Restricted Projections to Hyperplanes in ${\mathbb{R}}^n$

Abstract

We study dimensions of sets projected to an $(n-2)$-dimensional family of hyperplanes in ${\mathbb{R}}^n$ under curvature conditions. Let $n\ge 3$ and $\hat{I}\pounds \subset S^{n-1}$ be an $(n-2)$-dimensional $C^2$ manifold such that $\hat{I}\pounds$ has non-vanishing geodesic curvature ($n=3$)/sectional curvature $>1$ ($n\ge 4)$. Let $Z\subset {\mathbb{R}}^n$ be analytic with $\dim Z\le n-2$ and $0<s<\dim Z$. Then \begin{equation*} \dim \{x \in Σ: \dim π_{T_xS^{n-1}}(Z) < s\} \le s \end{equation*} where $\ddot{I}{€}_{T_x{S}^{n-1}}$ is the orthogonal projection from ${\mathbb{R}}^n$ to the tangent space $T_x{S}^{n-1}$. In particular, for ${\mathcal{H}}^{n-2}$-a.e. $x\in \hat{I}\pounds$, $\dim \ddot{I}{€}_{T_x{S}^{n-1}}(Z)=\dim Z$. When $n=3$ and $\dim Z<1$, the quantitative estimate improves the one obtained by Gan-Guo-Guth-Harris-Maldague-Wang. For the case $\dim Z>n-2$, if in addition $\ddot{I}{€}_{T_y{S}^{n-1}}(Z)\le n-2$ for some $y\in S^{n-1}$, we show that $\dim \ddot{I}{€}_{T_x{S}^{n-1}}(Z)=\min \{\dim Z,n-1\}$ for ${\mathcal{H}}^{n-2}$-a.e. $x\in \hat{I}\pounds$.

30 pages, 2 figures