OPTIMAL RIGIDITY ESTIMATES FOR MAPS OF A COMPACT RIEMANNIAN MANIFOLD TO ITSELF

Let M be a smooth, compact, connected, oriented Riemannian manifold, and let \imath : 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(TqM,Tf(q)M) of orientation preserving linear isometries (in an L p sense) is already W " p close to a global isometry of M . More precisely we prove, for p \in (1, oo), the optimal estimate inf \phi E Isom + ( M ) \imath \circ f- \imath \circ\phi pW 1 ,p \leq CE p ( f ), where E p ( f ) := \intMdistp(df(q),SO(TqM,Tf(q)M))dvolM and where Isom+(M) denotes the group of orientation preserving isometries of M . This is a Riemannian counterpart of the Euclidean rigidity estimate of Friesecke, James, and Mu"\ller [ 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 E p ( f k )- 0 to an optimal quantitative estimate. The proof relies on the weak Riemannian Piola identity of Kupferman, Maor, and Shachar, a uniform C " \alpha approximation through the harmonic map heat flow, and a linearization argument which reduces the estimate to the Riemannian version of Korn's inequality.