Julien Roth
email: julien.roth@u-pem.fr

A NEW RESULT ABOUT ALMOST UMBILICAL HYPERSURFACES OF REAL SPACE FORMS

Keywords: 2010 Mathematics Subject Classification. 53A42, 53C20, 53C21 Almost-umbilical hypersurfaces, space forms, constant scalar curvature. 1

In this short note, we prove that an almost umbilical compact hypersurface of a real space form with almost Codazzi umbilicity tensor is embedded, diffeomorphic and quasiisometric to a round sphere. Then, we derive a new characterization of geodesic spheres in space forms.

Introduction

Let (M n , g) be a connected and oriented compact Riemannian manifold isometrically immersed into the simply-connected real space form M n+1 (δ) of constant curvature δ. Let B be the second fundamental form of the hypersurface and H its mean curvature. Since we consider only hypersurfaces, we take B as the real-valued second fundamental form. We denote by τ = B -Hg the traceless part of the second fundamental form, also called umbilicity tensor. We say that M is totally umbilical if τ = 0.

It is a well-known fact that a compact (without boundary) totally umbilical hypersurface of a simply connected real space form is a geodesic sphere. In the present note, we will investigate the natural question of the stability of this rigidity result. In other words, if a compact hypersurface of a real space form is almost umbilical, is this hypersurface close to a sphere? In what sense? Shiohama and Xu proved in [START_REF] Shiohama | Rigidity and sphere theorems for submanifolds[END_REF][START_REF] Shiohama | Rigidity and sphere theorems for submanifolds II[END_REF] that if ||τ || n is small enough, then M n is homeomorphic to the sphere S n . Later on, we obtain quantitative results about the closeness of almost hypersurfaces to spheres in [START_REF] Grosjean | Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces[END_REF][START_REF] Roth | A remark on almost umbilical hypersurfaces[END_REF]. For instance, we prove in [START_REF] Roth | A remark on almost umbilical hypersurfaces[END_REF], always for hypersurfaces of space forms, that if ||B -kg|| ∞ is sufficientely small, for a positive constant k, then the hypersurface is quasi-isometric to a sphere of radius 1 k . In [START_REF] Grosjean | Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces[END_REF] and also in [START_REF] Roth | A remark on almost umbilical hypersurfaces[END_REF], we obtain similar results with as less restrictive assumption replacing the L ∞ -norm by the L r -norm (r > n). In particular, the hypersurface is diffeomorphic to the sphere S n . The proximity to the sphere is stronger than for the results of Shiohama and Xu but in counterpart, the assumption on the umbilicity is also stronger. Indeed, the hypothesis implies that the umbilicity tensor is close to zero and in addition that the mean curvature is close to a constant. We add that in a very recent paper [START_REF] Scheuer | Quantitative oscillation estimates for almost-umbilical closed hypersurfcaes in Euclidean space[END_REF], Scheuer obtain the following nice result. If M is embedded into R n+1 and mean convex, then ||τ || ∞ sufficiently small implies that M is strictly convex and Hausdorff close to a geodesic sphere of appropriate radius n λ1 , where λ 1 is the first positive eigenvalue of the Laplacian on M .

The aim of the present paper is to give a result comparable to those of [START_REF] Grosjean | Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces[END_REF] and [START_REF] Roth | A remark on almost umbilical hypersurfaces[END_REF] with an alternative condition to the assumption that the mean curvature is almost constant. Starting from the remark that a hypersurface of a real space form has constant mean curvature if and only if its umbilicity tensor is Codazzi, we will prove that a hypersurface of a real space form which is almost umbilical with almost Codazzi umbilicity tensor is close to a sphere in the same sense as in [START_REF] Roth | A remark on almost umbilical hypersurfaces[END_REF] (see Theorem 3.1). Then, we prove a new rigidity result for geodesic spheres in real space forms (Corollary 4.4).

Preliminaries

Before stating the results, we will introduce some useful notations and recalls. First, we recall that the results obtained in [START_REF] Roth | A remark on almost umbilical hypersurfaces[END_REF] and [START_REF] Grosjean | Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces[END_REF] are consequences of pinching results for the first eigenvalue of the Laplacian proved in [START_REF] Colbois | A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space[END_REF] and [START_REF] Grosjean | Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces[END_REF]. A key tool for these pinching results is the Michael-Simon's extrinsic Sobolev inequality for submanifolds of the Euclidean space [START_REF] Michael | Sobolev and mean-value inequalities on generalized submanifolds of R n[END_REF] and its generalization by Hoffman and Spruck for any ambient manifold [START_REF] Hoffman | Sobolev and isoperimetric inequalities for Riemannian submanifolds[END_REF]. We begin by recalling the conditions under which these Sobolev inequalities are valid. Let (N n+1 , g) be a Riemannian manifold with sectional curvature bounded by above, say K N b 2 , with b real or purely imaginary and n 2. Let 0 < α < 1, we denote by H V (N, α) the set of all connected, oriented and compact Riemannian manifolds without boundary (M n , g) isometrically immersed into N and satisfying the two following conditions

(1) b 2 (1 -α) -2/n (ω -1 n V ol(M )) 2/n 1, (2) 2ρ 0 inj M (N ),
where inj M (N ) is the injectivity radius of N restricted to M and ρ 0 is given by

ρ 0 =    b -1 sin -1 b(1 -α) -1/n (ω -1 n V ol(M )) 1/n if b is real, (1 -α) -1/n (ω -1 n V ol(M )) 1/n if b is imaginary.
Under these hypotheses, Hoffman and Spruck showed that for any C 1 function f on M , the following extrinsic Sobolev inequality holds

(3)

M f n n-1 dv g n-1 n K(n, α) M (|∇f | + |Hf |) dv g ,
where K(n, α) depends only on n and α (not on b). We remark that for Euclidean and hyperbolic spaces, that is δ 0, (1) and ( 2) are trivially satisfied, whereas for spheres (δ > 0) (1) and (2) resume to V ol(M )

(1-α)ωn δ n/2
, where ω n is the volume of n-dimensional unit sphere. An immediate consequence of this inequality is that 1 K(n, α)||H|| ∞ V ol(M ) 1/n by taking f = 1. This extrinsic Sobolev inequality is of crucial importance to obtain pinching results for the first eigenvalue of the Laplacian (see [START_REF] Colbois | A pinching theorem for the first eigenvalue of the Laplacian on hypersurfaces of the Euclidean space[END_REF][START_REF] Grosjean | Eigenvalue pinching and application to the stability and the almost umbilicity of hypersurfaces[END_REF]). An other important fact under these assumptions is that the diameter of the hypersurface is bounded from above in terms of the mean curvature. Namely, Topping (for the Euclidean space [START_REF] Topping | Relating diameter and mean curvature for submanifolds of Euclidean space[END_REF]) and Wu and Zheng (for any ambient manifold [START_REF] Wu | Relating diameter and mean curvature for Riemannian submanifolds[END_REF]) proved that if Equations ( 1) and (2) hold, then there exists a constant C(n, α) depending only on n and α so that

(4) diam(M ) C(n, α) M |H| n-1 dv g .
For the statements of our results, we introduce the following subset of H V (M n+1 (δ), α).

Let q > n and A > 0, we denote by H V (n, δ, α, q, A) the subset of all manifolds in

H V (M n+1 (δ), α) satisfying max M ||H|| ∞ V ol(M ) 1/n , ||B|| q V ol(M ) 1/n A if δ 0 and max M ||H|| ∞ V ol(M ) 1/n , ||B|| q V ol(M ) 1/n , ||H||∞ √ ||H|| 2 ∞ +δ A if δ < 0. Note that the quantity ||H|| 2
∞ + δ is positive, even if δ is negative and hence, its square root has a sense. For instance, we can see this fact from the upper bound for the first positive eigenvalue of the Laplacian 0 < λ 1 (∆) n(||H|| 2 ∞ + δ), obtained by Heinze [START_REF] Heinzte | Extrinsic upper bound for λ 1[END_REF]. Note also that this last condition is invariant by any dilatation of the metric of the ambiant space.

We also introduce the following useful functions. Let r(x) = d(p, x) is the distance function to a base point p (in the sequel, p will be the center of mass of M ). We denote by Z the position vector defined by Z = s δ (r)∇r, where ∇ is the connection of M n+1 (δ). Moreover the functions c δ and s δ are defined by

c δ (t) =    cos( √ δt) if δ > 0 1 if δ = 0 cosh( |δ|t) if δ < 0 and s δ (t) =      1 √ δ sin( √ δt) if δ > 0 t if δ = 0 1 √ |δ| sinh( |δ|t) if δ < 0
A last notion will be useful for the sequel, the extrinsic radius. We recall that the extrinsic radius R(M ) of M is defined by

R(M ) = inf{ρ > 0| ∃x ∈ M n+1 (δ) s.t. φ(M ) ⊂ B(x, r)},
where φ is the immersion of M into M n+1 (δ). By a slight abuse of notation, we denote it R(M ) but, this radius depends not only on M but also on the immersion φ. Since in this paper, the considered immersion will be fixed, this notation does not lead to any ambiguity. Finally, even if, this is not optimal, we remark that R(M ) diam(M ).

Almost umbilical hypersurfaces

Now, we have all the ingredients to state the main result of this note, which gives a new result about the closeness to spheres for almost umbilical hypersurfaces.. Theorem 3.1. Let (M n , g) ∈ H V (n, δ, α, q, A). There exists ε 0 ∈]0, 1[ depending on n, q and A so that if ε ε 0 and

||τ || ∞ ε||H|| ∞ and ||d ∇ τ || ∞ ε||H|| 2 ∞ 2nC(n, α)A n ,
then M is embedded and (M, g) is ε-quasi-isometric to the round sphere

S p, s -1 δ 1 √ ||H|| 2 ∞ +δ
, where p is the center of mass of M . In particular, M is diffeomorphic to the sphere S n .

Before giving the proof of this theorem, we prove the following lemma Lemma 3.2. Let (M n , g) be a hypersurface of M n+1 (δ). For any tangent vector fields X, Y , we have

d ∇ τ (X, Y ) = Y (H)X -X(H)Y.
Proof: We compute the curvature R(X, Y )ν. We have

R(X, Y )ν = ∇ X ∇ Y ν -∇ Y ∇ X ν -∇ [X,Y ] ν = -∇ X A(Y ) + ∇ Y A(X) + A([X, Y ]) = -∇ X τ (Y ) + ∇ Y τ (X) + τ ([X, Y ]) -X(H)Y + Y (H)X +H∇ Y X -H∇ X Y + H[X, Y ] = -d ∇ τ (X, Y ) -X(H)Y + Y (H)X,
where we have used the fact ∇ Y X -∇ X Y +[X, Y ] = 0 since ∇ is torsion-free. Moreover, since M lies into a space of constant curvature, R(X, Y )ν = 0, which concludes the proof of the lemma.

Remark 3.3. We deduce immediately from this lemma that M have constant mean curvature if and only if τ is Codazzi.

Proof of Theorem 3.1: Let ε > 0. We set η = Since by assumption M ∈ H V (n, δ, α, A), the diameter is bounded in terms the mean curvature by [START_REF] Cheng | Hypersurfaces with constant scalar curvature[END_REF]. Namely, we have

diam(M ) C(n, α) M |H| n-1 dv g . Hence, we get |H(p) -H(x)| nηC(n, α)||H|| n-1 ∞ V ol(M ). Thus, we have |H 2 -||H|| 2 ∞ | 2nηC(n, α)||H|| n ∞ V ol(M ). Since η = ε||H|| 2 ∞ 2nC(n,α)A n and the assumption ||H|| n ∞ V ol(M ) A n , we get |H 2 -||H|| 2 ∞ | ε||H|| 2 ∞ .
This, together with the other assuption ||τ || ∞ ε||H|| ∞ leads to the conclusion that if ε ε 0 , where ε 0 is a constant depending on n, q and A given by [6, Theorem 1.3], then M is εquasi-isometric to the the sphere S p, s -1

δ 1 √ ||H|| 2 ∞ +δ
, where, p is the center of mass of M . In particular, M is diffeomorphic to S n . Moreover, in the proof of [6, Theorem 1.3], the diffeomorphism is explicitely given. Namely, this diffeomorphism is the map

F : M -→ S(p, ρ) x -→ exp p ρ X |X| , where ρ = s -1 δ 1 √ ||H|| 2 ∞ +δ
is the radius of the sphere, φ is the immersion of M into M n+1 (δ) and X = exp -1 p (φ(x)) is the position vector. Since F which is of the form F = G • φ is a diffeomorphism, then φ is necessarely injective. Thus, the immersion φ is an embedding. This concludes the proof. This square is required since the diameter of M appears by the use of the mean value inequality.

A new characterization of geodesic spheres

From Theorem 3.1, we can obtain a new characterization of geodesic spheres in real space forms. We are motivated by the well-known Alexandrov theorem and the Yau conjecture [START_REF] Yau | Seminar on Differential Geometry, Problem Section[END_REF]. Indeed, the Alexandrov theorem [START_REF] Alexandrov | A characteristic property of spheres[END_REF] states that a compact CMC hypersurface embedded into the Euclidean space, the hyperbolic space or the half-sphere is a geodesic sphere. Many generalizations of this result have been proved after. For instance, Ros proved that the same holds for higher order mean curvatures [START_REF] Ros | Compact hypersurfaces with constant higher order mean curvatures[END_REF][START_REF] Ros | Compact hypersurfaces with constant scalar curvature and a congruence theorem[END_REF]. In particular, in the Euclidean space, the second mean curvature H 2 = σ 2 (B) defined as the elementary symmetric homogeneous polynomial of the second fundamental form is (up to a multiplicative constant) the scalar curvature. Precisely, we have scal = n(n -1)H 2 . More generally, in M n+1 (δ), we have scal = n(n -1)(H 2 + δ). In the famous Problem section of [START_REF] Yau | Seminar on Differential Geometry, Problem Section[END_REF], Yau conjectured that the embedding is not necessary for the Alexandrov theorem for the scalar curvature. This conjecture is still open, even if several partial answers have been given. We can cite for instance, the cases where the hypersurface is convex [START_REF] Cheng | Hypersurfaces with constant scalar curvature[END_REF], stable [START_REF] Alencar | On the first eigenvalue of Linearized operator of the r-th mean curvature of a hypersurface[END_REF], of cohomogeneity 2 [START_REF] Okayasu | On compact hypersurfaces in a Euclidean space with constant scalar curvature[END_REF], locally conformally flat [START_REF] Cheng | Complete hypersurfaces in a Euclidean space R n+1 with constant scalar curvature[END_REF] or with pinched second fundamental form [START_REF] Li | Hypersurfaces with constant mean curvature in space forms[END_REF]. In [START_REF] Roth | Rigidity results for geodesic sphere in space forms, Differential Geometry[END_REF] and [START_REF] Roth | Une nouvelle caractérisation des sphères godésiques dans les espaces modèles[END_REF], we prove it with the assumption that the mean curvature is almost constant. In this section, we give another partial answer to this conjecture with a different additional assumption. Namely, we will show that hypersurfaces with constant scalar curvature and almost Codazzi umbilicity tensor are geodesic spheres. This characterization is valid in the three ambient space forms (Euclidean space, hyperbolic space and half-sphere).

First, we give this technical lemma. Lemma 4.1. Let (M n , g) ∈ H V (n, δ, α, q, A) and s a positive constants. Let ε > 0 and assume that

H -||H|| ∞ ε||H|| ∞ et ||scal -s|| ∞ ε||H|| 2 ∞ , then ||τ || ∞ D||H|| ∞ ε,
where D > 1 is an explicit constant depending on n, ||H|| ∞ , δ and A.

Remark 4.2.

(1) The constant D does not depend on s. Moreover, we will see in the proof that s is then close to n

(n -1)(||H|| 2 ∞ + δ). (2) If δ 0, D does not depend on ||H|| ∞ .
Proof: The proof of this lemma comes directly from the Hisung-Minkowski formulae. We recall that the Hsiung-Minkowksi formulae are integral formulae involving two consecutive higher order mean curvatures. In particular, we have the first two ones ( 5)

M H Z, ν + c δ (r) dv g = 0, (6)
M H 2 Z, ν + c δ (r)H dv g = 0.
Since we assume that |scal -s| < ε, and scal = n(n -1)(H 2 + δ), we get easily

(7) H 2 - s n(n -1) -δ < 1 n(n -1) ε||H|| 2 ∞ .
For more convenience, we will denote

h 2 = s n(n-1) -δ and ||H|| ∞ = h. Then, from (6) 0 =
M H 2 Z, ν + c δ (r)H dv g = M h 2 Z, ν + c δ (r)H dv g + M (H 2 -h 2 ) Z, ν dv g = h 2 h M h Z, ν dv g + M c δ (r)Hdv g + M (H 2 -h 2 ) Z, ν dv g = h 2 h M H Z, ν dv g + h 2 h M (h -H) Z, ν dv g + M c δ (r)hdv g + M c δ (r)(H -h)dv g + M (H 2 -h 2 ) Z, ν dv g .
Now, we use the other Hsiung-Minkowski formula (5) to get

0 = - h 2 h M c δ (r)dv g + h 2 h M (h -H) Z, ν dv g + M c δ (r)hdv g + M c δ (r)(H -h)dv g + M (H 2 -h 2 ) Z, ν dv g = h - h 2 h M c δ (r)dv g + h 2 h M (h -H) Z, ν dv g + M c δ (r)(H -h)dv g + M (H 2 -h 2 ) Z, ν dv g .
Then, since |Z| = s δ (r) and using the assumption |H -h| hε and ( 7), we deduce

h - h 2 h M c δ (r)dv g h 2 ε M s δ (r)dv g + εh M c δ (r)dv g + εh 2 n(n -1) M s δ (r)dv g .
Using the fact that |H 2 | H 2 , we deduce from the assumptions on h and h 2 that

h 2 |H 2 | + 1 n(n -1) εh 2 H 2 + 1 n(n -1) εh 2 h 2 + 2εh 2 + 1 n(n -1) εh 2 h 2 (1 + 3ε).
Hence, we get

h - h 2 h M c δ (r)dv g εh M c δ (r)dv g + 1 + 3ε + 1 n(n -1) εh 2 M s δ (r)dv g .
Therefore, unsing the fact that ε < 1, we get

h 2 -h 2 1 + h 1 + 3ε + 1 n(n -1) M s δ (r)dv g M c δ (r)dv g εh 2 1 + 5h M s δ (r)dv g M c δ (r)dv g εh 2 .
From now on, we will discuss the three cases δ = 0, δ > 0 and δ < 0.

First case: δ = 0. In this case, we have s δ (r) = r and c δ (r) = 1, then, M s δ (r)dvg

M c δ (r)dvg
R. Since R is the extrinsic radius, we have this obvious relation with the interior diameter R diam(M ) and therefore, by the result of Topping

M s δ (r)dv g M c δ (r)dv g R diam(M ) C(n, α) M |H| n-1 dv g C(n, α) A n h .
Second case: δ > 0. In this case, c δ (r) = cos( √ δr) and s δ (r) = 1 √ δ sin( √ δr) and we have

M s δ (r)dvg M c δ (r)dvg
t δ (R) R, since we have assumed that M is contained in a sphere a radius π 4 √ δ . Thus, like in the case δ = 0, we have M s δ (r)dv g M c δ (r)dv g

C(n, α)

A n h .

Third case: δ < 0. In this case, c δ (r) = cosh( |δ|) and s δ (r) = 1 √ |δ sinh( |δ|). Hence, we get immediately that M s δ (r)dvg

M c δ (r)dvg 1 √ |δ| .
Then, in the three cases, we have

h 2 -h 2 Eεh 2 ,
where

E =      1 + 5C(n, α)A n if δ 0, 1 + 5h √ |δ| if δ < 0,
is a constant depending on n, α, δ, h and A.

Now, we recall the

Gauss formula. For X, Y, Z, W ∈ Γ(T M ), R(X, Y, Z, W ) = R(X, Y, Z, W ) + SX, Z SY, W -SY, Z SX, W
where R and R are respectively the curvature tensor of M and M n+1 (δ). By taking the trace in X and Z and for W = Y , we get

Ric(Y ) = Ric(Y ) -R(ν, Y, ν, Y ) + nH SY, Y -S 2 Y, Y ,
where S is the shape operator. Since, the ambient space is of constant sectional curvature δ, by taking the trace a second time, we have

scal = n(n -1)δ + n 2 H 2 -|S| 2 ,
or equivalently

scal = n(n -1) H 2 + δ -|τ | 2 .
Hence, we have

||τ || 2 = n(n -1)(H 2 -H 2 ) n(n -1) |H 2 -h 2 | + |h 2 -h 2 | n(n -1) 2h 2 ε + Eh 2 ε Dh 2 ε,
where we have set D = n(n -1)(2 + E). This concludes the proof of the lemma. Now, from this lemma, we can prove the following result.

Theorem 4.3. Let (M n , g) ∈ H V (n, δ, α, q, A). There exists D > 1 depending on n, α, δ, ||H|| ∞ and A and there exists η 0 ∈]0, 1[ depending on n,q and A and so that if η η 0 and

||scal -s|| ∞ η||H|| 2 ∞ D and ||d ∇ τ || ∞ η||H|| 2 ∞ 2nC(n, α)AD ,
then M is embedded and (M, g) is η-quasi-isometric to the round sphere S p, s -1

δ s n(n-1)
, where p is the center of mass of M . In particular, M is diffeomorphic to the sphere S n .

Proof: First, the constant D of the theorem is the one computed in Thus, (8) and ( 9) are exactely the hypotheses of Theorem 3.1. Then, we conclude that (M, g) is η-quasi-isometric to the round sphere S p, s -1

δ s n(n-1)
, where p is the center of mass of M . In particular, M is diffeomorphic to S n . In addition Theorem 3.1 insures that M is embedded into M n+1 (δ).

We deduce easily the following corollary which is a new characterization of geodesic spheres and gives a new partial answer to the Yau conjecture. Proof: This Corollary is a direct consequence of Theorem 4.3 together with the Alexandrov theorem for the scalar curvature proved by Ros [START_REF] Ros | Compact hypersurfaces with constant scalar curvature and a congruence theorem[END_REF]. Indeed, from the assumptions, we can apply Theorem 4.3 and get that M is embedded. Since M is assumed to have constant scalar curvature, by [START_REF] Ros | Compact hypersurfaces with constant scalar curvature and a congruence theorem[END_REF], M is a geodesic sphere. Moreover, the radius of this sphere is determined by the scalar curvature and is s -1 δ s n(n-1) .

ε||H|| 2 ∞

 2 2nC(n,α)A n . From the previous Lemma and the assumption ||d ∇ τ || ∞ η, we deduce that |X(H)| η for any unitary vector X. Thus, we have ||∇H|| nη. Now, let p ∈ M be a point where the maximum of |H| is achieved. Then, for any x ∈ M , by the mean value inequality, we have |H(p) -H(x)| nηd(p, x) nηdiam(M ).

Remark 3 . 4 . 2 ∞

 342 The first condition ||τ || ∞ ε||H|| ∞ is invariant by homothety. But, the second condition ||d ∇ τ || ∞ ε||H|| 2nC(n,α)A n is not invariant by homothety because of the square on ||H|| ∞ .

2 ∞ 2 ∞ 2 ∞

 222 Lemma 4.1. The second assupmtion ||d ∇ τ || ∞ η||H|| 2nC(n,α)AD implies by the computations of the proof of Theorem 3.1 that H -||H|| ∞ η||H||∞ D . Thus, we can apply Lemma 4.1 with ε = η D to get (8) ||τ || ∞ D||H|| ∞ ε = η||H|| ∞ . Moreover, since D > 1, we have (9) ||d ∇ τ || ∞ η||H|| 2nC(n, α)AD η||H|| 2nC(n, α)A .

Corollary 4 . 4 . 2 ∞

 442 Let (M n , g) ∈ H V (n, δ, α, q, A) and s a positive constant. There exists D > 1 depending on n, α, δ, ||H|| ∞ and A and there exists η 0 ∈]0, 1[ depending on n, q and A so that if η η 0 andscal = s and ||d ∇ τ || ∞ η||H|| 2nC(n, α)A ,then M is a geodesic sphere of radius s -1 δ s n(n-1) .


Abdou K Diallo

Liana Annunziata

Stéphane Fouquay

Guillaume Michaud

Frédéric Simon

Jean-Michel Brusson

Sophie M Guillaume
email: sophie.guillaume@univ-rennes1.fr

Jean-François Carpentier
email: jean-francois.carpentier@univ-rennes1.fr

Ring-opening metathesis polymerization of cyclooctene derivatives with chain transfer agents derived from glycerol carbonate

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Functionalized telechelic poly(butadiene)s (PBDs) are extensively used in the polyurethane industry as cross-linking agents in the preparation of products ranging from sealants and adhesives to rocket propellant binders. 1 Such telechelic polymers are often synthesized by radical or anionic polymerization methods which usually require demanding conditions and induce varying amounts of 1,2-linkages in the polymer backbone. 2 This leads to average number of functional groups per polymer chain (F n ) values that greatly deviate from 2, and it also limits the elastomeric potential of the ultimate materials. [START_REF] Odian | Principles of polymerizations[END_REF] Recent advances in rutheniumcatalyzed ring-opening metathesis polymerization (ROMP) have provided new opportunities for the preparation of such telechelic polymers with F n = ca. 2, when functionalized chaintransfer agents (CTAs) are employed. In comparison to tungsten-and molybdenum-based catalysts, Grubbs' catalysts based on ruthenium complexes are widely used in ROMP thanks to their tolerance toward protic functional groups, oxygen, and moisture. [START_REF] Trnka | [END_REF] When symmetric acyclic substituted alkenes are employed as CTAs, only the -telechelic polymers are produced. Thus, dihydroxy-telechelic poly(butadiene)s (HTPBDs) have been obtained via the Ru-catalyzed ROMP of cyclooctene (COE), 5 1,5-cyclooctadiene (COD) 68 or 1,5-dimethyl-1,5-cycloctadiene (DMCOD) 9 in the presence of cis-1,4-bis(acetoxy)-2-butene as CTA, followed by a post-polymerization deprotection step of the acetoxy groups. Similarly, the synthesis of dicarboxy-and diamino-telechelic PBDs has been reported from cis-1,4-bis (2-tert-butoxycarbonyl)-2-butene and cis-1,4-di-tert-butyl-2-butene-1,4-dicarbamate as CTAs, respectively. 10 Hillmyer and co-workers have described the use of unprotected maleic acid as a CTA in the ROMP of COE mediated by Grubbs' 2 nd -generation catalyst (G2), to afford ,-dicarboxy-telechelic PCOE. 11 Also, high molar mass dicyano-and dichloro-telechelic PBDs have been synthesized via G2-catalyzed ROMP of COD in the presence of 1,4dicyano-, 1,8-dicyano-4-octene or 1,4-dichloro-2-butene as difunctional CTAs. 12,13 On the other hand, there are only a few reports on Ru-catalyzed ROMP in the presence of heteroatom-substituted vinyl substrates as CTAs. Grubbs and coworkers prepared endfunctionalized poly(norbornene)s (PNBs) using Cl 2 (PCy 3 ) 2 Ru=CHPh as the ROMP catalyst and allyl acetate as CTA. 14 α-Acetoxy,-vinyl-PNB was thus recovered as the major product along with small amounts of bis-acetoxy and bis-vinyl end-functionalized PNBs. Acryloyl species were used as CTAs for ring-opening cross metathesis (ROCM) of COD or trisubstituted cycloolefins mediated by G2 catalyst. 15 With COD as monomer, α,diacrylate-poly(COD)s (PCODs) were synthesized whereas with the trisubstituted cycloolefins, the acrylate moiety was eventually found only onto the less-substituted chainend.

Valorization of glycerol, a cheap and largely available renewable resource derived from biodiesel production, has become a topical challenge for (agro)chemical industries.

Among possible useful green chemicals derived thereof, the inexpensive glycerol carbonate (4-hydroxymethyl-1,3-dioxolan-2-one; hereafter referred to as cyclocarbonate or GC) has been investigated as a non-volatile reactive solvent and novel component for a variety of materials. 1618 In a prior study, we have reported the use of GC as a CTA in the "immortal" ring-opening polymerization (ROP) of trimethylene carbonate (TMC) leading to α-GC endfunctionalized poly(trimethylene carbonate) (PTMC). 19 Subsequent transformation of the -hydroxy chain-end into a GC moiety eventually afforded ,-di(GC) telechelic PTMC.

We have shown that this latter polymer can undergo a carbonate ring-opening polyaddition with a diamine to afford poly(hydroxyurethane)s (PHUs), 19 a class of the so-called Non-Isocyanate PolyUrethanes (NIPUs) of high topical interest. 2026 Also, taking advantage of the high tolerance of ROMP catalysts toward functional groups, we have recently communicated on the first synthesis of mono-and di-cyclocarbonate end-functionalized polyolefins through the ROMP of COE in the presence of vinyl and acryloyl derivatives of GC as CTAs, respectively. 27 In order to better understand the mechanism of these ROMP reactions in the presence of GC derivatives, so as to improve the overall efficiency and the scope of the process and to eventually access to room-temperature liquid materials (much more useful for formulation of end-products such as NIPUs), we have now explored in details the ROMP of COE and of a variety of 3-and 5-substituted COE derivatives (Schemes 1, 3, 5). Herein, we describe the selective synthesis and the characterization of the resulting mono-and di-telechelic PCOEs and provide mechanistic insights into the ROMP process, notably by taking into account the substitution pattern of the cycloolefin.

Experimental Section

Chemicals. All the syntheses and manipulations of air-and moisture-sensitive materials were carried out in Schlenk-type glassware on a dual-manifold Schlenk line. Cyclooctene (COE), [(IMesH 2 )(Cy 3 P)RuCl 2 (=CHPh)] (Grubbs' 2 nd -generation catalyst, G2) and 4-vinyl-1,3dioxolan-2-one (1) were purchased from Sigma-Aldrich. COE was first degassed and dried over CaH 2 overnight and then distilled before use. All other reagents were used as received.

(2-Oxo-1,3-dioxolan-4-yl)methyl acrylate (2), 27 3-or 5-alkyl-substituted COEs (3-Me-COE, 28 3-Et-COE, 28 3-n-Hex-COE, 28 5-Et-COE 29 ) and functionalized COEs (5,6-epoxy-COE, 5-OH-COE, 5-O=COE) 30 were synthesized according to the reported literature procedures.

Characterization.

NMR spectra were recorded on AM-400 and AM-500 Brüker FTIR spectra of the polymers were acquired on a Shimadzu IR-Affinity-1 equipped with an ATR (32 scans, resolution = 4 cm 1 ).

General ROMP Procedure. All polymerizations were performed similarly according to the following typical procedure (Table 1, entry 1). The only differences lie in the nature of the CTA and in its initial concentration ([CTA] 0 ). Under argon atmosphere, COE (1.4 mL, 10.8 mmol) and dry THF (5.0 mL) were placed in a 100 mL flask. Then, under stirring, 4-vinyl-1,3-dioxolan-2-one (1) (50.0 µL, 0.54 mmol) was syringed in and the flask was immersed in an oil-bath at 40 °C immediately before adding, via a cannula, a THF (2 mL) solution of G2 catalyst (5.0 mg, 5.0 µmol). The reaction mixture turned highly viscous within 2 min. The viscosity then slowly decreased over the next 10 min, and after several hours the reaction medium became completely fluid. After the desired reaction time (typically 2 or 24 h), the solvent was concentrated under vacuum. The polymer was then recovered upon precipitation in excess methanol (100 mL) (thereby allowing removal of the catalyst), filtration and drying at 23±2 °C under vacuum (94% yield). All polymers were recovered as colorless materials, readily soluble in chloroform and THF, and insoluble in methanol.

Results and Discussion

ROMP of COE and 3-R-COEs (R = Me, Et, n-hexyl) with various concentrations of vinyl-GC (1) and acryloyl-GC (2) using Grubbs' 2 nd -generation catalyst (G2) was first explored in THF at 40 °C over 24 h (Table 1, Scheme 1). The high isolated yields (9498%)

and the absence of monomer in the 1 H NMR spectra of the crude reaction products were coherent with quantitative monomer consumption. In all cases, the experimental molar mass of the polymers (M n,NMR or M n,SEC ) was found to vary inversely proportionally to the amount of CTA (compare entries 1/2, 3/4... 15/16); this evidenced an efficient chain-transfer process.

Scheme 1. Synthesis of ,-di(GC)-PCOEs from the ROMP of COE using G2 catalyst and 1

or 2 as CTA.

All the PCOEs recovered using 1 or 2 as CTA for the ROMP of unsubstituted COE over 24 h (Table 1, entries 14) displayed the same structure featuring GC end-groups at both termini (F n = 2); this is illustrated in Figure 1 to the carbonyl stretching in the GC moiety (ESI, Fig. S1). The telechelic PCOEs prepared from CTA 2 (Table 1, entries 34) all displayed the same structure with two GC end-groups, as previously reported (ESI, Fig. S2S4). 27 The polymer functionality and molar mass value in the ROMP of COE with 1 and 2 as CTAs were monitored by 1 H NMR and SEC over the reaction course, respectively. In both cases, the molar mass of the recovered PCOEs was found to decrease with time, before reaching a plateau (ESI, Fig. S5). This is consistent with the swift increase of the viscosity of the reaction mixture observed at the initial stage of the polymerization, followed by its gradual decrease over longer reaction times. ‡ When the ROMP of COE was performed in the presence of 1 as CTA, a vinyl end-group along with a GC at the other terminus (F n = 1) was obtained in the polymers recovered from the reaction carried out over 2 h, while after 24 h, PCOEs were then end-capped with one GC end-group at both chain-ends (F n = 2). This clearly indicated that the nature of end-groups in the final polymer changes as the reaction proceeds. As depicted in Figure 2, a similar time evolution of the selectivity for the monoand di-functionalized PCOEs was observed in the presence of 2 as CTA, but over a significantly shorter time period. After 10 min of reaction, 100% of -vinyl,-GC-PCOE were formed; then, the amount of this latter product decreased concomitantly with the increase of the amount of ,-di(GC)-PCOE up to 100% selectivity throughout a period of 2 h. In comparison with vinyl-GC (1), this clearly showed that the electron-poor acrylate-GC

(2) is a more active CTA. The functionality of PCOEs is therefore highly dependent on both the reaction time and nature of the CTA. ‡ Upon adding the G2 catalyst to the reagents, the reaction mixture turned very viscous within 2 min (presumably indicative of the formation of high molar mass PCOE) and then gradually fluidized over time (as a result of cross-metathesis of PCOE with CTA; ESI, Fig. S5). and ,-di(GC)-PCOE (bifunctional, ) during the ROMP of COE promoted by G2 catalyst in the presence of 2 as CTA (Table 1, entry 3).

THF and dichloromethane (DCM) -two common solvents for metathesis-were selected to evaluate the solvent influence on the ROMP of COE in the presence of 2 as CTA.

Monitoring of the reaction by 1 H NMR showed that, in DCM, at a 2000 (COE):100 (2):1 (G2) ratio, a 40:60 mixture of -vinyl,--GCand -di(GC)-PCOEs formed after 24 h (ESI, Fig. S6). However, in THF, selective formation of the -di(GC)-PCOE was observed after only 2 h of reaction (Fig. 2). Hence, a change of the solvent also had a significant impact on the polymer functionality. In fact, this result clearly indicated that the rate of the crossmetathesis reaction between PCOE and CTA 2 was higher in THF as compared to DCM.

This trend is in line with literature data on ROMP and cross-metathesis. 31 Such an acceleration of the rate of ROMP has been accounted for by the increased stability of the four-coordinate ruthenium intermediate and/or the assistance of the solvent in the displacement of coordinated PCy 3 ligand from the metal center to generate the catalytically active species. 32 Scheme 2 depicts the proposed mechanistic scenario leading to the formation of endfunctionalized PCOEs from the ROMP of COE in the presence of functionalized olefins 1 or 2 as CTA. After the initial, fast ROMP of COE (leading to high molar mass PCOEs and hence to a highly viscous reaction medium, vide supra), the Ru-alkylidene-polymer terminated with a vinyl group undergoes a cross-metathesis with the CTA to provide a lower molar mass end-differentiated (i.e., vinyl/GC) product; this polymer can further react over the reaction course with another CTA to form the symmetrically GC end-capped products.

Besides the gradual decrease in viscosity of the reaction medium observed over time, the ethylene released during such cross-metathesis events was unambiguously evidenced by a GC-MS analysis (ESI, Fig. S7); this further supports this proposed mechanism. A more efficient route to selective ROMP involving the selective generation of -vinyl,-GC-polycycloolefins is using 3-alkyl-substituted COEs (3-R-COEs, Scheme 3, Table 1, entries 516). In all cases with R = methyl, ethyl, hexyl, the recovered P(3-R-COE)s displayed a single structure with two different chain-end groups: a GC moiety at one terminus and a vinyl on the other extremity. As illustrative example, Figure 3 shows the 1 H NMR and

R + CH 2 L n Ru R RuL n R RuL n X R R RuL n R R RuL n R R RuL n R R n-2 R L n Ru n CTA R X observed product n R CH 2 L n Ru R RuL n R RuL n
Not observed Scheme 4. Proposed selectivity model for the ROMP of 3-alkyl-substituted COEs mediated by Grubbs' 2 nd -generation catalyst.

The thermal profiles of these -vinyl,-GC-P(3-R-COE) polymers, as determined by Differential Scanning Calorimetry (DSC) analysis, did not exhibit crystallization or melting transitions (in contrast to the corresponding -vinyl,-GC-PCOEs; Table 2, entry 1), 27 but only a glass transition (Table 2, entries 24; ESI, Fig. S10, S15, and S21). As anticipated, the alkyl group in the polymer backbone has a noticeable influence on the thermal behavior: the larger the alkyl chain, the lower the T g . These T g values are consistent with those previously reported by Wagener and co-workers for polyethylene-type polymers wherein a methyl, 34,35 ethyl 35 or n-hexyl 35,36 was placed on the ninth backbone carbon (via acyclic diene metathesis (ADMET) of the 6-alkyl-substituted 1,10-undecadiene).

Seeking to get ,-di(GC) polymers which are liquids at room-temperature (for easier implementation in formulations towards Non-Isocyanate PolyUrethanes, NIPUs), 1926 the ROMP of a differently substituted COE derivative, namely 5-Et-COE, was carried out in the presence of 2, in a similar approach as that described above for other COEs (Scheme 5; Table 1, entry 17). The recovered material was indeed a liquid (T g = 65 °C) and functionalized at both termini. Figure 4 shows the 1 H and 13 C{ 1 H} NMR spectra of such telechelic ,-di(GC)-P(5-Et-COE). The 1 H signals of the GC moieties are observed at δ 4.324.94 ppm, in a 1:1 ratio with the signals for the internal olefinic hydrogens H7 and H6 at δ 7.04 and 5.84 ppm, respectively (Fig. 4a). Correspondingly, the 13 C{ 1 H} NMR spectrum exhibits no signal corresponding to a potential vinylic chain-end group (expected at  114.1 and 143.1 ppm; Fig. 3b).

Scheme 5. Synthesis of ,-di(GC)-P(5-Et-COE) from the ROMP of COE using Grubbs' 2 nd -generation catalyst (G2) and 2 as CTA. The ROMP of COE derivatives substituted with polar functional groups (5,6-epoxy, 5-OH and 5-O=) in the presence of the more reactive CTA 2 were also effectively achieved (Table 1, entries 1820). The polymers were isolated in good yields (typically 9498%) and their structures were determined by 1 H and 13 C{ 1 H} NMR spectroscopy (see ESI, Fig.

S2425, S2729

). The polymers recovered from the ROMP of 5-OH-COE and 5,6-epoxy-COE after 24 h reaction exhibited two GC end-groups, one at each terminus (F n = 2). With the ketone-functionalized COE, a 60:40 mixture of mono-and di-GC terminated P(5-O=COE) was typically observed. Attempts to access selectively the telechelic -di(GC)-P(5-O=COE) by using longer reaction times or higher catalyst loadings failed. We assume that the ketone group in 5-O=COE was probably responsible for a faster catalyst deactivation, eventually preventing the full cross-metathesis of vinyl chain-ends.

The functional telechelic -di(GC)-P(5-OH-COE) and -di(GC)-P(5-O=COE)

are solid materials at room temperature, while the -di(GC)-P(5,6-epoxy-COE) is a liquid (T g = 48 °C, ESI Fig. S26).

Conclusion

The ROMP of COE derivatives in the presence of acryloyl and vinyl derivatives of GC provides an efficient entry towards a variety of mono-and di-cyclocarbonate polyolefin materials. The selectivity for mono-vs. di-functionalized telechelic polymers is highly dependent on the nature of the solvent, the reaction time, the CTA, and the cycloolefin substrate. Noteworthy, the selectivity of the ROMP of substituted COEs reported here is largely influenced by the substituent position on the cycloolefin. All these observations have been rationalized by a simple mechanistic scenario involving at the initial stage of the reaction the formation of monofunctionalized polymers ultimately converted into the analogous

2 )

 2 spectrometers at 298 K in CDCl 3 . A relaxation delay of 3 s was used during the acquisition to afford quantitative 1 H and13 C NMR spectra. The chemical shifts are referenced to tetramethylsilane (TMS) and were determined using the residual proton ( 1 H) and carbon ( 13 C) resonances of the deuterated solvent. Molar mass (M n and M w ) and dispersity (Ð M = M w /M n ) values of polymers were determined by Size Exclusion Chromatography (SEC) using a Polymer Laboratories PL-GPC 50 instrument equipped with two ResiPore Mixed E 300 × 7.5 mm columns and a refractive index detector. The samples were eluted with THF at 30 °C at 1.0 mL•min -1 and the columns were calibrated using 11 monodisperse polystyrene standards (range of M n = 580 to 380,000 g•mol 1 ). The reported molar mass values are uncorrected for the possible difference in hydrodynamic volumes of the polymers vs. polystyrene. Molar mass values determined by 1 H NMR spectroscopy (M n,NMR ) were calculated from the integral value ratio of the signals of end-groups hydrogens to internal olefin hydrogens. The theoretical molar mass of polymers was calculated from the formula: M n,theo = M(monomer) × [monomer] 0 /([CTA] 0 /F n ) × conversion(monomer), where F n is the functionality of the polymer (i.e., the number of GC groups per macromolecular chain; 1 < F n < Differential scanning calorimetric (DSC) analysis was performed on a Setaram DSC 131 apparatus, calibrated with indium, at a rate of 10 °C min 1 , under continuous flow of helium (25 mL min 1 ), using aluminum capsules. Reported values were obtained from the second run.

  which shows the 1 H NMR and13 C{1 H} NMR spectra of the PCOE prepared from 20 equiv. of 1. The signals corresponding to the GC endgroups are found at δ 4.125.04 ppm, in a 1:1 ratio with the signals for the vinylidene hydrogens (H4, δ 5.49 ppm; H5, δ 5.92 ppm), as expected. In addition, the 13 C{ 1 H} NMR spectrum shows the signals at δ 154.8 ppm and δ 77.9 and 69.3 ppm assigned to the carbonyl and the two other carbons of the 5-membered GC moiety, respectively. FT-IR spectroscopy confirmed these results with the presence of the vibration band at  1810 cm 1 corresponding

Figure 1 .

 1 Figure 1. 1 H NMR (a, top) and 13 C{ 1 H} (b, bottom) NMR spectra (400 and 100 MHz, CDCl 3 , 298 K) of an ,-di(GC)-PCOE synthesized from the ROMP of COE mediated by G2 catalyst in the presence of 1 as CTA (Table1, entry 1) (* stands for residual solvent

Figure 2 . 1 H

 21 Figure 2. 1 H NMR monitoring of the formation of α-vinyl,-GC-PCOE (monofunctional, )

Scheme 2 .

 2 Scheme 2. Simplified mechanisms for the formation of end-differentiated and symmetrically

13 C{ 1 Scheme 3 .

 1313 Scheme 3. Synthesis of -vinyl,-GC-P(3-R-COE)s from the ROMP of 3-alkyl-substituted COEs using Grubbs' 2 nd -generation catalyst (G2) and 1 or 2 as CTA.

Figure 3 .

 3 Figure 3. 1 H NMR (a, top) and 13 C{ 1 H} NMR (b, bottom) spectra (400 and 100 MHz, CDCl 3 , 298 K) of an -vinyl,-GC-P(3-Et-COE) synthesized from the ROMP of 3-Et-COE in the presence of 2 as CTA (Table 1, entry 9) (* stands for residual solvent resonances).

Figure 4 .

 4 Figure 4. 1 H NMR (top) and 13 C{ 1 H} NMR (bottom) spectra (400 MHz and 100 MHz, CDCl 3 , 298 K) of an α,-di(GC)-P(5-Et-COE) synthesized from the ROMP of 5-Et-COE in

Table 1

 1

, entry 9) (* stands for residual solvent resonances).

Table 1 ,

 1 entry 17; * stands for residual solvent resonances).

Table 1 .

 1 ROMP of COE and substituted COE derivatives mediated by Grubbs' 2 nd -generation catalyst in the presence of 1 and 2 as CTA over 24 h. Number of GC group per macromolecular chain. b Theoretical M n value calculated for complete chain-transfer, from the relation: M n,theo = M(monomer) × [monomer] 0 /([CTA] 0 /F n ) × conversion(monomer) (= 100% for all these experiments). c Experimental M n value determined by1 H NMR (refer to the Experimental Section). d Experimental M n and dispersity values determined by SEC (refer to the Experimental Section).

	Entry	R	CTA	[R-COE] 0 / [CTA] 0 /[G2] 0	F n	a	M n,theo b (g.mol 1 )	M n,NMR (g.mol 1 ) c	d M n,SEC (g.mol 1 )	Ð M	d
	1	H	1	2000:100:1	2		4400	13 400	9000	1.75
	2	-	1	2000:30:1	2		14 600	nd	17 400	1.43
	3	-	2	2000:100:1	2		4400	10 200	6600	1.56
	4	-	2	2000:30:1	2		14 600	30 200	19 200	1.63
	5	3-Me	1	2000:100:1	1		2500	5100	7000	1.42
	6	-	1	2000:30:1	1		8200	15 200	17 800	1.42
	7	-	2	2000:100:1	1		2500	4800	6200	1.46
	8	-	2	2000:30:1	1		8200	15 000	19 000	1.57
	9	3-Et	1	2000:100:1	1		2800	9800	6500	1.39
	10	-	1	2000:30:1	1		9100	29 300	19 700	1.52
	11	-	2	2000:100:1	1		2800	8000	9700	1.50
	12	-	2	2000:30:1	1		9100	nd	17 700	1.50
	13	3-n-Hexyl	1	2000:100:1	1		3900	4900	7300	1.30
	14	-	1	2000:30:1	1		13 100	14 600	20 600	1.38
	15	-	2	2000:100:1	1		3900	9900	9700	1.42
	16	-	2	2000:30:1	1		13 100	nd	23 600	1.41
	17	5-Et	2	2000:100:1	2		5600	4600	3500	1.75
	18	5,6-epoxy	2	2000:100:1	2		5000	13 600	6200	150
	19	5-OH	2	2000:100:1	2		5100	14 300	15 500	1.52
	20	5-O=	2	2000:100:1	<2	<5000	nd	6000	1.87

a

Table 2 .

 2 Thermal properties of -vinyl,-GC-P(3-R-COE)s obtained by Grubbs' 2 ndgeneration-catalyzed ROMP of 3-R-COEs in the presence of 1 as CTA, as determined by DSC (10 °C.min 1 , second heating cycle).

	R	T g (°C)	T m (°C)
	H	not observed	60
	Me	64	-
	Et	69	-
	n-Hexyl	77	-

Acknowledgments

This work was financially supported by Bostik and Total Cies (postdoctoral grants to A.K.D. and L.A.).

difunctionalized ones. The molecular and structural features of the recovered polymers were confirmed by NMR, FTIR and SEC analyses, and their thermal properties were investigated by DSC. Interestingly, polymers having a 3-or 5-alkyl substituent on their backbone, no matter the nature of their end-groups (i.e., mono-or di-GC-functionalized) and their molar mass, feature a low glass transition temperature and are liquid materials at room temperature. This makes them promising precursors towards NIPUs, via eventual reaction with various polyamines. 1926 Experiments in these directions are presently in progress in our laboratories and the results will be reported in due course.

Electronic Supplementary Information

Detailed characterization data of polymers.

For the Table of Content entry

Mono-and di-(glycerol carbonate) telechelic polyolefins have been prepared by ROMP of cyclooctene derivatives in the presence of a chain-transfer agent