Winding map and local injectivity

The winding map ${W}$ is a humble example that is conjectured to be extremal in a long-standing open problem. Its planar version is defined in polar coordinates ${(r,\theta)}$ by

$\displaystyle (r,\theta) \mapsto (r,2\theta)$

All this map does it stretch every circle around the origin by the factor of two — tangentially, without changing its radius. As a result, the circle winds around itself twice. The map is not injective in any neighborhood of the origin ${r=0}$.

The 3D version of the winding map has the same formula, but in cylindrical coordinates. It winds the space around the ${z}$-axis, like this:

In the tangential direction the space is stretched by the factor of ${2}$; the radial coordinate is unchanged. More precisely: the singular values of the derivative matrix ${DW}$ (which exists everywhere except when ${r=0}$) are ${2,1,1}$. Hence, the Jacobian determinant ${\det DW}$ is ${2}$, which makes sense since the map covers the space by itself, twice.

In general, when the singular values of the matrix ${A}$ are ${\sigma_1\ge \dots \ge\sigma_n}$, the ratio ${\sigma_n^{-n} \det A}$ is called the inner distortion of ${A}$. The word “inner” refers to the fact that ${\sigma_n}$ is the radius of the ball inscribed into the image of unit ball under ${A}$; so, the inner distortion compares this inner radius of the image of unit ball to its volume.

For a map, like ${W}$ above, the inner distortion is the (essential) supremum of the inner distortion of its derivative matrices over its domain. So, the inner distortion of ${W}$ is ${2}$, in every dimension. Another example: the linear map ${(x,y)\mapsto (3x,-2y)}$ has inner distortion ${3/2}$.

It is known that there is a constant ${K>1}$ such that if the inner distortion of a map ${F}$ is less than ${K}$ almost everywhere, the map is locally injective: every point has a neighborhood in which ${F}$ is injective. (Technical part: the map must be locally in the Sobolev class ${W^{1,n}}$.) This was proved by Martio, Rickman, and Väisälä in 1971. They conjectured that ${K=2}$ is optimal: that is, the winding map has the least inner distortion among all maps that are not locally injective.

But at present, there is still no explicit nontrivial lower estimate for ${K}$, for example we don’t know if inner distortion less than ${1.001}$ implies local injectivity.