Towards the Willmore conjecture

We develop a variety of approaches, mainly using integral ge- ometry, to proving that the integral of the square of the mean curvature of a torus immersed in R3 must always take a value no less than 2! 2. Our partial results, phrased mainly within the S3-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 recentresultofRos(20)confirmingtheWillmoreconjectureforsurfacesin- variant under the antipodal map, and a strengthening of the expected results for flat tori. The value 2! 2 arises in this work in a number of different ways - as the volume (or renormalised volume) of S3, SO(3) or G2,4, and in terms of the