2024/02/09 by Sergio Conti, Conti, Sergio, Georg Dolzmann +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.48550/arxiv.2402.06448
openalex publication_date 2024/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let M be a smooth, compact, connected, oriented Riemannian manifold, and let \imath: M → \mathbb 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, Tf(q) M) of orientation preserving isometries (in an Lp sense) is already W1,p close to a global isometry of M. More precisely we prove for p ∈ (1,∞) the optimal linear estimate infϕ∈ Isom+(M) ‖ \imath ∘ f - \imath ∘ ϕ‖W1,pp ≤ C Ep(f) where Ep(f) := ∫M \rm distp(df(q), SO(Tq M, Tf(q) M)) d\rm volM 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. \bf 55 (2002), 1461--1506] to Riemannian manifolds. It also extends the Riemannian stability result of Kupferman-Maor-Shachar [Arch. Ration. Mech. Anal. \bf 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 C1,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.