---
title: "h-Principle and Rigidity for C^{1,α} Isometric Embeddings"
authors: ["Sergio Conti", "Camillo De Lellis", "László Székelyhidi"]
year: 2012
venue: "Abel Symp."
arxiv: "0905.0370"
doi: null
source: "https://arxiv.org/abs/0905.0370"
topics: [curiosity]
seed_rank: 1235
seed_batch: "curiosity-2026-08-25"
reviewed: "2026-08-25"
pool: "maths-foundations"
relevance_score: 10
lineage: h-principle
cites:
  - title: "Holonomic approximation and Gromov's h-principle"
    url: "https://arxiv.org/abs/math/0101196"
    year: 2001
    arxiv: "math/0101196"
    doi: null
  - title: "C^1 isometric imbeddings"
    url: "https://doi.org/10.2307/1969840"
    year: 1954
    arxiv: null
    doi: "10.2307/1969840"
  - title: "The Euler equations as a differential inclusion"
    url: "https://arxiv.org/abs/math/0511502"
    year: 2009
    arxiv: "math/0511502"
    doi: null
see:
  - "1051-finite-element-exterior-calculus"
  - "1052-discrete-conformal-maps-and-ideal-hyperbolic-polyhedra"
  - "1149-min-max-theory-and-the-willmore-conjecture"
---

# h-Principle and Rigidity for C^{1,α} Isometric Embeddings

## One-sentence takeaway

A crumpled paper can be C¹ isometric; raise the Hölder exponent past 2/3 and rigidity slams shut — convex integration versus the Weyl problem, with Onsager's conjecture sitting in the same commutator.

## Why it's lovely

Why you might love this: Nash–Kuiper says you can C¹-isometrically embed a torus into an arbitrarily small ball in R³. C² is rigid (Weyl). Conti–De Lellis–Székelyhidi locate the cliff: convex integration still works for C^{1,α} with α small, while Borisov rigidity holds for α > 2/3. The same mollification-and-commutator estimates that prove Onsager for Euler appear here for the Gauss curvature. Geometry leftover of Willmore: flexibility of maps, not min-max of surfaces.

## Problem

An isometric immersion is a map whose derivative pulls the Euclidean metric back to a given Riemannian metric. In codimension one the PDE is overdetermined. Nash–Kuiper produce a C¹ zoo of solutions by corrugation; classical differential geometry says a C² ovaloid is unique. Where, in Hölder regularity, does the h-principle die and rigidity begin?

## Design

A stage of the Nash–Kuiper iteration is rewritten with a smoothing operator (Nash–Moser style) so only finitely many derivatives are tracked. Metric error is split into primitive rank-one terms; each step adds a corrugation. Commutator estimates on convolutions control the quadratic error. The same estimates, run backwards, show that C^{1,α} isometric immersions with α > 2/3 of positively curved surfaces have bounded extrinsic curvature, hence are convex and rigid.

## Evidence

Local and global h-principles for α below an explicit dimensional threshold; rigidity theorem for α > 2/3 recovering Borisov with a short analytic proof. Proceedings: Abel Symposia vol. 7 (2012) 83–116. Holonomic approximation (Eliashberg–Mishachev) is cited, not minted — toolkit, not the crumpled-sphere paper. The Euler-as-differential-inclusion sibling is cited, not minted.

## Limitations

The exact critical exponent is still not known (1/2 is the folklore guess; 2/3 is the rigidity side). Codimension one. Not a visualization of the C¹ flat torus (Borrelli–Jabrane–Lazarus–Thibert is a sibling, not on arXiv in this sweep).

## Implications for Broadside

Willmore (1149) is the least-wrinkled torus; this is the most-wrinkled isometric torus. FEEC and discrete conformal maps (1051, 1052) live on the rigid, high-regularity side. Convex integration is the other end of the same geometric pipeline: when the PDE is treated as a differential inclusion, wild maps exist, and they look like turbulence.

## Bottom line

Mint the cliff. Flexibility below, rigidity above, Onsager in the commutator.

## Links

- arXiv: [0905.0370](https://arxiv.org/abs/0905.0370)
- Proceedings: Abel Symposia vol. 7 (2012) 83–116