Optimal rigidity estimates for maps of a compact Riemannian manifold to itself

Abstract

Let $M$ be a smooth, compact, connected, oriented Riemannian manifold, and let $ı:M\to {\mathbb{R}}^d$ be an isometric embedding. We show that a Sobolev map $f:M\to M$ which has the property that the differential $df(q)$ is close to the set $SO(T_{q}M,T_{f(q)}M)$ of orientation preserving isometries (in an $L^p$ sense) is already $W^{1,p}$ close to a global isometry of $M$. More precisely we prove for $p\in (1,\infty )$ the optimal linear estimate $$\underset{\varphi \in {\mathrm{Isom}}_{+}(M)}{\inf }\Vert ı\circ f-ı\circ \varphi {\Vert }_{W^{1,p}}^p\le C{E}_p(f)$$ where $$E_p(f):={\int }_M{\mathrm{dist}}^p(df(q),SO(T_{q}M,T_{f(q)}M))d{\mathrm{vol}}_M$$ and where ${\mathrm{Isom}}_{+}(M)$ denotes the group of orientation preserving isometries of $M$. This extends the Euclidean rigidity estimate of Friesecke-James-Müller [Comm. Pure Appl. Math. {\bf 55} (2002), 1461--1506] to Riemannian manifolds. It also extends the Riemannian stability result of Kupferman-Maor-Shachar [Arch. Ration. Mech. Anal. {\bf 231} (2019), 367--408] for sequences of maps with $E_p(f_k)\to 0$ to an optimal quantitative estimate. The proof relies on the weak Riemannian Piola identity of Kupferman-Maor-Shachar, a uniform $C^{1,\alpha }$ approximation through the harmonic map heat flow, and a linearization argument which reduces the estimate to the well-known Riemannian version of Korn's inequality.