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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.06448 |
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| _version_ | 1866911774167531520 |
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| author | Conti, Sergio Dolzmann, Georg Müller, Stefan |
| author_facet | Conti, Sergio Dolzmann, Georg Müller, Stefan |
| contents | Let $M$ be a smooth, compact, connected, oriented Riemannian manifold, and let $\imath: M \to \mathbb R^d$ be an isometric embedding. We show that a Sobolev map $f: M \to M$ 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 $p \in (1,\infty)$ the optimal linear estimate $$\inf_{ϕ\in \mathrm{Isom}_+(M)} \| \imath \circ f - \imath \circ ϕ\|_{W^{1,p}}^p \le C E_p(f)$$ where $$ E_p(f) := \int_M {\rm dist}^p(df(q), SO(T_q M, T_{f(q)} M)) \, d{\rm vol}_M$$ and where $\mathrm{Isom}_+(M)$ 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 $E_p(f_k) \to 0$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal rigidity estimates for maps of a compact Riemannian manifold to itself Conti, Sergio Dolzmann, Georg Müller, Stefan Analysis of PDEs Differential Geometry Let $M$ be a smooth, compact, connected, oriented Riemannian manifold, and let $\imath: M \to \mathbb R^d$ be an isometric embedding. We show that a Sobolev map $f: M \to M$ 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 $p \in (1,\infty)$ the optimal linear estimate $$\inf_{ϕ\in \mathrm{Isom}_+(M)} \| \imath \circ f - \imath \circ ϕ\|_{W^{1,p}}^p \le C E_p(f)$$ where $$ E_p(f) := \int_M {\rm dist}^p(df(q), SO(T_q M, T_{f(q)} M)) \, d{\rm vol}_M$$ and where $\mathrm{Isom}_+(M)$ 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 $E_p(f_k) \to 0$ 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. |
| title | Optimal rigidity estimates for maps of a compact Riemannian manifold to itself |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2402.06448 |