Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal rigidity estimates for maps of a compact Riemannian manifold to itself

Sergio Conti, Georg Dolzmann, Stefan Müller

Abstract

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.

Optimal rigidity estimates for maps of a compact Riemannian manifold to itself

Abstract

Let be a smooth, compact, connected, oriented Riemannian manifold, and let be an isometric embedding. We show that a Sobolev map which has the property that the differential is close to the set of orientation preserving isometries (in an sense) is already close to a global isometry of . More precisely we prove for the optimal linear estimate where and where denotes the group of orientation preserving isometries of . 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 to an optimal quantitative estimate. The proof relies on the weak Riemannian Piola identity of Kupferman-Maor-Shachar, a uniform 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.
Paper Structure (20 sections, 20 theorems, 173 equations)

This paper contains 20 sections, 20 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Strategy of proof
  4. Preliminaries and notation
  5. Some linear algebra
  6. Second fundamental form
  7. Sobolev spaces on manifolds
  8. Lipschitz approximation
  9. The Piola identity and almost harmonicity
  10. Harmonic map heat flow and uniform $C^{1,\alpha}$ approximation
  11. Local existence and regularity for $W^{1,\infty}$ initial data
  12. Refined estimates for almost harmonic initial data
  13. Linearization of the metric deficit equation and rigidity estimates close to the identity
  14. From maps to vector fields
  15. The metric deficit equation and its linearization
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $U \subset \mathbb{R}^n$ be open, bounded and connected, with Lipschitz boundary. Then there exists a constant $C$, depending on $U$ and $n$, such that for every $f \in W^{1,2}(U; \mathbb{R}^n)$ there exists a constant matrix $R \in SO(n)$ with

Theorems & Definitions (40)

  • Theorem 1.1: FrieseckeJamesMueller2002-CPAM, Theorem 3.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Theorem 4.1: Weak extrinsic Piola identity, KupfermanMaorShachar2019, Theorem 2
  • ...and 30 more