ was defined! Presumably $S$ means the image of $M$.
Ought one to write $W(S)$ instead of $W(M)$, to be clear about a lack of dependence on a choice of parametrization?