Reshetnyak Rigidity for Riemannian Manifolds

Abstract

We prove two rigidity theorems for maps between Riemannian manifolds. First, we prove that a Lipschitz map f : M → N between two oriented Riemannian manifolds, whose differential is almost everywhere an orientation-preserving isometry, is an isometric immersion. This theorem was previously proved using regularity theory for conformal maps; we give a new, simple proof, by generalizing the Piola identity for the cofactor operator. Second, we prove that if there exists a sequence of mapping f n : M → N, whose differentials converge in L p to the set of orientation-preserving isometries, then there exists a subsequence converging to an isometric immersion. These results are generalizations of celebrated rigidity theorems by Liouville (1850) and Reshetnyak (1967) from Euclidean to Riemannian settings. Finally, we describe applications of these theorems to non-Euclidean elasticity and to convergence notions of manifolds.

On Asymptotic Rigidity and Continuity Problems in Nonlinear Elasticity on Manifolds and Hypersurfaces

“…In the recent work by Kupferman-Maor-Shachar [22], the Resetnyak rigidity theorem was generalized to Lipschitz maps f : M → N between Riemannian manifolds. It is raised as an open question in [22] whether the quantitative inequality of Proposition 3.1 admits generalizations to mappings (not necessarily Lipschitz) between Riemannian manifolds.…”

“…Remark 3.4. In the recent preprint [1], Alpern-Kupferman-Maor further extended their asymptotic rigidity theorem in [22] to hypersurfaces in space forms. As remarked in [1], a quantitative result in the form of a geometric rigidity theorem may help to extend the asymptotic rigidity theorem therein to arbitrary ambient Riemannian manifolds.…”

“…It remains an open question whether the analogous results for mappings between Riemannian manifolds remain valid; cf. Kupferman-Maor-Shachar [22]. In this paper, as a first step towards the above open question, we establish a geometric rigidity estimate from a d-dimensional Riemannian manifold into spheres of dimension d ≥ 2.…”

Variational convergence of discrete geometrically-incompatible elastic models

“…If F(F) = 0, then it follows from the above argument that dF ∈ SO(g, e) almost everywhere. It follows by [LP10, Lemma 3.1] (see also [KMS19]) that F is smooth, hence dF ∈ SO(g, e) everywhere and therefore g is flat. ■ Remark: An alternative proof of the second part can be obtained using the explicit formula for Q dist 2 (A, SO(g, e)) calculated in [Šil01, Example 4.2].…”

A geometric perspective on the Piola identity in Riemannian settings

“…The second proof was presented in [5]. It is based on a characterization of the cofactor as the derivative of the determinant.…”