From Convex Integration to Flat Tori

These lectures are devoted to a general method for solving differential relations: the Gromov Convex Integration Theory. We explore its simple version for first order differential relations with special attention to its geometrical and analytical foundations (Lecture 1). We introduce the notion of h-principle and then prove by using Convex Integrations that the h-principle holds for ample and open relations. We derive from this instance of hprinciple many historical results in immersion-theoretic topology including the Smale eversion of the 2-sphere (Lecture 2). One special feature of Convex Integration is that it applies to solve certain classes of closed relations. The most interesting case is the one of the isometric relation with, as corollary, the celebrated Nash-Kuiper theorem on C isometric embeddings. We show that the parametric h-principle holds for the isometric relation and, as a consequence, that the eversion of the 2-sphere can be realized by a regular homotopy of C isometric immersions (Lecture 3).