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

Let M be a smooth, compact, connected, oriented Riemannian manifold, and let : M → Rᵈ be an isometric embedding. We show that a Sobolev map f: M → M which has the property that the differential $df(q)$ is close to the set SO(Tq M, Tf(q) M) of orientation preserving isometries (in an Lᵖ sense) is already W1,p close to a global isometry of M. More precisely we prove for p ∈ (1,∞) the optimal linear estimate infφ ∈ Isom₊(M) \| ∘ f - ∘ φ\|_W1,pᵖ ≤ C Eₚ(f) where Eₚ(f) := ∫M distᵖ(df(q), SO(Tq M, Tf(q) M)) \, d volM and where Isom₊(M) denotes the group of orientation preserving isometries of M. This extends the Euclidean rigidity estimate of Friesecke-James-M\"uller [Comm. Pure Appl. Math. { 55} (2002), 1461--1506] to Riemannian manifolds. It also extends the Riemannian stability result of Kupferman-Maor-Shachar [Arch. Ration. Mech. Anal. { 231} (2019), 367--408] for sequences of maps with Eₚ(fₖ) → 0 to an optimal quantitative estimate. The proof relies on the weak Riemannian Piola identity of Kupferman-Maor-Shachar, a uniform C1,α 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.