3 The normal map

The Nash-Kuiper process for curves

Vincent Borrelli, Sa¨ıd Jabrane, Francis Lazarus and Boris Thibert January 18, 2013

An isometric immersion of a Riemannian manifold into an Euclidean space is a C¹ map f : (M^m, g) −→ E^q = (R^q,h., .i) such that f^∗h., .i = g.

Such a map preserves the length of curves that is:

Length(f ◦γ) =Length(γ)

for every rectifiable curve γ : [a, b] −→ M^m. In a local coordinate system x= (x1, ..., xm) the isometric condition gives rise to a system ofsm= ^m(m+1)₂ equations

1≤i≤j≤n,

∂f

∂xi

(x), ∂f

∂xj

(x)

=gx

∂

∂xi

, ∂

∂xj

.

Thus generically, isometric maps are expected to exist –at least locally– if the target space has dimension greater thans_m. In 1926, Janet [10] proved that any analytic Riemannian surface (M², g) admit local isometric analytic immersions in E^s2. Shortly after, this result was generalized by Cartan [5]

for analytic Riemannian manifold of dimensionm: local isometric analytic maps do exist if the dimension of the target Euclidean space is at leastsm. Thirty years after the result of Janet, J. Nash showed in an outstanding ar- ticle [15] that everyC^kRiemannian manifold (3≤k≤+∞) can be mapped C^k isometrically into an Euclidean space E^q with q = 3s_m+ 4m if M^m is compact andq = (m+1)(3sm+4m) if not. This result was then improved by Gromov [12] and G¨unther [9] who proved thatq= max{s_m+2m, s_m+m+5}

is enough for the compact case.

Another amazing result of J. Nash is the discovery that, in a C¹ setting, the barrier formed by the Janet dimension can be completely destroyed: in the compact case, if a Riemannian manifold admits an immersion into some E^q, q ≥m+ 1, then it admits a C¹ isometric immersion into the same Eu- clidean space (Nash [14] proved the caseq≥m+ 2 and Kuiper [13] the case

q =m+ 1). As a consequence every compact riemannian surface admits a C¹ isometric immersion inE³ but in general, for obvious curvature reasons, the immersion can not be enhanced to be C².

Beyond the breaking of the dimensional barrier, there is another phenomenon which is utterly baffling in the Nash-Kuiper result: not only C¹ isometric maps do exist but they are plentiful! In fact, there is a C¹ isometric map near every strictly short map. A mapf₀ : (M^m, g)−→ E^q is called strictly shortif it strictly shortens distances, that is, if the difference g−f₀^∗h., .i is a metric. The Nash-Kuiper approach reveals that if f0 is a strictly short embedding, then for every > 0 there exists a C¹ isometric embedding f : (M^m, g)−→E^q such that

kf−f0k_C0 ≤

wherek.k_C0 denotes the supremum norm overM^m(this manifold is assumed to be compact for the simplicity of the presentation). For instance, for every >0,there is a C¹ isometric embedding of the unit sphere inside a ball of radius.

Recently [4], we have converted the Nash-Kuiper proof into an algorithm, using the Gromov convex integration theory ([12], [16], [7]). We have imple- mented this algorithm and produced numerical pictures of a C¹ isometric embeddingf∞ of the square flat torusE²/Z² insideE³ that isC⁰ close to a strictly short embeddingf₀ ofE²/Z² as a torus of revolution. Our algorithm generates a sequence of maps

f₀, f_1,1, f_1,2, f_1,3, f_2,1, f_2,2, f_2,3, ...

defined recursively thatC¹ converges towardf∞.The geometry of the limit map consists merely of the behavior of its tangent planes or, equivalently, of the properties of its Gauss mapn∞:E²/Z−→ S² ⊂E³. From the algo- rithm, one can extract a formal expression of that Gauss map as an infinite product ofcorrugation matricesapplied to the initial Gauss map off₀.One major obstacle to the understanding of n∞ lies in the inherent complexity of the coefficients of these corrugation matrices. The main theorem of [4]

(the Corrugation Theorem) describes their asymptotic behaviour.

In this article, we propose to study the normal map of isometric maps result- ing from a convex integration process in the simpler situation of isometric immersions of the circle E/Z into E². In this case, the isometric problem

in itself is totally trivial but the way the Nash-Kuiper process solves it, produces a sequence of curves

f0, f1, f2, ...

whose limit f∞ has a non trivial geometry. Of course, in that one dimen- sional setting, some of the difficulties inherent to the dimension two vanish.

In particular, if the initial curve f₀ :E/Z−→ E² 'C is parametrized with constant speed and is radially symmetric (see the definition below) all com- putations can be completely carried out and lead to an explicit formula for the normal mapn∞ of the limit curve f∞.

Theorem 1.– Let n_k be the normal map of f_k.We have

∀x∈E/Z, nk(x) =e^{iαkcos(2πNkx)}nk−1(x)

where α_k∈]0,^π₂[ is the amplitude of the loop used in the convex integration to build fk−1 from fk and Nk ∈ 2N^∗ is the number of corrugations of fk

(precise definitions below). In particular, the normal mapn∞of f∞ has the following expression

∀x∈E/Z, n∞(x) =

+∞

Y

k=1

e^{iαkcos(2πNkx)}

! n0(x).

The above expression of the normal mapn∞ is reminiscent of aRiesz prod- uct, that is a product of the form

h(x) =

+∞

Y

k=1

(1 +α_kcos(2πN_kx)).

It is a fact that an exponential growth ofN_k, known as Hadamard’s lacunary condition, results in a fractional Hausdorff dimension of the Riesz measure¹ µ:=h(x)dx[11].

The normal mapn∞can be thought of as a 1-periodic map fromRtoC.In

§3 we perform its Fourier series expansion. Its spectrum, whose structure is very similar to the spectrum of a Riesz product, is obtained as a limit of an

1Letdimsupµ(resp. diminfµ) denotes the supremum (resp. the infimum) of the Haus- dorff dimension of the Borel sets of positiveµ-measure. Ifd=dimsupµ=diminfµ then the measureµis said to have Hausdorff dimensiond.

iterative process starting with the spectrum of the initial mapn0.Precisely, let

∀x∈E/Z, n_k(x) =X

p∈Z

a_p(k)e^2iπpx

denotes the Fourier series expansion of the normal mapn_k. We derive from the above theorem the following inductive formula (cf. Lemma 3):

Fourier series expansion of n_k.–We have

∀p∈Z, a_p(k) =X

n∈Z

u_n(k)ap−nN_k(k−1) where un(k) =iⁿJn(αk).

In the above formula, J_n denotes the Bessel function of order n(see [1] or [17]):

α7−→Jn(α) = 1 π

Z π

0

cos(nu−αsinu)du.

The Fourier expansion of n_k gives the key to understand the construction of the spectrum (ap(k))p∈Z from the spectrum (ap(k−1))p∈Z. The k-th spectrum is obtained by collecting an infinite number of shifts of the previous spectrum. The n-th shift is of amplitude nN_k and weighted by u_n(k) = iⁿJn(αk).Since

|J_n(α_k)| ↓0

the weight is decreasing withn(see the figure of§3).

In the Nash-Kuiper process there is a infinite number of degrees of freedom in the construction of the sequence (f_k)k∈N.In particular, given any sequence of positive numbers (δ_k)k∈N increasing toward 1, the process produces a sequence such that

kf_k⁰ −f_k−1⁰ k_C0 ≤C^tep

δ_k−δk−1. Thus, if

X pδ_k−δk−1<+∞

the sequence (f_k)k∈Nis C¹ converging toward a C¹ limitf∞.Moreover if X pδ_k−δk−1N_k <+∞

thenf∞is C² (see Proposition 5). Regarding the intermediary regularities, we prove the following:

Theorem 2.– Assume that

X pδ_k−δk−1 <+∞ and X p

δ_k−δk−1N_k= +∞.

Let 0< η <1 andS_k :=P_k

l=1

pδ_l−δl−1N_l. If X(δ_k−δk−1)^1−η2 S^η_k <+∞

thenf∞ is C^1,η.

In the simplified one dimensional approach followed in this article, the se- quence (Nk)k∈N can be chosen freely. This is no longer possible in the general case: some constraints appear that force the N_ks to be increasing.

The control of the growth of the N_ks is then the key to understand the C^1,η regularity of the limit map. In the original proof of Nash, the chosen sequence forδ_k was 1−2^−(k+1).For such a choice, the numerical result we have obtained for the square flat torus seems to suggest that the sequence (Nk)k∈Nis exponentially growing (see also the theoretical arguments of [6]).

This gives the motivation for the following corollary.

Corollary 3.– Let 0< γ <1 and δk:= 1−e^−γ(k+1). If there exists β >0 such that

∀k∈N, N_k≤N₀e^βk thenf∞ is C^1,η for any η >0 such that

η < γ 2β.

The question of the C^1,η regularity of isometric maps resulting from the Nash-Kuiper process is addressed in [2], [3] and [6]. The optimalC^1,η reg- ularity of an isometric immersion of a Riemannian surface in E³ is still an open question.

1 The convex integration process for curves

The convex integration process.–Letf₀: [0,1]→R² be aC^∞map and let

h: [0,1] −→ C^∞(R/Z,C) x 7−→ h(x, .)

be a C^∞ family of loops such that

∀x∈[0,1], Z 1

0

h(x, s)ds=f₀⁰(x).

LetN ∈N^∗the any natural number. We define a newC^∞mapf : [0,1]−→

R² by the formula

∀x∈[0,1], f(x) :=f₀(0) + Z x

0

h(s,{N s})ds

where{N s}denotes the fractional part ofN s.We call such a formula giving a new mapf from the data off0 andh aconvex integration. We sometimes write

f :=IC(f₀, h, N).

The new mapf has a derivative whose image obviously lies inside the image ofh since

∀x∈[0,1], f⁰(x) =h(x,{N x}).

Moreover,f remainsC⁰ close tof₀.Indeed, it can be shown that kf−f0k_C0 =O

1 N

(see [4] for instance).

Curves with given speeds.– Letf₀ : [0,1]−→E²'Cbe a regular curve (∀x∈[0,1], f₀⁰(x)6= 0) and let r: [0,1]−→R^∗₊ be anyC^∞ map such that

∀x∈[0,1], r(x)>kf₀⁰(x)k.

Leth defined by

h(x, s) :=r(x) (cos(α(x) cos 2πs)t0(x) + sin(α(x) cos 2πs)n0(x)) witht0 := _kf^f000

0k,n0:=it0 and α(x)∈]0, κ[ is such that r(x)J₀(α(x)) =kf₀⁰(x)k

where J0 denotes the Bessel function of the first kind and of order 0 and κ'2.4 denotes the first zero ofJ₀.Since the Bessel functionJ₀ is decreasing

on the interval [0, κ] and J0(0) = 1, there is a unique α(x) that solves the above implicite equation. Note that

Z 1 0

h(x, s)ds=r(x)J₀(α(x))t₀(x)

therefore the above implicit condition on α(x) implies that the average of h(x, .) is f₀⁰(x). The map f obtained by convex integration from f0 and h has speedkf⁰kequal to the given functionrand is arbitrarilyC⁰ close tof₀. Closed curves with given speeds.– Iff0is defined overE/Zrather than [0,1] the curvef obtained fromf₀ andh by convex integration is not closed in general. This defect can be easily corrected by the following modification of the convex integration formula:

∀x∈[0,1], f(x) :=f0(0) + Z x

0

h(s,{N s})ds−x Z 1

0

h(s,{N s})ds.

For short we writef :=IC(ff ₀, h, N).The C⁰ closeness implies that

Z 1 0

h(x, s)ds

=O 1

N

so that the correction can be made arbitrarily small. We still have kf − f0k_C0 =O _N¹

but nowkf⁰k is only approximately equal tor(x),precisely

∀x∈E/Z, kf⁰(x)k=

h(x,{N x})− Z 1

0

h(s,{N s})ds

and therefore, for all x∈E/Z,we have |kf⁰(x)k −r(x)|=O _N¹ .

Nash and Kuiper process.– In the spirit of the Nash and Kuiper proof, the way to obtain a mapf :E/Z−→E²'C with speed the given function r is to produce a sequence of closed curves (f_k)k∈N^∗ by iteratively applying the modifying convex integration formula so that to reduce step by step the isometric defaultr− kf₀⁰k.

Let (δk)k∈N^∗ be a sequence of increasing positive number converging to- ward 1, we set

∀k∈N^∗,∀x∈E/Z, r²_k(x) :=kf₀⁰(x)k²+δ_k r²(x)− kf₀⁰(x)k² .

Note that for everyx ∈E/Z, the sequence rk(x) is increasing towardr(x).

We definefk to be IC(ff k−1, hk, Nk) with

hk(x, s) :=rk(x)e^iαk(x) cos 2πstk−1(x) whereα_k(x) =J₀⁻¹_kf⁰

k−1(x)k rk(x)

andtk−1is the normalized derivative offk−1. Eachfk has a speed which is approximatelyrk:

kf_k⁰(x)k −rk(x) =O

1 N_k

.

Since the sequencer_k(x) is strictly increasing for everyx∈E/Z,the number Nk can be chosen large enough such that

∀x∈E/Z, r_k+1(x)>kf_k(x)k.

This is crucial to definef_k+1 asIC(ff _k, h_k+1, N_k+1).If the sequence (δ_k)k∈N^∗

is chosen so that

X pδ_k−δ_k−1<+∞

and if (N_k)k∈N^∗is rapidly diverging then the sequencef_k:=ICf(fk−1, h_k, N_k) isC¹ converging toward aC¹ limitf∞with speed kf_∞⁰ k=r. This is proven further in the text in the particular case of closed curves with constant speed.

The general case, slightly more technical in nature, is left to the reader.

Closed curves with constant speed.– From now on, in order to get the most pleasant computations we consider the simplified case wherer≡1 and f₀:E/Z→E² is aC^∞ map such that:

• (Cond1) it is of constant speedr₀:=kf₀⁰k<1

• (Cond2) it is radially symmetric, that is: f₀⁰(x+¹₂) =−f₀⁰(x).

In all what follows, we will also assume that the Nash-Kuiper sequence of C^∞maps derived from f₀:

fk:=IC(ff k−1, hk, Nk), k∈N^∗

is such that hk(x, s) = rke^iαk(x) cos 2πstk−1(x) and Nk ∈ 2N^∗. Note that (Cond1) implies that every functionr_k =p

r²₀+δ_k(1−r₀²) is constant.

Proposition 1.– For every k∈N^∗, fk is of constant speed rk and radially symmetric. In particular,

f_k =IC(fk−1, h_k, N_k).

The functionsαk are also constant and equal to J₀⁻¹ _r

k−1

r_k

.

Proof.– By induction. Assume thatfk−1 satisfies (Cond1) and (Cond2).

In particular fk−1 is of constant speed rk−1 and thus the function αk = J₀⁻¹

rk−1

rk

is constant. Since N_k∈2N^∗, we have h_k(x+1

2,{N_k(x+1

2)}) =−h_k(x,{N_kx}) and consequently

Z 1 0

hk(s,{N s})ds= 0.

It ensues that

IC(fk−1, h_k, N_k) =IC(ff k−1, h_k, N_k)

and therefore f_k is of constant speed kf_k⁰(x)k =kh_k(x,{N x})k = r_k. It is also radially symmetric sincefk(x) =h(x,{N_kx}). .

· · ·

The convex integration process applied to circle (left f0, center f1,right f_∞)

2 C

convergence

It turns out that the sequence (δ_k)k∈N mainly determines the sequence (α_k)_k∈N.

Lemma 1 (Amplitude Lemma).– We have α_k∼

q

2(1−r²₀)p

δ_k−δk−1

where ∼denotes the equivalence of sequences. We also have αk ≤ 1

r0

q

2(1−r₀²)p

δk−δk−1.

Proof.– By definition αk = J₀⁻¹(^rk−1_r

k ). Recall that the Taylor expansion ofJ0(α) up to order 2 is

w= 1− α²

4 +o(α²).

Lety= 1−w andX =α², we havey= ^X₄ +o(X) thus X= 4y+o(y) and soX∼4y. We finally get

α∼2√

1−w and α_k∼2 r

1− rk−1

r_k . Sincer²₀+ (1−r₀²) = 1,we have

r²_k=r²₀+δ_k(1−r₀²) = 1 + (δ_k−1)(1−r²₀) so

r²_k−r²_k−1= (δ_k−δk−1)(1−r₀²) and

1−r_k−1²

r_k² = (δ_k−δk−1)(1−r²₀)

1−(1−δk)(1−r²₀) ∼(δ_k−δk−1)(1−r²₀).

In an other hand 1−r²_k−1

r²_k =

1−rk−1

r_k 1 +rk−1

r_k

∼2

1−rk−1

r_k

.

Thus

1−rk−1

r_k

∼ 1

2(δ_k−δk−1)(1−r₀²).

and

α_k∼2 r

1−rk−1

rk

∼ q

2(1−r₀²)p

δ_k−δ_k−1.

The Taylor expansion ofJ0 up to order 4 shows that w≤1−α²

4 +α⁴ 64 =

1− α²

8 2

(because it is alternating) and hence α²_k ≤8

1−

rrk−1

r_k

. Thus

α²_k ≤ 8

√rk

√rk−√ rk−1

≤ 8

√r_k(√ r_k+√

rk−1)(rk−rk−1)

≤ 8

√rk(√ rk+√

rk−1)(rk+rk−1) r²_k−r²_k−1

≤ 2

r₀² r_k²−r_k−1² sincer0 < rk−1 < rk.We deduce

αk≤ 1 r0

q

2(r²_k−r²_k−1) = 1 r0

q

2(1−r₀²)p

δk−δk−1.

Let (A_k)k∈N^∗ be the sequence of functions defined by

∀x∈E/Z, A_k(x) :=

k

X

l=1

α_lcos(2πN_lx).

Lemma 2.– For every x∈E/Z,we have:

f_k⁰(x) =e^iAk(x)rk

r₀f₀⁰(x).

Proof.– Letnk−1 :=itk−1.From

f_k⁰(x) = r_k(cos(α_kcos(2πN_kx))tk−1(x) + sin(α_kcos(2πN_kx))nk−1(x))

= rke^{iαkcos(2πNkx)} 1 rk−1

f_k−1⁰ (x) we deduce by induction : f_k⁰(x) =e^iAk(x)r_r^k

0f₀⁰(x).

Proposition 2.– If P p

δk−δk−1 <+∞ then

i) the sequence(A_k)k∈Nis normally converging andA∞:= limk→+∞A_k is continuous.

ii) the sequence (fk)k∈N^∗ is C¹ converging toward f∞ := limk→+∞fk

and

∀x∈R/Z, f_∞⁰ (x) =e^iA∞(x) 1 r₀f₀⁰(x).

Proof.– From the Amplitude Lemma we deduce that Xα_k<+∞

thus the sequence (A_k)k∈N is normally converging and A∞:= lim

k→+∞A_k is continuous. Moreover, from the relation

f_k⁰(x) =e^iAk(x)r_k r₀f₀⁰(x) we also deduce that f_k⁰

k∈N is normally converging toward e^iA∞(x)1

r0

f₀⁰(x).

Since (f_k(0))k∈N obviously converges, we obtain that the sequence (f_k)k∈N

isC¹ converging towardf∞:= limk→+∞f_k.

Corollary 1.– Let γ >0and δ_k:= 1−e^−γ(k+1).Then sequence (δ_k)k∈N^∗ is increasing toward 1 andp

δ_k−δk−1∼√

δ₀e^−γ2k.In particular, f∞ isC¹.

3 The normal map

From now on, we assume

X pδ_k−δk−1<+∞

so that the sequence (f_k)k∈N is C¹ converging toward its limit f∞. The following theorem is a straightforward consequence of the results of the pre- ceding section:

Theorem 1.– Let nk be the normal map of fk.We have

∀x∈E/Z, n_k(x) =e^{iαkcos(2πNkx)}n_k−1(x)

In particular, the normal mapn∞ of f∞ has the following expression

∀x∈E/Z, n∞(x) =e^iA∞(x)n0(x).

We deduce from this theorem the following result about Fourier expansion ofn_k.

Lemma 3 (Fourier expansion of n_k).– For all k∈Nwe denote by

∀x∈E/Z, n_k(x) =X

p∈Z

ap(k)e^2iπpx the Fourier expansion ofn_k. We have

∀p∈Z, a_p(k) =X

n∈Z

u_n(k)ap−nN_k(k−1)

where un(k) =iⁿJn(α_k) (Jn denotes the Bessel function of order n).

Proof.– From the Jacobi-Anger identity e^izcosθ =

+∞

X

n=−∞

iⁿJn(z)e^inθ we deduce

e^{iαkcos(2πNkx)}=

+∞

X

n=−∞

iⁿJ_n(α_k)e^2iπnNkx =

+∞

X

n=−∞

u_n(k)e^2iπnNkx. Since the Fourier coefficients of a product of two fonctions are given by the discrete convolution product of their coefficients, the product

n_k(x) =e^{iαkcos(2πNkx)}nk−1(x) can be written

n_k(x) =

+∞

X

n=−∞

un(k)e^2iπnNkx

! _+∞

X

p=−∞

ap(k−1)e^2iπpx

!

=

+∞

X

p=−∞

+∞

X

n=−∞

un(k)ap−nNk(k−1)

! e^2iπpx.

Therefore

a_p(k) =

+∞

X

n=−∞

u_n(k)ap−nN_k(k−1).

A schematic picture of the various spectra (ap(k))_p∈Z withNk =b^k. Remark.– The analogy with Riesz products suggests that the Hausdorff dimension of the graph of the normal map n∞ could be fractional. Note that the relevant part of this map is the 1-periodic function

R3x7−→A∞(x) =

+∞

X

k=1

α_kcos(2πN_kx)∈R.

In the simple case where α_k = a^k and N_k = b^k with 0 < a < 1 < b , the mapA∞ is a Weiestrass function². Ifab >1,it is known that its graph has a fractional Hausdorff dimension. The exact value of this dimension is still an open question. It is believed to be equal to 2 +^ln_ln^a_b (see [8]).

4 C

regularity

Proposition 3.– We have

kf_k⁰ −f_k−1⁰ k_C0 ≤Cte1

pδk−δk−1

2Cf. Lemma 1 and the lines above Corollary 3 in the introductory part of this article for a motivation for such choice forαk andNk.

withCte1=p

7(1−r²₀).

Proof.– For every pointx∈E/Z, we have

kf_k⁰ −f_k−1⁰ k² =kf_k⁰k²+kf_k−1⁰ k²−2kf_k⁰kkf_k−1⁰ kcos(α_kcos 2πN_kx) sinceα_kcos 2πN_kxis the angle betweenf_k⁰(x) andf_k−1⁰ (x).An upper bound for this angle isα_k =J₀⁻¹(w) wherew=rk−1/r_k∈]0,1[ since

r_k =kf_k⁰(x)k and rk−1=kf_k−1⁰ (x)k.

Recall that from the Amplitude Lemma we have the following inequality α_k²

2 ≤4(1−√ w).

By using the upper boundα_k, we obtain

kf_k⁰ −f_k−1⁰ k² ≤ r_k²+r_k−1² −2rk−1r_kcosα_k

≤ r_k²−r_k−1² + 2rk−1(rk−1−r_kcosα_k).

Since

cosα_k≥1−α_k² 2 we have

rk−1(rk−1−r_kcosα_k) ≤ r_k−1² −r_krk−1+rk−1r_kα²_k 2

≤ r_k−1² −r_krk−1+ 4rk−1r_k

1−

rrk−1

r_k

≤ r_k−1² + 3rk−1rk−4rk−1

√rkrk−1

≤ r_k−1² + 3r²_k−4rk−1

q

r_k−1² ( since rk−1< r_k)

≤ 3(r²_k−r²_k−1).

Therefore

kf_k⁰ −f_k−1⁰ k² ≤7 kf_k⁰k²− kf_k−1⁰ k² . Now

kf_k⁰k²− kf_k−1⁰ k² = r²_k−r_k−1²

= (δ_k−δk−1)(1−r₀²).

Finally

kf_k⁰ −f_k−1⁰ k_C0 ≤Cte₁p

δ_k−δk−1

withCte1 =p

7(1−r²₀).

For everyk ∈N, we denote by M_k(g) the supremum over E/Z of the k-th derivative g^(k) of g : E/Z −→ C (if k = 0, it is understood that g⁽⁰⁾ = g) and we definekgk_Ck to be the sumM₀(g) +...+M_k(g).

Corollary 2.– We have

kf_k−fk−1k_C1 ≤2Cte1

pδ_k−δk−1

withCte1=p

7(1−r²₀).

Proof.– From the theorem we deduce by a mere integration kf_k−fk−1k_C0 ≤Cte1

pδk−δk−1

thus the result since

kf_k−fk−1k_C1 =kf_k−fk−1k_C0+M1(f_k−fk−1).

Proposition 4.– For every x∈E/Z,we have

f_k⁰⁰(x) = (−2πα_kN_ksin 2πN_kx+r_k−1scal_k−1(x))ir_ke^{iαkcos 2πNkx}t_k−1(x) where scalk denotes the signed curvature offk.Moreover

r_kscal_k(x) =r₀scal₀(x)−2π

k

X

l=1

α_lN_lsin(2πN_lx).

Proof.– We have f_k⁰⁰(x) = ∂

∂x r_ke^{iαkcos 2πNkx}tk−1(x)

= ∂

∂x(rk(cos(αkcos 2πNkx)tk−1(x) + sin(αkcos 2πNkx)nk−1(x))

= r_k rk−1

∂

∂x cos(α_kcos 2πN_kx)f_k−1⁰ (x) + sin(α_kcos 2πN_kx)if_k−1⁰ (x)

= −2iπα_kN_ksin(2πN_kx)r_ke^{iαkcos 2πNkx}tk−1(x) + r_k

rk−1

cos(α_kcos 2πN_kx)f_k−1⁰⁰ (x) + sin(α_kcos 2πN_kx)if_k−1⁰⁰ (x) Sincefk−1 is of constant speedrk−1 we have

f_k−1⁰⁰ (x) =rk−1scalk−1(x)if_k−1⁰ (x)

therefore

f_k⁰⁰(x) = −2iπα_kNksin(2πNkx)rke^{iαkcos 2πNkx}tk−1(x) +r_krk−1scalk−1(x)ie^{iαkcos 2πNkx}tk−1(x) . Finally,

f_k⁰⁰(x) = (−2πα_kN_ksin 2πN_kx+rk−1scalk−1(x))ir_ke^{iαkcos 2πNkx}tk−1(x).

Becausef_k is of constant arc length we also have

f_k⁰⁰(x) =rkscalk(x)if_k⁰(x) =rkscalk(x)irke^{iαkcos 2πNkx}tk−1(x).

From this we deduce

r_kscal_k(x) =rk−1scalk−1(x)−2πα_kN_ksin(2πN_kx) and by induction

r_kscal_k(x) =r₀scal₀(x)−2π

k

X

l=1

α_lN_lsin(2πN_lx).

Proposition 5.– If P

k∈N^∗

pδ_k−δk−1N_k<+∞ then f∞ is C².

Proof.– Since we already know that the sequence (f_k)k∈NC¹ converges, it is enough to prove that (f_k⁰⁰)_k∈

Nis a Cauchy sequence. From f_k⁰⁰(x) =r_kscal_k(x)if_k⁰(x)

we deduce

kf_k⁰⁰(x)−f_k−1⁰⁰ (x)k_C0 ≤ kr_kscal_k(x)f_k⁰(x)−rk−1scalk−1(x)f_k−1⁰ (x)k_C0

≤ kr_k−1scalk−1(x)f_k⁰(x)−rk−1scalk−1(x)f_k−1⁰ (x)k_C0

+|r_kscal_k(x)−rk−1scalk−1(x)|kf_k⁰(x)k_C0

≤ rk−1|scal_k−1(x)|kf_k⁰(x)−f_k−1⁰ (x)k_C0

+r_k|r_kscal_k(x)−r_k−1scal_k−1(x)|.

Since

rkscalk(x) =r0scal0(x)−2π

k

X

l=1

αlNlsin(2πNlx)

we have

|r_kscal_k(x)−rk−1scalk−1(x)| ≤2πα_kN_k and

rk|scal_k(x)| ≤r0|scal₀(x)|+ 2π X

l∈N^∗

αlNl.

In particular ther_k|scal_k(x)|are uniformly bounded by M :=kr₀scal0(x)k_C0+ 2π X

k∈N^∗

αkNk.

Note thatM <+∞.Indeedα_k∼p

2(1−r²₀)p

δ_k−δk−1 therefore X

k∈N^∗

pδ_k−δk−1N_k<+∞ =⇒ X

k∈N^∗

α_kN_k <+∞.

We deduce

kf_k⁰⁰(x)−f_k−1⁰⁰ (x)k_C0 ≤Mkf_k⁰(x)−f_k−1⁰ (x)k_C0+ 2παkNk. Letp < q, we thus have

kf_q⁰⁰(x)−f_p⁰⁰(x)k_C0 ≤ M

q

X

k=p

pδk−δk−1+ 2π

q

X

k=p

αkNk

≤ M

∞

X

k=p

pδ_k−δk−1+ 2π

∞

X

k=p

α_kN_k.

Hence f_k⁰⁰

k∈N is a Cauchy sequence.

Theorem 2.– Assume that

X pδ_k−δk−1 <+∞ and X p

δ_k−δk−1N_k= +∞.

Let 0< η <1 andS_k :=Pk l=1

pδ_l−δ_l−1N_l. If

X(δ_k−δk−1)^1−η2 S^η_k <+∞

thenf∞ is C^1,η.

Proof.– Let 0< η <1.We are going to use the interpolation inequality kfk_C1,η ≤C^tekfk^1−η

C¹ kfk^η

C²

to show that (kf_k−fk−1k_C1,η)k∈N^∗ is a Cauchy sequence. From the above sections, we have

kf_k−fk−1k_C1 ≤2Cte1

pδ_k−δk−1

and

M2(fk−fk−1) ≤ M2(fk) +M2(fk−1)

≤ M₀(r_k scal_k)M₁(f_k) +M₀(rk−1 scalk−1)M₁(f_k)

≤ M₀(scal_k) +M₀(scalk−1)

≤ 2M0(scal0) + 4π

k

X

l=1

α_lN_l.

From the Amplitude Lemma we deduce M₂(f_k−fk−1) ≤ 2M₀(scal₀) +^4π

√

2(1−r²₀) r0

P_k

l=1

pδ_l−δl−1N_l

≤ 2M₀(scal₀) +^4π

√2(1−r²₀) r0 S_k. So

kf_k−f_k−1k_C2 ≤2Cte₁p

δ_k−δ_k−1+ 2M₀(scal₀) +4πp

2(1−r²₀) r0

S_k.

Since lim_k→+∞S_k= +∞, for klarge enough we have kf_k−fk−1k_C2 ≤Cte₂S_k. for some constantCte₂. We now have

kf_k−f_k−1k^1−η_C1 kf_k−f_k−1k^η_C2 ≤Cte₃(δ_k−δ_k−1)^1−η2 S_k^η

withCte₃ = (2Cte₁)^1−ηCte^η₂.

Corollary 3.– Let 0< γ <1 and δ_k:= 1−e^−γ(k+1). If there exists β >0 such that

∀k∈N, Nk≤N0e^βk thenf∞ is C^1,η for any η >0 such that

η < γ 2β.

Proof.– We have

δ_k−δk−1 =δ₀e^−γk thus

S_k=

k

X

l=1

pδ_l−δ_l−1N_l≤p δ₀N₀

k

X

l=1

e^(β−γ2)l<p

δ₀N₀e^β−γ21−e(^β−γ2)^(k+1) 1−e^β−γ2 Suppose first thatβ > γ

2.We then have : S_k≤Cte₄e^(β−γ2)k. Finally

(δ_k−δk−1)^1−η2 S_k^η ≤Cte5e^{−γ1−η2 k}e^{η(β−γ2)k}. Now

−γ1−η 2 +η

β−γ 2

<0 if and only if

η < γ 2β. Therefore, under that condition

X(δ_k−δk−1)^1−η2 S^η_k <+∞

hence the corollary in the case whereβ > γ

2.We left to the reader the easier caseβ ≤ γ

2.

References

[1] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, 1965.

[2] J. Borisov,C^1,α-isometric immersions of Riemmannian spaces, Dokl.

Akad. Nauk. SSSR 163 (1965), 11-13, translation in Soviet. Math. Dokl.

6 (1965), 869-871.

[3] J. Borisov, Irregular C^1,β-surfaces with an analytic metric, Siberian Math. J. 45, 1 (2004), 19-52.

[4] V. Borrelli, S. Jabrane, F. Lazarus, B. Thibert, Flat tori in three-dimensional space and convex integration, PNAS 2012, vol 109, 7218-7223.

[5] E. Cartan,Sur la possibilit´e de plonger un espace riemannien donn´e dans un espace euclidien, Ann. Soc. Pol. Math., 6 (1927), 1-7.

[6] S. Conti, C. De Lellis and L. Szkelyhidi,H-principle and rigidity forC^1,α-isometric embeddings, to appear in the Proceedings of the Abel Symposium 2010, arXiv:0905.0370v1.

[7] Y. Eliahsberg, N. Mishachev, Introduction to the h-principle, Graduate Studies in Mathematics, Vol 48, AMS, Providence, 2002.

[8] K. Falconer,Fractal Geometry, Wiley, 2003.

[9] M. G¨unther,On the pertubation problem associated to isometric em- beddings of Riemannian manifolds, Ann. Global Anal. Geom. 7(1989), 69-77.

[10] M. Janet, Sur la possibilit´e de plonger un espace riemannien donn´e dans un espace euclidien, Ann. Soc. Pol. Math., 5 (1926), 38-43.

[11] J.-P. Kahane, Jacques Peyri`ere et les produits de Riesz, arXiv.org/abs/1003.4600v1.

[12] M. Gromov, Partial Differential Relations, Ergebnisse der Mathe- matik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 9. Springer-Verlag, Berlin, 1986.

[13] N. Kuiper,OnC¹-isometric imbeddings, Indag. Math., 17 (1955), 545- 556.

[14] J. Nash,C¹-isometric imbeddings, Ann. of Math. (2), 60 (1954), 383- 396.

[15] J. Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (1), 63 (1956), 20-63.

[16] D. Spring,Convex Integration Theory, Bikhauser, 1998.

[17] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cam- bridge University Press, 1995.