I would like to characterize the smooth maps $\varphi: \mathbb{R}^n \rightarrow \mathbb{R}^k$, $n\geq k$, with the following property:

For every $x\in \mathbb{R}^n$ there exists a positive number $r_0=r_0(x)$ such that for every positive number $r\leq r_0$ there holds $$ \mathrm{vol}_k \bigl( \varphi(B_r(x)) \bigr) \geq \omega_k r^k. $$

Here $B_r(x)\subset \mathbb{R}^n$ is the ball of radius $r$ around $x$ and $\omega_k$ denotes the $k$-volume of the unit ball in $\mathbb{R}^k$. These maps are in a certain sense ``submersions which locally expand volume''.

Some comments. An obvious necessary condition is that the $k$-Jacobian of $\varphi$ at every point is at least one: $$ J_k \varphi (x) := \max_{V\in \mathrm{Gr}_k(\mathbb{R}^n)} |\det D\varphi(x)|_V | = \sqrt{ \det \bigl( D\varphi(x) D\varphi(x)^* \bigr)} \geq 1, \quad \forall x\in \mathbb{R}^n. $$ Here $\mathrm{Gr}_k(\mathbb{R}^n)$ is the Grassmannian of $k$-planes in $\mathbb{R}^n$ and $A^*$ denotes the adjoint of the linear map $A:\mathbb{R}^n \rightarrow \mathbb{R}^k$ with respect to the Euclidean inner product.

This condition is clearly sufficient when $k=n$ (by the inverse mapping theorem and the area formula).It is also sufficient when $k=1$ (in which case $J_1 \varphi$ is just the norm of the gradient of the scalar function $\varphi:\mathbb{R}^n \rightarrow \mathbb{R}$): this is a nice exercise for students (hint: use the gradient flow of $\varphi$).

However, simple examples show that this condition is not sufficient when $n>k>1$.

The above condition becomes sufficient when we add the following requirement: let $\mathcal{V}(x)$ be the set of all $k$-planes $V\subset \mathbb{R}^n$ such that $|\det D\varphi(x)|_V |\geq 1$ (by the above condition the above set is not empty). Then we require the "multi-valued distribution" $\mathcal{V}$ to be ``integrable'', in the sense that $\mathbb{R}^n$ has a smooth $k$-dimensional foliation such that at every $x\in \mathbb{R}^n$ the leaf through $x$ is tangent to some $k$-plane in $\mathcal{V}(x)$.

The proof that this condition is sufficient uses the following fact, of which I do not know an elementary proof (see my MO question Existence of a large leaf in a foliation of the ball and its answers): a smooth $k$-foliation of the unit ball in $\mathbb{R}^n$ which is close enough to the affine foliation has a leaf whose $k$-volume is at least $\omega_k$. Then the claim follows from the area formula.

However, this sufficient condition seems to be far from necessary.

Does anybody have an idea on how to narrow the gap between necessary and sufficient? Or some bibliographical suggestions?


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