DescriptionThis project will deal with a contemporary result in Differential Geometry. We describe one option in some detail. The Willmore conjecture is a conjecture about the Willmore energy of a torus, named after the late Durham mathematician Tom Willmore. The statement of the conjecture is as follows: Let v : M \to R^3 be a smooth immersion of a compact, orientable surface. Let H : M \to R be the mean curvature (the arithmetic mean of the principal curvatures \kappa_1 and \kappa_2 at each point). In this notation, the Willmore energy W(M) of M is given by W(M) = \int_{M} H^{2}.It is not hard to prove that the Willmore energy W(M) is bounded from below by 4\pi, with equality if and only if M is an embedded round sphere. Calculation of W(M) for a few examples suggests that there should be a better bound for surfaces with genus g(M) > 0. In particular, calculation of W(M) for tori with various symmetries led Willmore to propose in 1965 the following conjecture, which now bears his name: for any smooth immersed torus M in R^3, W(M) is bounded from below by 2\pi^2. On February 27, 2012, Fernando Coda Marques (Rio de Janeiro) and Andre Neves (London) announced a proof of the Willmore conjecture on a preprint posted in http://arxiv.org/abs/1202.6036, using the min-max theory of minimal surfaces. In this project, you will study some of the elements that enter into the work Coda and Neves. These include the direct method in the calculus of variations, the index theory for minimal surfaces, and regularity theory for critical points of coercive problems, and geometric measure theory. We have a limit of four students on this project. PrerequisitesAnalysis III/IV and Differential Geometry III.ResourcesManfredo do Carmo : Differential Geometry of curves and surfaces, Prentice HallT. Colding and C. De Lellis, The min-max construction of minimal surfaces, Surveys in Differential Geometry VIII , International Press, (2003), 75 - 107. H. I. Choi and R. Schoen: The space of minimal embeddings of a surface into a threedimensional manifold of positive Ricci curvature, Invent. Math. 81 (1985), 387 - 394. R. Bryant, A duality theorem for Willmore surfaces. J. Differential Geom. 20 (1984), 23 - 53. email: W Klingenberg |