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Full flexibility of isometric immersions of metrics with low Hölder regularity in Poznyak theorem's dimension

Marta Lewicka

TL;DR

This work proves that any 2D Riemannian metric with Hölder regularity $\mathcal{C}^{r,\beta}$ admits isometric immersions into $\mathbb{R}^4$ with $\mathcal{C}^{1,\alpha}$ regularity for every $\alpha<\min\{\frac{r+\beta}{2},1\}$, achieving full flexibility in the Poznyak setting. The authors develop a refined convex integration framework comprising a Stage theorem that yields quantitative defect decay and controlled Hessian growth, combined with a Nash–Kuiper scheme to obtain $\mathcal{C}^{1,\alpha}$ isometric immersions arbitrarily close to short immersions. Key technical innovations include a defect-decomposition into primitive rank-one pieces, oscillatory integration-by-parts lemmas, and a Källén-type iteration to damp non-oscillatory defects, all adapted to the 2D Monge–Ampère–type nonlinear system and higher codimension. The results bridge flexibility between the Monge–Ampère framework and classical isometric-immersion theory, resolving rigidity phenomena seen in codimension one and extending Poznyak’s theorem to metrics with low Hölder regularity. The methods have broad implications for geometric analysis and nonlinear PDEs, demonstrating that rigidity can be entirely overcome in dimension 4 for 2D metrics with low regularity.

Abstract

A classical result by Poznyak asserts that any smooth $2$-dimensional Riemannian metric $g$, posed on the closure of a simply connected domain $ω\subset\mathbb{R}^2$, has a smooth isometric immersion into $\mathbb{R}^4$. Using techniques of convex integration, we prove that for any $2$-dimensional $g\in\mathcal{C}^{r,β}$, an isometric immersion of regularity $\mathcal{C}^{1,α}(\barω,\mathbb{R}^4)$ for any $α<\min\{\frac{r+β}{2},1\}$, may be found arbitrarily close to any short immersion. The fact that this result's regularity reaches $\mathcal{C}^{1,1-}$ for $g\in \mathcal{C}^2$, which is referred to as "full flexibility", should be contrasted with: (i) the regularity $\mathcal{C}^{1,1/3-}$ achieved by Cao, Hirsch and Inauen for isometric immersions into $\mathbb{R}^{3}$ and the lack of flexibility (rigidity) of such isometric immersions with regularity $\mathcal{C}^{1, 2/3+}$ proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity $\mathcal{C}^{1,1-}$ obtained by Källen for isometric immersions into higher codimensional space; and (iii) the regularity $\mathcal{C}^{1,\frac{1}{d(d+1)/k}-}$ achieved by the author in the general case of $d$-dimensional metrics and $(d+k)$-dimensional immersions for the closely related Monge-Ampère system.

Full flexibility of isometric immersions of metrics with low Hölder regularity in Poznyak theorem's dimension

TL;DR

This work proves that any 2D Riemannian metric with Hölder regularity admits isometric immersions into with regularity for every , achieving full flexibility in the Poznyak setting. The authors develop a refined convex integration framework comprising a Stage theorem that yields quantitative defect decay and controlled Hessian growth, combined with a Nash–Kuiper scheme to obtain isometric immersions arbitrarily close to short immersions. Key technical innovations include a defect-decomposition into primitive rank-one pieces, oscillatory integration-by-parts lemmas, and a Källén-type iteration to damp non-oscillatory defects, all adapted to the 2D Monge–Ampère–type nonlinear system and higher codimension. The results bridge flexibility between the Monge–Ampère framework and classical isometric-immersion theory, resolving rigidity phenomena seen in codimension one and extending Poznyak’s theorem to metrics with low Hölder regularity. The methods have broad implications for geometric analysis and nonlinear PDEs, demonstrating that rigidity can be entirely overcome in dimension 4 for 2D metrics with low regularity.

Abstract

A classical result by Poznyak asserts that any smooth -dimensional Riemannian metric , posed on the closure of a simply connected domain , has a smooth isometric immersion into . Using techniques of convex integration, we prove that for any -dimensional , an isometric immersion of regularity for any , may be found arbitrarily close to any short immersion. The fact that this result's regularity reaches for , which is referred to as "full flexibility", should be contrasted with: (i) the regularity achieved by Cao, Hirsch and Inauen for isometric immersions into and the lack of flexibility (rigidity) of such isometric immersions with regularity proved by Borisov and then by Conti, de Lellis and Szekelyhidi; (ii) the regularity obtained by Källen for isometric immersions into higher codimensional space; and (iii) the regularity achieved by the author in the general case of -dimensional metrics and -dimensional immersions for the closely related Monge-Ampère system.

Paper Structure

This paper contains 17 sections, 18 theorems, 347 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical existence and flexibility results
  3. The dimension $d=2$ and rigidity results
  4. Recent results on flexibility
  5. The Monge-Ampère system
  6. Flexibility of the Monge-Ampère system
  7. The main technical contributions of this paper and an outline of proofs
  8. Notation
  9. Acknowledgement
  10. The defect decomposition lemmas and two Step constructions
  11. Propagation of normal vectors lemmas
  12. The initial Stage in the Nash-Kuiper scheme
  13. The oscillatory defect decomposition lemmas
  14. The Källén iteration technique
  15. Stage construction: the proof of Theorem \ref{['thm_STA']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $g\in \mathcal{C}^{r,\beta}(\bar{\omega}, \mathbb{R}^{2\times 2}_{\mathrm{sym}})$ be defined on the closure of an open set $\omega\subset\mathbb{R}^2$ diffeomorphic to $B_1$, for some $r+\beta>0.$ Then, for every $u\in \mathcal{C}^1(\bar{\omega},\mathbb{R}^4)$ satisfying: for every $\epsilon>0$, and for every regularity exponent $\alpha$ in: there exists $\tilde{u}\in \mathcal{C}^{1,\alpha}(

Figures (2)

  • Figure 1: Progression of the principal frequencies in (\ref{['md_def']}).
  • Figure 2: Progression of frequencies and the corresponding intermediate fields $U$, $\bar{U}$ in the three corrugation modification from $u_k$ to $u_{k+1}$. Note the distinction of cases $k=0$ and $k=1\ldots K-1$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 17 more