A characterization of submanifolds of $\mathbb{R}^{m \times n}$ satisfying optimal rigidity estimates
Let $K \subset \mathbb{R}^{m \times n}$ be a compact $C^1$-submanifold with boundary, $p \in (1,\infty)$ and $Q := (0,1)^n$. We prove that $K$ satisfies a rigidity estimate of the form $\|Du - (Du)_{Q}\|_{L^p} \leq C \|\mathrm{dist}_K(Du)\|_{L^p}$, $u \in W^{1,p}(Q,\mathbb{R}^m)$, if and only if $K$ satisfies sequential rigidity and for each $A \in K$, the tangent space to $K$ at $A$ satisfies exact rigidity. We further prove that this rigidity estimate is stable under small graphical perturbations of $K$. The key technical ingredient is proving that outside some small bad set where the maximal function of $\mathrm{dist}_K^p(Du)$ is large, the size of the superlevel sets of $\lvert Du - (Du)_Q\rvert$ decays exponentially. This is achieved by an adaptation of the John-Nirenberg inequality for $BMO$-functions.