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 →ℝ^d 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(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 ∈ (1,∞) the optimal linear estimate inf_ϕ∈Isom_+(M)∘ f - ∘ϕ_W^1,p^p ≤ C E_p(f) where E_p(f) := ∫_M dist^p(df(q), SO(T_q M, T_f(q) M)) d vol_M and where 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. 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 E_p(f_k) → 0 to an optimal quantitative estimate. The proof relies on the weak Riemannian Piola identity of Kupferman-Maor-Shachar, a uniform C^1,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.