Convex integration

One of the methods developed by M. Gromov to prove the $h$-principle. The essence of this method is contained in the following statement: If the convex hull of some path-connected subset $A_0\subset {\mathbf{R}}^n$ contains a small neighbourhood of the origin, then there exists a mapping $f:S^1\to {\mathbf{R}}^n$ whose derivative sends $S^1$ into $A_0$. This is equivalent to saying that the differential relation for mappings $f:S^1\to {\mathbf{R}}^n$ given by requiring $f^{\prime }(\theta )\in A_0$ for all $\theta \in S^1$ satisfies the $h$-principle. More generally, the method of convex integration allows one to prove the $h$-principle for so-called ample relations $\mathcal{R}$. In the simplest case of a $1$-jet bundle $X^{(1)}$ over a $1$-dimensional manifold $V$, this means that the convex hull of $F\cap \mathcal{R}$ is all of $F$ for any fibre $F$ of $X^{(1)}\to X$ (notice that this fibre is an affine space). The extension to arbitrary dimension and higher-order jet bundles is achieved by studying codimension-one hyperplane fields $\tau$ in $V$ and intermediate affine bundles $X^{(r)}\to X^{\perp }\to X^{(r-1)}$ defined in terms of $\tau$.

One particular application of convex integration is to the construction of divergence-free vector fields and related geometric problems.

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