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 E^{n }(n>m) can be C^{1}isometrically 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 C^{1 } isometrically embedded into an arbitrarily small ball in Euclidean 3-space. And, there exist C^{1 }isometric embeddings of flat tori into E^{3 }(a flat torus is a quotient E^{2}/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 C^{2 }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.