Flat tori in three-dimensional space and convex integration

Vincent Borrellia,1, Saïd Jabranea, Francis Lazarusb, and Boris Thibertc

aInstitut Camille Jordan, Université Lyon I, 69622 Villeurbanne, France; bCentre National de la Recherche Scientifique, Laboratoire Grenoble Image Parole Signal Automatique, 38402 Grenoble, France; and cLaboratoire Jean Kuntzmann, Université de Grenoble, 38041 Grenoble, France

Edited by Yakov Eliashberg, Stanford University, Stanford, CA, and approved March 9, 2012 (received for review November 9, 2011)

It is well-known that the curvature tensor is an isometric invariant and Mishachev (5). To prove that the h-principle holds in many of C 2 Riemannian manifolds. This invariant is at the origin of the situations, Gromov introduced several powerful methods for rigidity observed in Riemannian geometry. In the mid 1950s, Nash solving partial differential relations. One of which, convex inte- amazed the world mathematical community by showing that this gration theory (5–7), provides a quasi-constructive way to build rigidity breaks down in regularity C 1. This unexpected flexibility sequences of embeddings converging toward isometric embed- has many paradoxical consequences, one of them is the existence dings. Nevertheless, because of its broad purpose, this theory of C 1 isometric embeddings of flat tori into Euclidean three-dimen- remains far too generic to allow for a precise description of the sional space. In the 1970s and 1980s, M. Gromov, revisiting Nash’s resulting map. results introduced convex integration theory offering a general In this article, we convert convex integration theory into an framework to solve this type of geometric problems. In this re- explicit algorithm. We then provide an implementation leading search, we convert convex integration theory into an algorithm to images of an embedded square flat torus in three-dimensional that produces isometric maps of flat tori. We provide an implemen- space. This visualization has led us in turn to discover a unique tation of a convex integration process leading to images of an em- geometric structure. This structure, described in the corrugation bedding of a flat torus. The resulting surface reveals a C 1 fractal theorem below, reveals a remarkable property: The normal vec- structure: Although the tangent plane is defined everywhere, tor exhibits a fractal behavior. the normal vector exhibits a fractal behavior. Isometric embeddings MATHEMATICS of flat tori may thus appear as a geometric occurrence of a structure General Strategy that is simultaneously C 1 and fractal. Beyond these results, our The general strategy (1) starts with a strictly short embedding, i.e., implementation demonstrates that convex integration, a theory an embedding of the square torus that strictly shrinks distances. still confined to specialists, can produce computationally tractable To build an isometric embedding, this initial map is corrugated solutions of partial differential relations. along the meridians with the purpose of increasing their length (Fig. 1). This corrugation is performed while keeping a strictly geometric torus is a surface of revolution generated by short map, achieving a smaller isometric default in the vertical Arevolving a circle in three-dimensional space about an axis direction. The isometric perturbation in the horizontal direction coplanar with the circle. The standard parametrization of a geo- is also kept under control by choosing the number of oscillations metric torus maps horizontal and vertical lines of a unit square to sufficiently large. Corrugations are then applied repeatedly in latitudes and meridians of the image surface. This unit square can various directions to produce a sequence of maps, reducing step- also be seen as a torus; the top line is abstractly identified with the by-step the isometric default. Most importantly, the sequence of bottom line and so are the left and right sides. Because of its local oscillation numbers can be chosen so that the limit map achieves Euclidean geometry, it is called a square flat torus. The standard a continuously differentiable isometry onto its image. parametrization now appears as a map from a square flat torus One-Dimensional Convex Integration. The above general strategy into the three-dimensional space having as its image a geometric leaves a considerable latitude in generating the corrugations. This torus. Although natural, this map distorts the distances: The great flexibility is one of the surprises of the Nash–Kuiper result lengths of latitudes vary whereas the lengths of the corresponding because it produces a plethora of solutions to the isometric em- horizontal lines on the square remain constant. bedding problem. It is a remarkable fact that Gromov’s convex It was a long-held belief that this defect could not be fixed. integration theory provides both the deep reason of the presence In other words, it was presumed that no isometric embedding of corrugations and the analytic recipe to produce them. Never- of the square flat torus—a differentiable injective map that pre- theless, this theory does not give preference to any particular serves distances—could exist into three-dimensional space. In the corrugation and is not constructive in that respect. Here we refine mid 1950s Nash (1) and Kuiper (2) amazed the world mathema- the traditional analytic approach of corrugations, adding a geo- tical community by showing that such an embedding actually metric point of view. exists. However, their proof relies on an intricated construction A corrugation is primarily a one-dimensional process. It aims that makes it difficult to analyze the properties of the isometric 3 to produce, from an initial regular smooth curve f 0 : ½0; 1 → E embedding. In particular, these atypical embeddings have never (as usual E3 denotes the three-dimensional Euclidean space), been visualized. One strong motivation for such a visualization is a new curve f whose speed is equal to a given function the unusual regularity of the embedding: A continuously differ- 0 r : ½0; 1 → R with r>‖f ‖E 3 . In the general framework of con- entiable map that cannot be enhanced to be twice continuously þ 0 vex integration (5, 7), one starts with a one-parameter family of differentiable. As a consequence, the image surface is smooth enough to have a tangent plane everywhere, but not sufficient to admit extrinsic curvatures. Author contributions: V.B., S.J., F.L., and B.T. performed research and wrote the paper. In the 1970s and 1980s, Gromov, revisiting the results of Nash The authors declare no conflict of interest. and others such as Phillips, Smale, or Hirsch, extracted the under- This article is a PNAS Direct Submission. lying notion of their works: the h-principle (3, 4). This principle 1To whom correspondence should be addressed. E-mail: vincent.borrelli@math states that many partial differential relation problems reduce to .univ-lyon1.fr. ’ purely topological questions. The raison d être of this counterin- This article contains supporting information online at www.pnas.org/lookup/suppl/ tuitive phenomenon was later brought to light by Eliashberg doi:10.1073/pnas.1118478109/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1118478109 PNAS Early Edition ∣ 1of6 Downloaded by guest on September 27, 2021 Z Z Z 1 1 u þ j 1 u þ j Sj ¼ h ;u du ¼ h ;u dudx: 0 0 N N Ij N

1 ∂ ‖ − ‖ 3 ≤ ‖ h ‖ 0 It ensues that Sj sj E N 2 ∂t C . The lemma then follows 2 ‖ − ‖ 3 ≤ ‖ ‖ 0 from the obvious inequalities Sn sn E N h C and ‖ − ‖ 3 ≤ ∑n ‖ − ‖ 3 fðtÞ f 0ðtÞ E j¼0 Sj sj E .

Here we set hðt; uÞ ≔ rðtÞeiαðtÞ cos 2πu; [3]

0 iθ ≔ θ t θ n t ≔ f 0 n 0 1 → E3 where e cos þ sin with ‖ 0‖ , : ½ ; is a f 0 E 3 smooth unit vector field normal to the initial curve and the function α is determined by the barycentric condition [1]. We claim that our convex integration formula captures the natural geometric notion of a corrugation. Indeed, if the initial curve f 0 is planar then the signed curvature measure Fig. 1. The first four corrugations. 0 μ ≔ kds ¼ kðtÞ‖f ðtÞ‖E 3 dt loops h : ½0; 1 × R∕Z → E3 satisfying the isometric condition of the resulting curve is connected to the signed curvature mea- μ ≔ ‖hðt; uÞ‖E 3 ¼ rðtÞ, for all ðt; uÞ ∈ ½0; 1 × R∕Z, and the bary- sure 0 k0ds of the initial curve by the following simple formula centric condition Z 0 1 μ ≔ μ0 þ½α cosð2πNtÞ − 2πNα sinð2πNtÞdt: 0 ∀t ∈ ½0; 1;f0ðtÞ¼ hðt; uÞdu: [1] 0 Our corrugation thus modifies the curvature in the simplest way by sine and cosine terms with frequency N. 0 This last condition expresses the derivative f 0ðtÞ as the barycenter of the loop hðt; ·Þ. One then chooses the number N of oscillations Two-Dimensional Convex Integration. The classical extension of of the corrugated map f and set convex integration to the two-dimensional case consists in apply- Z t ing the one-dimensional process to a one-parameter family of fðtÞ ≔ f 0ð0Þþ hðu; NuÞdu: [2] curves that foliates a two-dimensional domain. Given a strictly 0 2 2 3 short smooth embedding f 0 : E ∕Z → E of the square flat 2 2 2 Here, Nu must be considered modulo Z. It appears that not only torus, a nowhere vanishing vector field W : E ∕Z → E , and 0 ≔ E2∕Z2 → R ‖f ðtÞ‖E 3 ¼ ‖hðt; NtÞ‖E 3 ¼ rðtÞ as desired, but f can also be made a function r þ, the aim is to produce a smooth E2∕Z2 → E3 arbitrarily close to the initial curve f 0, see Fig. 2. map f : whose derivative in the direction W has the target norm r. The natural generalization of our one-dimen- Lemma 1. We have sional process leads to the following formula: Z 1 ∂ s h φ ≔ φ iθðφðt;uÞ;uÞ [4] ‖f − f 0‖ 0 ≤ 2‖h‖ 0 þ ‖ ‖ 0 ; fð ðt; sÞÞ f 0ðtV Þþ rð ðt; uÞÞe du; C N C ∂t C 0 0 ‖ ‖ 0 ‖ ‖ 3 φ where g C ¼ supp∈D gðpÞ E denotes the C norm of a function where ðt; sÞ denotes the point reached at time s by the flow of W g : D → E3. issuing from tV . The vector V is chosen so that the line of initial conditions RV ⊂ E2∕Z2 is a simple closed curve transverse to the iθ θ t θ n t Proof: Let t ∈ ½0; 1. We put n ≔ ½Nt (the integer part of Nt) and flow. We also use the notation e ¼ cos þ sin , where is 1 j jþ 0 ≤ ≤ − 1 n the normalized derivative of f 0 along W and n is a unit normal to set Ij ¼½N ; N for j n and In ¼½N ;t. We write the embedding f 0. Similarly as above, θðq; uÞ ≔ αðqÞ cos 2πNu, α 2 2 n n is determined by the barycentric condition [1] and q ∈ E ∕Z . − 0 − 0 fðtÞ fð Þ¼ Sj and f 0ðtÞ f 0ð Þ¼ sj The resulting map f is formally defined over a cylinder. In general ∑ ∑ 2 2 j¼0 j¼0 f does not descend to the flat square torus E ∕Z . This defect is rectified by adding a term that smoothly spreads out the gap pre- 1 with S ≔∫ hðv; NvÞdv and s ≔∫ ∫ 0 hðx; uÞdudx. By the venting the map to be doubly periodic. j Ij j Ij change of variables u ¼ Nv − j, we get for each j ∈ ½0;n− 1 Basis of the Embeddings Sequence. The iterated process leading to an isometric embedding requires to start with a strictly short em- bedding of the square flat torus E2∕Z2 → E3 f init : :

The metric distortion induced by f init is measured by a field of bilinear forms Δ : E2∕Z2 → ðE2 ⊗ E2Þ obtained as the point- wise difference:

Δ ≔ 2 − 3 ð·; ·Þ h·; ·iE f inith·; ·iE : Fig. 2. The black curve is corrugated with nine oscillations. Note that the 3 3 right endpoints of the curves do not coincide. The corrugated gray curve As usual, f inith·; ·iE ¼hdf initð·Þ;dfinitð·ÞiE denotes the pullback can be made arbitrarily close to the black curve by increasing the number of the Euclidean inner product by f init. Notice that f init is strictly of oscillations. short if and only if the isometric default Δ is a metric, i.e., a map

2of6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1118478109 Borrelli et al. Downloaded by guest on September 27, 2021 − 3 ρ ℓ ⊗ ℓ ρ ℓ ⊗ ℓ ρ ℓ ⊗ ℓ from the square flat torus into the positive cone of inner products gk f k;0h·; ·iE ¼ k;1 1 1 þ k;2 2 2 þ k;3 3 3 of the plane. The convexity of this cone implies the existence of [6] ℓ … ℓ ≥ 3 linear forms of the plane 1; ; S, S , such that ρ S (the k;js being positive functions). We would like to build a map Δ ρ ℓ ⊗ ℓ − 3 ¼ ∑ j j j f k;1 with the requirement that its isometric default gk f k;1h·; ·iE j¼1 is roughly equal to the sum of the last two terms ρ 2ℓ2 ⊗ ℓ2 þ ρ 3ℓ3 ⊗ ℓ3. To this end we introduce the inter- ρ k; k; for nonnegative functions j. By a convenient choice of the initial mediary metric ℓ map f init and of the js, the number S can be set to three. In prac- ℓ ≔ ∕‖ ‖ 2 tice we set the linear forms ð·Þ hUðjÞ UðjÞ ; ·iE to the nor- μ ≔ 3 ρ ℓ ⊗ ℓ [7] j k;1 f 0h·; ·iE þ k;1 1 1; malized duals of the following constant vector fields: k;

1 1 and observe that the above requirement amounts to ask that f k;1 Uð1Þ ≔ e1;Uð2Þ ≔ ðe1 þ 2e2Þ;Uð3Þ ≔ ðe1 − 2e2Þ; μ 5 5 is quasi-isometric for k;1. Although natural, it turns out that per- forming a two-dimensional convex integration along the constant 2 where e1;e2 is the canonical basis of E . For later use, we set 1 μ ð Þ vector field Uð Þ does not produce a quasi-isometric map for k;1. Instead, we consider the following nonconstant vector field: V ð1Þ ≔ e2;Vð2Þ ≔−2e1 þ e2;Vð3Þ ≔ 2e1 þ e2: ≔ 1 ζ 1 Note that the parallelogram spanned by UðjÞ and V ðjÞ is a fun- W k;1 Uð Þþ k;1V ð Þ; damental domain for the Z2 action over E2. As an initial map we ζ choose the standard parametrization of a geometric torus. It is where the scalar k;1 is such that the field W k;1 is orthogonal to 1 μ easy to check that the range of the isometric default Δ lies inside V ð Þ for the metric k;1. With this choice the integral curves φ R 1 E2∕Z2 the positive cone ðt; :Þ of W k;1 issuing from the line V ð Þ of define a diffeomorphism φ : R∕Z × ½0; 1 → ðR∕ZÞVð1Þ × ½0; 1Uð1Þ. 3 We now build a new map F 1 by applying to f 0 a two-dimen- C ρ ℓ ⊗ ℓ ρ 0 ρ 0 ρ 0 k; k; ¼ ∑ j j jj 1 > ; 2 > ; 3 > sional convex integration (see Eq. 4) along the integral curves j¼1 φðt; :Þ, i.e., MATHEMATICS spanned by the ℓ ⊗ ℓ s, j ∈ f1; 2; 3g, whenever the sum of the Z j j s φ ≔ 1 φ iθðφðt;uÞ;uÞ [8] minor and major radii of this geometric torus is strictly less than Fk;1ð ðt; sÞÞ f k;0ðtV ð ÞÞ þ rð ðt; uÞÞe du: one (Fig. S1). 0 We define a sequence of metrics ðg Þ converging toward k k∈N μ the Euclidean inner product, The isometric condition in the direction W k;1 for the metric k;1 ‖ ‖2 μ 8 is dFk;1ðW k;1Þ E 3 ¼ k;1ðW k;1;Wk;1Þ. By differentiating [ ] ≔ 3 δ Δ [5] gk f h·; ·iE þ k ; ‖ ‖2 2 init with respectp toffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis we get dFk;1ðW k;1Þ E 3 ¼ r , hence we must δ 1 − −γk γ 0 μ with k ¼ e for some fixed > . We then construct a se- choose r ¼ k;1ðW k;1;Wk;1Þ. Furthermore, the map f k;0 is μ quence of maps ðf kÞk∈N such that every f k is quasi-isometric for strictly short for k;1 because 3 ≈ gk, i.e., f h·; ·iE gk. In other words, each f k, seen as a map from k 2 ‖ ‖2 ρ ‖ℓ 1 ‖2 ‖ ‖2 the square flat torus to Euclidean three space, has an isometric r ¼ df k;0ðW k;1Þ E 3 þ k;1 1ðUð ÞÞ E 2 > df k;0ðW k;1Þ E 3 : default approximately equal to e−γkΔ. θ ≔ α 2π The map f k is obtained from f k−1 by a succession of corruga- We finally set ðq; uÞ ðqÞ cos Nk;1u, where Nk;1 is the fre- ℓ 1 …ℓ tions. Precisely, if Sk linear forms k; ; k;Sk are needed for the quency of our corrugations. convex decomposition of the difference Note that the map Fk;1 is properly defined over a cylinder, but

Sk does not descend to the torus in general. We eventually glue the − 3 ρ ℓ ⊗ ℓ gk f −1h·; ·iE ¼ k;j k;j k;j; two cylinder boundaries with the following formula, leading to a k ∑ E2∕Z2 j¼1 map f k;1 defined over ,

then Sk convex integrations will also be needed to (approxi- ∘ φ ≔ ∘ φ − − ∘ φ 1 [9] ρ ∈ 1 … f k;1 ðt; sÞ Fk;1 ðt; sÞ wðsÞ · ðFk;1 f k;0Þ ðt; Þ; mately) cancel every coefficient k;j, j f ; ;Skg. As a key point of our implementation, we manage to set each number where w : ½0; 1 → ½0; 1 is a smooth S-shaped function satisfying Sk to three and to keep unchanged our initial set of linear forms 0 0 1 1 ðkÞ 0 ðkÞ 1 0 ∈ N ℓ ℓ ℓ ℓ ℓ ℓ wð Þ¼ , wð Þ¼ , and w ð Þ¼w ð Þ¼ for all k . f k;1; k;2; k;3g¼f 1; 2; 3g. We therefore generate a se- To cancel the last two terms in [6], we apply two more corruga- quence μ tions in a similar way. For every j, the intermediary metric k;j involves f k;j−1 and the jth coefficient of the isometric default f ; f 1 1;f1 2;f1 3; f 2 1;f2 2;f2 3; … init ; ; ; ; ; ; ≔ − 3 Dk;j gk f k;j−1h·; ·iE . Notice that the three resulting maps f 1, f 2, and f 3 are completely determined by their numbers with an infinite succession of three terms blocks. Each map is the k; k; k; of corrugations N 1, N 2, and N 3. result of a two-dimensional convex integration process applied to k; k; k; the preceding term of the sequence. We eventually obtain the de- ∈ 1 2 3 ≔ ∈ N Theorem 1. For j f ; ; g, there exists a constant Ck;j independent sired sequence of maps, setting f k f k;3 for k . of Nk;j (but depending on f k;j−1 and its derivatives) such that

Ck;j Reduction of the Isometric Default. The aim of each convex integra- ‖ − ‖ 0 ≤ i. f k;j f k;j−1 C ρ ρ ρ Nk;j tion process is to reduce one of the coefficients k;1, k;2,or k;3 pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ck;j ‖ − ‖ 0 ≤ 7 ‖ρ ‖ 0 without increasing the two others. This goal is achieved with a ii. df k;j df k;j−1 C N þ k;j C careful choice for the field of directions along which we apply k;j Ck;j the corrugations. Suppose we are given a map f 0 ≔ f −1 3 whose ‖μ − 3 ‖ 0 ≤ k; k ; iii. k;j f k;jh·; ·iE C : C Nk;j isometric default with respect to gk lies inside the cone :

Borrelli et al. PNAS Early Edition ∣ 3of6 Downloaded by guest on September 27, 2021 0 The first point ensures that f k;j is C close to f k;j−1, whereas this constraint, we introduce some more notations. Let the second point keeps the increase of the differentials under ρ Δ ≔ ρ ρ ρ control and the last point guarantees that f k;j is quasi-isometric minð Þ min f 1ðpÞ; 2ðpÞ; 3ðpÞg; μ p∈E 2∕Z 2 for k;j. ρ 9 where the js are, as above, the coefficients of the decomposition Main arguments of the proof: We deduce from [ ] that Δ ℓ ⊗ ℓ of over the j js. We also denote by errk;j the norm ‖μ − 3 ‖ 0 ‖ − ‖ 0 ≤ 2‖ ∘ φ − ∘ φ‖ 0 k;j f h:; :iR C of the isometric default of f k;j. By Theorem f k;j f k;j−1 C Fk;j f k;j−1 C : k;j 1, this number can be made as small as we want provided that the ≔ ∘ number of corrugations Nk;j is chosen large enough. We then apply Lemma 1 to the right-hand side with f Fk;j φðt; ·Þ and f 0 ≔ f −1 ∘ φðt; ·Þ to obtain (i). For (ii), it is sufficient k;j Lemma 2. If ‖ − ‖ 0 to bound df k;jðXÞ df k;j−1ðXÞ C for X ¼ VðjÞ and X ¼ W k;j. ∂φ pffiffiffi Because ∂t ¼ cφV ðjÞ for some nonvanishing function cφ,the 15 3 ∂ ∘φ δ − δ ρ Δ ðf k;j Þ ðerrk;1 þ errk;2 þ errk;3Þ < ð kþ1 kÞ minð Þ; ‖ − ‖ 0 ‖ − 8 norm df k;jðV ðjÞÞ df k;j−1ðV ðjÞÞ C is bounded by ∂t ∂ ∘φ ðf k;j−1 Þ ‖ 0 , up to a multiplicative constant. It is readily seen that ∂t C ≔ − 3 C ∂ ∘φ then D gkþ1 f 3h:; :iR lies inside . ðf k;j Þ k; ∂t results from a convex integration process applied to ∂ ∘φ ∂ ∘φ ∂ ∘φ ðf k;j−1 Þ ðf k;j Þ ðf k;j−1 Þ ‖ − ‖ 0 Proof: We want to show that ρ ðDÞ > 0 for j ∈ f1; 2; 3g.Let ∂t and Lemma 1 shows that ∂t ∂t C ¼ j 1 9 B ≔ g − f 3h:; :iR 3 . Because D ¼ðδ 1 − δ ÞΔ þ B,wehave Oð Þ.ForX ¼ W k;j, we differentiate [ ]withrespecttos k k; kþ k Nk;j by linearity of the decomposition coefficient ρ : and obtain j ρ ρ δ − δ Δ ρ ≥ δ − δ ρ Δ ρ jðDÞ¼ jðð kþ1 kÞ Þþ jðBÞ ð kþ1 kÞ minð Þþ jðBÞ: ‖ − ‖ 0 ≤ ‖ Ψ ‖ 0 0 0 ‖Ψ‖ 0 df k;jðW k;jÞ df k;j−1ðW k;jÞ C d ðW k;jÞ C þjw jC C [10]

In particular, the condition ‖ρ ðBÞ‖ < ðδ 1 − δ Þρ ðΔÞ implies with Ψ ≔ F − f −1. We bound the second term of the right- j kþ k min k;j k;j ρ 0 1 jðDÞ > . Now, it follows by some easy linear algebra that hand side as for (i). Let J0ðαÞ ≔∫0 cosðα cos 2πuÞdu be the Bessel function of 0 order. On the one hand, for every nonnega- pffiffiffi α 1 α − 5 3 tive lower than the first positive root of J0 we have: þ J0ð Þ maxf‖ρ1ðBÞ‖ 0 ; ‖ρ2ðBÞ‖ 0 ; ‖ρ3ðBÞ‖ 0 g ≤ ‖B‖ 0 2 α α ≤ 7 1 − α ‖ Ψ ‖2 C C C 8 C J0ð Þ cosð Þ ½ J0ð Þ. On the other hand, d ðW k;jÞ E 3 ¼ 2 2 r þr0 −2rr0 cosðαcosð2πN sÞÞ,wherer0 ≔ ‖df −1ðW Þ‖E 3 . k;j k;j k;j and a computation shows that ‖B‖ 0 ≤ 3ðerr1 þ err2 þ err3Þ. α t C Because df k;j−1ðW k;jÞ¼rJ0ð Þ we obtain We are now in position to choose the corrugation numbers, and doing so, to settle a complete description of the sequence 2 2 2 ‖ Ψ ‖ ≤ 1 α − 2 α α ðf Þ ∈N ∈ 1 2 3 . d ðW k;jÞ E 3 r ½ þ J0ð Þ J0ð Þ cosð Þ k;j k ;j f ; ; g 2 2 2 2 ≤ 7r ½1 − J0ðαÞ ¼7ðr − r0 Þ: Choice of the Corrugation Numbers. Let ϵ > 0. At each step, we choose the corrugation number Nk;j large enough so that the fol- pffiffiffi 1∕2 lowing three conditions hold (j ∈ f1; 2; 3g): From [7], we deduce ‖dΨðW Þ‖ 0 ≤ 7‖UðjÞ‖E 2 ‖ρ ‖ 0 , k;j C k;j C ϵ ‖ − −1‖ 0 ≤ hence (ii). Once the differential of Ψ is under control, (iii)re- i. f k;j f k;j C 3.2 k pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 pffiffiffiffiffi 1 duces to a meticulous computation of the coefficients of 2 2 ii. ‖df − df −1‖ 0 ≤ δ − δ −1‖Δ‖ 0 þ 35‖ρ ðD Þ‖ 0 k;j k;j C k k C j k;j C μ − f ·; · 3 in the basis W ;V j . k;j k;jh iE ð k;j ð ÞÞ 4 pffiffi δ − δ ρ Δ iii. errk;j < 45 3 ð kþ1 kÞ minð Þ: Corrugation Numbers − 3 Here we have put, similarly as above, Dk;j ¼ gk f k;j−1h·; ·iE . We make a repeated use of Theorem 1 to show that the map f k;3 0 The first condition ensures the C closeness of f ∞ to f init. Thanks is quasi-isometric for gk and strictly short for gkþ1. Thereby, the 0 to Lemma 2, the third condition implies that the isometric default whole process can be iterated. Moreover, the C control of the − 3 C gkþ1 f k;3h·; ·iE lies inside the cone . It can be shown to also maps and the differentials, as provided by Theorem 1, allows in μ μ turn to control the C0 and C1 convergences of the sequence imply that the intermediary bilinear forms k;2 and k;3 are me- trics, an essential property to apply the convex integration process ðf k;3Þk∈N . With a suitable choice of the Nk;js, this sequence can 1 1 1 be made C converging, thus producing a C isometric map f ∞ in to f k;1 and f k;2. Finally, we can prove the C convergence of the the limit. By the C0 control of the sequence we also obtain a C0 resulting sequence with the help of the second condition. ϵ 0 Note that at each step, the map f is ensured to be a C1 em- density property: Given > , the Nk;js can be chosen so that k;j bedding if Nk;j is chosen large enough. This property follows 1 ‖ − ‖ 0 ≤ ϵ f ∞ f init C : from the two conditions (i) and (ii), because a C immersion, which is C1 close to a C1 embedding, must be an embedding.

1 Our Isometric Constraint. Compared to Nash’s and Kuiper’s proofs, C Fractal Structure we have an extra constraint. For each integer k the isometric de- The recursive definition of the sequence paves the way for a geo- − 3 C metric understanding of its limit. Because the resulting embed- fault gkþ1 f 3h·; ·iE must lie inside the convex hull spanned ℓ ⊗ ℓk; ∈ 1 2 3 ding is C1 and not C2, this geometry consists merely of the by the j j, j f ; ; g. The reason for this constraint is to avoid the numerous local gluings required by Nash’s and Kuiper’s behavior of its tangent planes or, equivalently, of the properties v proofs. Using a single chart substantially simplifies the implemen- of its Gauss map. We denote by k;j the normalized derivative ℓ n tation. More importantly, keeping the same linear forms j all of f k;j in the direction VðjÞ and by k;j the unit normal to f k;j. v⊥ ≔ v × n through the process enlightens the recursive structure of the We also set k;j k;j k;j. Obviously, there exists a matrix C ∈ 3 solution that was hidden in the previous methods. To deal with k;j SOð Þ such that

4of6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1118478109 Borrelli et al. Downloaded by guest on September 27, 2021 ⊥ t ⊥ t Y∞ ðv v n Þ ¼ C · ðv v −1 n −1Þ : k;j k;j k;j k;j k;j−1 k;j k;j ≔ 1 α 2π nðxÞ ½ þ k cosð NkxÞ; k¼1 Here, ðabcÞt stands for the transpose of the matrix with column a b c C α vectors , , and . We call k;j a corrugation matrix because it where ð kÞk∈N and ðNkÞk∈N are two given sequences. It is a fact (8) that an exponential growth of N , known as Hadamard’s la- encodes the effect of one corrugation on the map f k;j−1. Despite k its natural and simple definition, the corrugation matrix has cunary condition, results in a fractional Hausdorff dimension of n intricate coefficients with integro-differential expressions. The si- the Riesz measure nðxÞdx. A similar result for the normal ∞ of the embedding of the flat square torus seems hard to obtain. It is tuation is further complicated by some technicalities such as the likely that the graph of the Gauss map n∞ has Hausdorff dimen- elaborated direction field of the corrugation or the final stitching sion strictly larger than two. Yet, because the limit map is of the map used to descend to the torus. Remarkably, all these a continuously differentiable isometry onto its image, the em- difficulties vanish when considering the dominant terms of the bedded flat torus has Hausdorff dimension two. C two parts of a specific splitting of k;j. Implementation of the Convex Integration Theory C ∈ 3 Theorem 2. [Corrugation Theorem] The matrix k;j SOð Þ The above convex integration process provides us with an algo- can be expressed as the product of two orthogonal matrices rithmic solution to the isometric embedding problem for square L R k;j · k;j−1 where flat tori. This algorithm has for initial data three numbers ∈ N ϵ 0 γ 0 E2∕Z2 → E3 0 1 K , > , > , and a map f init : for which θ 0 θ the isometric default Δ is lying inside the cone C. From f a cos k;j sin k;j 1 init L @ 010A finite sequence of maps ðf Þ ∈ 1 ∈ 1 2 3 is iteratively con- k;j ¼ þ O k;j k f ;:::;Kg;j f ; ; g − θ 0 θ Nk;j sin k;j cos k;j structed. Each map f k;j is built from the map f k;j−1 by first apply- ing the convex integration formula [8] to obtain an intermediary 9 and map Fk;j. The gluing formula [ ] is further applied to Fk;j result- ∘ φ φ 0 1 ing in the composition f k;j , where is the flow of the vector β β 0 cos j sin j field W k;j. We finally get f k;j by composing with the inverse map R @ − β β 0 A ε γ k;j ¼ sin j cos j þ Oð k;jÞ of the flow. The number rules the amplitude of the isometric 001 default of each f 3 via the formula [5]. Formulas [8] and [9] are

k; MATHEMATICS completely explicit except for the corrugation number Nk;j which

ε ≔ ‖ 2 − 3 ‖ 2 is to be determined so that f fulfills the postconditions ex- and where k;j h:; :iE f k;jh:; :iE E is the norm of the iso- k;j β 1 metric default, j is the angle between V ðjÞ and V ðj þ Þ, and, pressed in the above choice of the corrugation numbers. Because θ α 2π there is no available formula, we obtain by binary search the smal- as above, k;jðp; uÞ¼ k;jðpÞ cos Nk;ju. lest integer Nk;j that satisfies these postconditions. The algorithm R v⊥ stops when the map f K;3 is constructed. Note that Theorem 1 in- Main arguments of the proof: The matrix k;j−1 maps ð k;j−1 v n t n × t n t sures that the algorithm terminates. The map f 3 satisfies k;j−1 k;j−1Þ to ð k;j−1 k;j−1 k;j−1 k;j−1Þ where k;j−1 K; ‖ − ‖ 0 ≤ ϵ f K;3 f init C and its isometric default is less than is the normalized derivative of f k;j−1 in the direction W k;j 3 − γ 1 K ‖Δ‖ 0 2 e C . Moreover, the limit f ∞ of the f K;3sisaC isometric (Fig. S2). This last vector field converges toward UðjÞ when the ϵ ‖ − ‖ 0 ≤ R map and f ∞ f K;3 C 2 K . Therefore, the algorithm produces isometric default tends to zero. Hence, k;j−1 reduces to a rota- an approximation f 3 of a solution of the underdetermined par- tion matrix of the tangent plane that maps V ðj − 1Þ to V ðjÞ. The K; L tial differential system for isometric maps, matrix k;j accounts for the corrugation along the flow lines.       ‖ − ∂ ∂ ∂ ∂ ∂ ∂ From the proof of Theorem 1 (ii)wehave df k;jðV ðjÞÞ f f 1 f f 0 f f 1 1 1 ∂ ; ∂ ¼ ; ∂ ; ∂ ¼ and ∂ ; ∂ ¼ ; df −1ðV ðjÞÞ‖E 3 ¼ Oð Þ. Therefore, modulo Oð Þ, the trans- x x x y y y k;j Nk;j Nk;j versal effect of a corrugation is not visible. In other words, a cor- 0 and this approximation is C close to the initial map f init. rugation reduces at this scale to a purely one-dimensional We have implemented the above algorithm to obtain a visua- phenomenon. Hence the simple expression of the dominant part lization of a flat torus shown in Fig. 3. The initial map f init is the of this matrix. Notice also that Theorem 1 (i)impliesthatthe standard parametrization of the torus of revolution with minor 1 1 perturbations induced by the stitching are not visible as well. and major radii, respectively, 10π and 4π. Each map of the se- n ≔ × The Gauss map ∞ of the limit embedding f ∞ lim f k;3 can quence ðf k;jÞk∈f1;:::;Kg;j∈f1;2;3g is encoded by a n n grid whose k→∞ ∕ ∕ node i1, i2 contains the coordinates of f k;jði1 n; i2 nÞ. Flows be expressed very simply by means of the corrugation matrices: and integrals are common numerical operations for which we have used Hairer’s solver (9) based on DOPRI5 for nonstiff dif- ∀k ∈ N ; ferential equations. To invert the flow φ of W we take advan- k;j Y∞ Y3 tage of the fact that the UðjÞ component of W k;j is constant. The nt 001 C v⊥ v n t R ∞ ¼ð Þ · ℓ;j · ð 0 k;0 k;0Þ : line V ðjÞ of initial conditions is thus carried parallel to itself k; i1 i2 ℓ 1 φ ¼k j¼ along the flow. It follows that the points ðn V ðjÞ; nÞ of a uniform sampling of the flow are covered by n lines running parallel to the The Corrugation Theorem gives the key to understand this initial conditions. We now observe that the same set of lines also × E2∕Z2 infinite product. It shows that asymptotically the terms of this covers the nodes of the n n uniform grid over . This last product resemble each other, only the amplitudes α , the fre- observation leads to a simple linear time algorithm for inverting k;j the flow. It is worth noting that the integrand in Eq. 8 essentially quencies Nk;j, and the directions are changing. In particular, depends upon the first order derivatives of f k;j−1 and upon the the Gauss map n∞ shows an asymptotic self-similarity: The accu- corrugation frequency Nk;j. The derivatives are accurately esti- mulation of corrugations creates a fractal structure. mated with a finite difference formula of order four. Regarding It should be noted that there is a clear formal similarity be- the corrugation frequency, we have observed a rapid growth as n γ 0 1 tween the infinite product defining ∞ and, in a one-dimensional from the four first values of Nk;j. For instance, for ¼ . ,we setting, the well-known Riesz products, have obtained the following:

Borrelli et al. PNAS Early Edition ∣ 5of6 Downloaded by guest on September 27, 2021 thus not be visible to the naked eye. To illustrate the metric im- provement we have compared the lengths of a collection of mer- idians, parallels, and diagonals on the flat torus with the lengths of their images by f 2;1. The length of any curve in the collection differs by at most 10.2% with the length of its image. By contrast, the deviation reaches 80% when the standard torus f init is taken in place of f 2;1. In practice, calculations were performed on a 8-core cpu (3.16 GHz) with 32 GB of RAM and parallelized C++ code. We used a 10;0002 grid mesh (n ¼ 10;000) for the three first cor- rugations and refined the grid to 2 milliards nodes for the last corrugation. Because of memory limitations, the last mesh was divided into 33 pieces. We then had to render each piece with a ray tracer software and to combine the resulting images into a single one as in Fig. 3. The computation of the final mesh took approximately 2 h. Two extra days were needed for the final im- age rendering with the ray tracer software Yafaray (10). Conclusion and Perspective Convex integration is a major theoretical tool for solving under- determined systems (4). After a substantial simplification, we obtained the implementation of a convex integration process, providing images of a flat torus. This visualization led us in turn to discover a geometric structure that combines the usual differ- entiability found in Riemannian geometry with the self-similarity of fractal objects. This C1 fractal structure is captured by an infinite product of corrugation matrices. In some way, these cor- Fig. 3. The image of a square flat torus by a C 1 isometric map. Views are rugations constitute an efficient and natural answer to the curva- from the outside and from the inside. ture obstruction (11) observed in the introduction, leading to an atypical solution. A similar process occurs in weak solutions of the Euler equation (12) and could be present in other natural phenomena (13, 14). N1 1 ¼ 611;N1 2 ¼ 69;311;N1 3 ¼ 20;914;595; ; ; ; Despite its high power, convex integration theory remains 6 572 411 478 N2;1 ¼ ; ; ; . relatively unknown to nonspecialists (15). We hope that our im- plementation will help to popularize this technique and will open In fact, computing these values is already a challenge because a a door to applications ranging from other isometric immersions, reasonable resolution of 10 grid samples per period of corrugation such as hyperbolic compact spaces, to solutions of underdeter- would require a grid with ð10 × 6;572;411;478Þ2 ≈ 4.31019 mined systems of nonlinear partial differential equations. Convex nodes. We have restricted the computations to a small neighbor- integration theory could emerge in a near future as a major op- 2 2 hood of the origin ð0; 0Þ ∈ E ∕Z to obtain the above values. erating tool in a very large spectrum of applied sciences. These values are therefore lower bounds with respect to the post- conditions for the choice of Nk;j. However, these postconditions ACKNOWLEDGMENTS. We thank Jean-Pierre Kahane for pointing out Riesz are only sufficient and we were able to reduce these numbers for products, Thierry Dumont for helpful discussions on mathematical softwares, the four first steps to, respectively, 12, 80, 500, and 9,000 after a Damien Rohmer for valuable advice on graphics rendering and the Calcul In- number of trials over the entire grid mesh (see Figs. 1 and 3). The tensif/Modélisation/Expérimentation Numérique et Technologique (CIMENT) project for providing access to their computing platform. Research partially pointwise displacement between the last map f 2;1 and the limit iso- supported by Région Rhône-Alpes [Projet Blanc-Créativité-Innovation metric map could hardly be detected as the amplitudes of the next (CIBLE)] and Centre National de la Recherche Scientifique [Projet Exploratoire corrugations decrease dramatically. Further corrugations would Premier Soutien (PEPS)].

1. Nash J (1954) C 1-isometric imbeddings. Ann Math 60:383–396. 9. Hairer E, Norsett SP, Wanner G (1993) Solving Ordinary Differential Equations I, Non- 1 2. Kuiper N (1955) On C -isometric imbeddings. Indag Math 17:545–556. stiff Problems, Series in Computational Mathematics (Springer, Berlin), 2nd Ed., 8. 3. Gromov M (1970) A topological technique for the construction of solutions of differ- 10. Yafaray (2010) Yet Another Free RAYtracer, http://www.yafaray.org, Version – ential equations and inequalities. Proc Intl Cong Math 2:221 225. 0.1.2 beta. 4. Gromov M (1986) Partial Differential Relations (Springer, Berlin). 11. Han Q, Hong JX (2006) Isometric Embedding of Riemannian Manifolds in Euclidean 5. Eliashberg Y, Mishachev N (2002) Introduction to the h-Principle, Graduate Studies in Spaces, Mathematical Surveys and Monographs (Am Math Soc, Providence, RI), 130. Mathematics (Am Math Soc, Providence, RI), 48. 12. Villani C (2010) Le paradoxe de Scheffer–Shnirelman revu sous l’angle de l’intégration 6. Geiges H (2003) h-Principle and Flexibility in Geometry, Memoirs of the American – Mathematical Society (Am Math Soc, Providence, RI), 164. convexe. Séminaire Bourbaki 332:101 134. – 7. Spring D (1998) Convex Integration Theory, Monographs in Mathematics (Birkhäuser, 13. Griffin LD (1994) The intrisic geometry of the cerebral cortex. J Theor Biol 166:261 273. Basel), 92. 14. Liang H, Mahadevan L (2009) The shape of a long leaf. Proc Natl Acad Sci USA 8. Kahane JP (2010) Jacques Peyrière et les produits de Riesz., http://arXiv.org/abs/1003. 106:22049–22054. 4600v1. 15. Berger M (2000) Encounter with a geometer, part I. Not Am Math Soc 47:183–194.

6of6 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1118478109 Borrelli et al. Downloaded by guest on September 27, 2021