Towards the Willmore conjecture

Abstract.

We develop a variety of approaches, mainly using integral geometry, to proving that the integral of the square of the mean curvature of a torus immersed in ${\mathbb R}^3$ must always take a value no less than $2\pi^2$. Our partial results, phrased mainly within the $S^3$-formulation of the problem, are typically strongest when the Gauss curvature can be controlled in terms of extrinsic curvatures or when the torus enjoys further properties related to its distribution within the ambient space (see Sect. 3). Corollaries include a recent result of Ros [20] confirming the Willmore conjecture for surfaces invariant under the antipodal map, and a strengthening of the expected results for flat tori.

The value $2\pi^2$ arises in this work in a number of different ways – as the volume (or renormalised volume) of $S^3, SO(3)$ or $G_{2,4}$, and in terms of the length of shortest nontrivial loops in subgroups of SO(4).