Mélanie Theillière (Université Claude Bernard Lyon 1) will speak in the geometry seminar on Tuesday 19 November 2019 at 11am in the Salle de Profs. Melanie’s title is Convex Integration without Integration and her abstract is below.
The Convex Integration Theory was developped by Gromov in 70s. This theory allows to solve differential constraints seen as subsets of the jet space and called Differential Relations. In the case of a relation of order 1, it allows to build a solution F from a section (x,f(x),L(x)) of the bundle J^1(M,W) -> M whose image lies inside the differential relation using an iteration of suitable integrations called “Convex Integrations”. Recently this theory led to explicit constructions of C^1-isometric embeddings. In this talk, we will propose an alternative formula to the Convex Integrations called Corrugation Process and we will introduce a kind of differential relation that we call Kuiper relation. For these relations, the formula is greatly simplified. As an application of this result, we will give an idea of the construction of a new immersion of RP^2 and we will state a Nash-Kuiper C^1-isometric embedding theorem in the case of totally real applications.