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 $\Sigma \subset S^{n-1}$ be an $(n-2)$-dimensional $C^2$ manifold such that $\Sigma$ 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 \Sigma : \dim \pi_{T_xS^{n-1}}(Z) < s\} \le s \end{equation*} where $\pi_{T_xS^{n-1}}$ is the orthogonal projection from $\mathbb{R}^n$ to the tangent space $T_xS^{n-1}$. In particular, for $\mathcal{H}^{n-2}$-a.e. $x \in \Sigma$, $\dim \pi_{T_xS^{n-1}}(Z) = \dim Z$. 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 $\pi_{T_yS^{n-1}}(Z) \le n-2$ for some $y \in S^{n-1}$, we show that $\dim \pi_{T_xS^{n-1}}(Z) = \min\{\dim Z, n-1\}$ for $\mathcal{H}^{n-2}$-a.e. $x \in \Sigma$.