Zoll-like manifolds in minimal surface theory

Professor Lucas Ambrozio (IMPA) will present his research in Analysis and Geometry.

A Riemannian metric is called Zoll if all of its non-trivial geodesics are periodic and have the same length. The first examples of such metrics where described long ago by Otto Zoll as spheres of revolution in R3, but it was only after the works of Funk and Guillemin that the most general Zoll deformations of the canonical metric of the unit sphere in R3 were described. In this talk, we will introduce and investigate a new generalisation of the Zoll condition on higher dimensional spheres, whereby periodic geodesics (= immersed circles that are critical point of the length functional) are replaced by embedded spheres of codimension one that are minimal (= critical points of the area functional). In particular, we will explain a new deformation theorem generalising Guillemin results, present some applications and explain why we think these new geometric objects deserve to be further investigated. This is part of a joint project with F. Marques (Princeton) and A. Neves (UChicago).

The talks are held using Zoom. Anyone interested in attending (outside the School of Mathematics) should email Ben Sharp at b.g.sharp@leeds.ac.uk to request zoom coordinates.