Namespaces
Variants
Actions

Convex integration

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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.

References

[a1] M. Gromov, "Partial differential relations" , Ergebn. Math. Grenzgeb. (3) , 9 , Springer (1986) MR0864505 Zbl 0651.53001
[a2] D. Spring, "Convex integration theory" , Monogr. Math. , 92 , Birkhäuser (1998) MR1488424 Zbl 0997.57500
How to Cite This Entry:
Convex integration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convex_integration&oldid=50351
This article was adapted from an original article by H. Geiges (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article