Difference between revisions of "Convex integration"
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| − | One of the methods developed by M. Gromov to prove the [[H-principle| | + | <!--This article has been texified automatically. Since there was no Nroff source code for this article, |
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| + | One of the methods developed by M. Gromov to prove the [[H-principle|$h$-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. | One particular application of convex integration is to the construction of divergence-free vector fields and related geometric problems. | ||
====References==== | ====References==== | ||
| − | <table>< | + | <table><tr><td valign="top">[a1]</td> <td valign="top"> M. Gromov, "Partial differential relations" , ''Ergebn. Math. Grenzgeb. (3)'' , '''9''' , Springer (1986) {{MR|0864505}} {{ZBL|0651.53001}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> D. Spring, "Convex integration theory" , ''Monogr. Math.'' , '''92''' , Birkhäuser (1998) {{MR|1488424}} {{ZBL|0997.57500}} </td></tr></table> |
Latest revision as of 17:00, 1 July 2020
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
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