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, T₅ (ₐ) M) of orientation preserving isometries (in an Lᵖ sense) is already W^1, p close to a global isometry of M. More precisely we prove for p (1, ) the optimal linear estimate ₈ₒ₎₌_+ (₌) \| f - \|ₖ^₁, ₏ᵖ C Eₚ (f) where Eₚ (f): = M distᵖ (df (q), SO (Tq M, T₅ (ₐ) 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 C^1, 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.