Optimal Rigidity Estimates for Maps of a Compact Riemannian Manifold to Itself

Let [Formula: see text] be a smooth, compact, connected, oriented Riemannian manifold, and let [Formula: see text] be an isometric embedding. We show that a Sobolev map [Formula: see text] which has the property that the differential [Formula: see text] is close to the set [Formula: see text] of orientation preserving linear isometries (in an [Formula: see text] sense) is already [Formula: see text] close to a global isometry of [Formula: see text]. More precisely we prove, for [Formula: see text], the optimal estimate [Formula: see text], where [Formula: see text] and where [Formula: see text] denotes the group of orientation preserving isometries of [Formula: see text]. This is a Riemannian counterpart of the Euclidean rigidity estimate of Friesecke, James, and Müller [ Comm. Pure Appl. Math., 55 (2002), pp. 1461–1506] and extends the Riemannian stability result of Kupferman, Maor, and Shachar [ Arch. Ration. Mech. Anal., 231 (2019), pp. 367–408] for sequences of maps with [Formula: see text] to an optimal quantitative estimate. The proof relies on the weak Riemannian Piola identity of Kupferman, Maor, and Shachar, a uniform [Formula: see text] approximation through the harmonic map heat flow, and a linearization argument which reduces the estimate to the Riemannian version of Korn’s inequality.