Grassmannian Frequency of Sobolev Dimension Distortion

Abstract

Let $f$ be a supercritical Sobolev map from a domain $\Omega$ in $\mathbb {R}^n$ into a complete separable metric space. For a fixed integer $m$, $0<m<n$, and $\alpha >m$, we estimate from above the Hausdorff dimension of the set of elements $V$ in the Grassmannian $G(n,m)$ (equipped with a Riemannian metric) such that $f(V\cap \Omega )$ has positive $\alpha$-dimensional Hausdorff measure. The proof relies heavily on the homogeneous structure of both $G(n,m)$ and the Stiefel manifold $O(n,m)$ of orthogonal injections of $\mathbb {R}^m$ into $\mathbb {R}^n$. A novel feature of the proof is a Morrey–Sobolev-type embedding theorem on the product manifold $O(n,m)\times \mathbb {R}^m$, valid for Sobolev maps which factor through the evaluation map $\Phi :O(n,m)\times \mathbb {R}^m\rightarrow \mathbb {R}^n$, $\Phi (\pi ,x)=\pi (x)$.