Conti, Sergio and Dolzmann, Georg and Mueller, Stefan (2024) Optimal Rigidity Estimates for Maps of a Compact Riemannian Manifold to Itself. SIAM JOURNAL ON MATHEMATICAL ANALYSIS, 56 (6). pp. 8070-8095. ISSN 0036-1410, 1095-7154
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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.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | NONLINEAR ELASTICITY; GEOMETRIC RIGIDITY; PLATES; rigidity estimates; elasticity; almost-isometric maps; geometric analysis |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Georg Dolzmann |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 10 Dec 2025 07:26 |
| Last Modified: | 10 Dec 2025 07:26 |
| URI: | https://pred.uni-regensburg.de/id/eprint/64555 |
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