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 $f : S ^ { 1 } \rightarrow \mathbf{R} ^ { n }$ whose derivative sends $S ^ { 1 }$ into $A _ { 0 }$. This is equivalent to saying that the differential relation for mappings $f : S ^ { 1 } \rightarrow \mathbf{R} ^ { n }$ 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 $X ^ { ( 1 ) } \rightarrow X$ (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 $X ^ { ( r ) } \rightarrow X ^ { \perp } \rightarrow X ^ { ( r - 1 ) }$ 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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