Flat tori in three-dimensional space
Vincent BORRELLI
In the mids 1950, Nash and Kuiper proved a bewildering theorem: any m-dimensional Riemannian manifold admitting an embedding in an Euclidean space Eⁿ (n>m) can be C¹isometrically embedded in the same Euclidean space. The result was then really a surprise, in part because of its numerous counterintuitive implications. For instance, it follows that a 2-dimensional sphere of radius one can be C¹ isometrically embedded into an arbitrarily small ball in Euclidean 3-space. And, there exist C¹ isometric embeddings of flat tori into E³ (a flat torus is a quotient E²/Λ where Λ = Ze1 ⊕ Ze2 is a lattice). A parametric version of the theorem also shows that the Smale eversion of the 2-sphere can be achieved isometrically, a result that outrageously contradicts the common intuition. In each case, the regularity of those amazing embeddings can not be enhanced to be C² since the existence of a curvature tensor would induce some direct obstructions.
In the 70-80's, Gromov turned the results of Nash into generic tools for solving undetermined system of partial differential equations: the Convex Integration theory. This theory offers a systematic approach and opens the door to a deeper understanding of Nash-Kuiper embeddings. In this talk, we shall focus on the geometric structure of embedded flat tori obtained by the convex integration process. We shall also present the first pictures of an embedded flat torus.