Claire Hotton

Guylaine Ducouret

Juliette Sirieix-Plénet

Thomas Bizien

Lionel Porcar

Natalie Malikova
email: natalie.malikova@sorbonne-universite.fr

Tuning structure and rheological properties of polyelectrolyte based hydrogels through counterion specic eects

Tuning at will the properties of gel forming systems is of key relevance for many biotechnologies, agricultural and biomedical applications. For polyelectrolyte-based gels, ion specic eects can be an attractive way for this purpose. This study investigates the counterion specic eect on the microscopic structure and the rheological properties of a physical hydrogel formed of ionene-type cationic polyelectrolytes. The focus is on two monovalent halide counterions (F -and Cl -) and a divalent counterion (SO 2- 4 ). A strong counterion specic eect appears within ionene based gels. In the 1 case of halide counterions, gelation is more eective for more weakly hydrated counterions. Indeed, strongly hydrated counterions maintain electrostatic repulsions between the chains and as a consequence gel formation is shifted towards higher concentrations (higher critical gelation concentration, CGC). The combination of the complementary small angle X-ray and neutron scattering (SAXS, SANS) techniques reveals a strong contribution of ion-ion correlations in the structure of the gel network. Contrary to chloride gels, which present a single correlation length characterizing the distance between the cross-linking nodes, uoride gels present an additional network of nodes.

This is accompanied by a very rapid increase of the elastic modulus of uoride gels, once CGC is reached. With divalent counterions, the gelation is even more remarkable with a lower CGC and a higher elastic modulus at equivalent polyelectrolyte concentrations. The presence of divalent counterions favours the association of chains, probably by a bridging eect. This evokes the "egg-box" model and the characteristic scaling of the elastic modulus with reduced gel concentration conrms this. However, only a narrow concentration window for gel forming exists for divalent counterions, before precipitation takes over due to too strong attractive chain-chain interactions.

Introduction

Hydrogels are a class of soft materials that have recently attracted signicant attention especially for agricultural, 1 super-absorbent, 2 biotechnologies 3 and biomedical applications. 4,5 In particular, physical hydrogels which present non-permanent cross-linking bonds exhibit responsiveness to external stimuli, allowing for the control of their physicochemical properties for further applications such as drug delivery systems.

68 In this study, physical hydrogels are based on polyelectrolytes, i.e. charged polymers. The use of polyelectrolytes allows us to tune the hydrogel properties by changing the nature and the valence of the counterion. Here, we used ionenes, a group of cationic polyectrolyte with pH independent charge based on quaternary ammonium charged centres separated by carbon chains 9 (Figure 1 (A)).

Ionenes possess a charge density that is regular along the chain and tuneable by synthesis which enables them to be considered as a model system. While conventional ionenes do not form gel, the introduction of aromatic and amide moieties into the ionene backbone open the possibility of hydrogelation, 10,11 see Figure 1 (B). Moreover, it has been shown in the literature that these modied ionenes exhibit antimicrobial properties 12 which made these systems suitable for biomedical applications. Hydrogels based on modied ionenes are formed through a combination of electrostatic repulsions, hydrogen bonding, [START_REF] Misawa | Hydrogels Based on Surfactant-Free Ionene Polymers with N,N'-(p-Phenylene)dibenzamide Linkages[END_REF] π-π stacking, anion-π and cation-π interactions, 11,13 see Figure 1 (C).

Ion-specic eects have been investigated in the literature for many years but are still a challenging topic.

14 Ion-specicity cannot be accounted for by purely electrostatic theories, the eects of individual ions are a complex combination of their valence, polarisability, geometry and hydration properties.

1416 Ion-specic eects on gelation mechanism have been already investigated in the literature in salty media.

1720 Only a few studies focus on the counter-ion specic eects in non-salty media. Mostly, the eect of the valence of the counterions has been highlighted, evoking the "egg-box" model for biopolymer gels containing divalent ions. 21,22 In conventional ionene solutions, striking counterion-specic eects have already been observed

2325 with divalent counterions but also between monovalent counteri- ons which is quite an original feature: in this context, the chemical nature and not only the valence of the counterion could have a drastic eect on the structure and the macroscopic properties of the gel. Moreover, considering that counterion-specic eects are more pronounced for the case of positively charged chains surrounded by anions, 26,27 modied ionene based hydrogels appear to be ideal systems to explore these eects. The gel-forming ionene used is the 6-X modied ionene (6-X) represented in Figure 1 (B). The chloride counterions (X = Cl -) are the ones obtained by synthesis. Counterion exchange is made by dialysis.

Fluoride (X = F -), another halide counterion, and a divalent counterion (X = SO 2-4 ) are investigated here. To determine the nanoscale structure of these gels, we used a combination of small angle X-ray and neutron scattering (SAXS and SANS) techniques by taking advan-tage of the dierent contrasts of the polyelectrolyte chains and the counterions available with these two beam sources, and at the same time by choosing the appropriate solvent (D 2 O or H 2 O). The macroscopic properties are investigated by rheological measurements. Experimental section 6-Cl modied ionene (6-Cl) and precursor synthesis

The 6-Cl and its precursors were synthesized by a two-step reaction with a protocol adjusted from the one described previously, 10 as detailed in. 13 The chemical reagents were all purchased from Sigma-Aldrich and used without additional purication. The solvents were purchased from Sigma-Aldrich or VWR.

First step: synthesis of the precursor 1,4-bis-[4-(chloromethyl)ben-zamido]benzene

The vials used were all dried in an oven at 80

• C 24h before use. p -phenylenediamine (1.08 g, 10 mmol, 1 eq) was dissolved in freshly distilled dichloromethane (100 mL) in a 250 mL round-bottom ask with a magnetic stirrer and placed under an inert argon atmosphere. The solution of p -phenylenediamine in the round-bottom ask was then cooled down to 0 • C via an ice-water bath and triethylamine (3.5 mL, 25 mmol, 2.5 eq) was added. A solution of 4-(chloromethyl)benzoyl chloride (3.78 g, 20 mmol, 2 eq) dissolved in distilled dichloromethane (70 mL) was inserted drop-wise with an isobar dropping funnel. After the addition, the icewater bath was removed, and the system was stirred at room temperature for 24h to ensure full conversion. The obtained white precipitate was vacuum ltrated and then washed with distilled water (2 x 80 mL). The product was dried under vacuum at 90

• C. The product was obtained in high yield (85%). Structural analysis was carried out by 1 H-NMR in DMSO-d 6 .

Second step: synthesis of 6-Cl ) solutions for 24h each. The excess salt was removed by 5 successive baths of ultra pure water. We checked that all the excess salt was removed by measuring the conductivity of the bath (the conductivity was equal to the one of ultra pure water after 5 successive baths). The contents of the dialysis cassette was then collected and freeze-dried to remove water.

The 6-X hydrogels were then formed by heating the 6-X modied ionene solution at 70

• C (X = Cl -, F -) or at 90
• C (X = SO 2-4 ) for 10 minutes followed by cooling to room temperature.

The CGC of the 6-X hydrogels were determined via vial inversion method and rheology measurements.

Small Angle X-ray Scattering (SAXS)

SAXS experiments were performed on the SWING beamline in SOLEIL synchrotron (Orsay, France) at a xed energy of 16 keV. Two sample-to-detector distances of 0.5 m and 6 m were used, allowing to have a scattering wave vector (q) range of 0.002 Å

-1 < q < 2 Å -1 .
Cylindrical glass capillaries of 1 mm in diameter were lled with the sample in the solution state (at 70

• C) and ame sealed. The hydrogels are then formed inside the capillaries by cooling down to room temperature. Two-dimensional scattering patterns were recorded on an Eiger 4M (Dectris) detector. The values of the scattering length density and the contrast of the main components of the system will be presented later in this article. The size of the symbols of the SAXS curves reects the errors bars. SAXS data are presented only in arbitrary units, passage to absolute units was unsactisfactory due to technical problems throughout the experiment (variations in capillary diameter).

Small Angle Neutron Scattering (SANS)

SANS experiments were performed on the D22 spectrometer at the Institute Laue-Langevin (ILL) (Grenoble, France) and on the PAXY facility at the Laboratoire Léon Brillouin (LLB) (CEA Saclay, France) under the condition of full contrast (in D 2 O solvent). On the D22 spectrometer, two dierent sample-to-detector distances of 3 and 17 m were used. The q range available was thus 0.002 Å -1 < q < 0.6 Å -1 . On PAXY, three dierent sample-todetector distances of 1, 3 and 5 m were used together with a neutron wavelength of 4, 5 and 8.5 Å respectively. The q range available was thus 0.006 Å -1 < q < 0.6 Å -1 . SANS measurements were carried out with samples inside quartz cells of path length of 2 mm.

The values of the scattering length density and the contrast of the main components of the system will be presented later in this article.

Rheology

Rheology measurements were conducted on a strain-controlled rheometer, ARES TA instruments (LS1) equipped with a cone-plate geometry of 20 mm in diameter, an angle of 0.04 rad and a gap of 0.043 mm at 25

• C. The samples were introduced into the rheometer one day after gel formation. The gel texture is not solid but rather soft and spreadable and the sample holder (plate geometry) was easily lled with the gel using a spatula. Then, the cone-plate gap was set to 0.12 mm and the excess of the sample at the edges of the cone was removed. Lastly, we xed the gap at 0.043 mm for the measurements. We observed the presence of a slight excess of the gel at the edges which indicated a good lling of the gap.

First, a short strain sweep at 1 Hz was performed to identify the linear viscoelastic plateau.

Then, a frequency sweep was performed in the linear viscoelastic regime at 1% in strain in a pulsation range from 1 to 100 rad/s. Finally, a strain sweep at 1 Hz was performed from 0.1% to 100% (ramp up) and from 100% to 0.1% (ramp down). The duration of the measurements was less than the sample drying time. When the measurement was longer, the geometry was protected by a hermetic cell to avoid the drying of the sample. Each measurement was at rst repeated twice with two extracts from the same sample to check the homogeneity of the system. For the same sample the measurement from the two extracts was very reproducible (less than 2% deviation). The nal data points are an average of two independent measurements for each sample (deviation between 5% and 10%). The elastic modulus values (G') are extracted from the average G' values of the at prole in the frequency sweeps.

Results and discussion

Gelation mechanisms of modied ionene hydrogels are multiple and complex processes that 11 and recently by our team in a previous article. 13 The main mechanisms proposed are hydrogen bond interactions between amide groups, π-π stacking interactions and (N + )-π cation interactions as well as electrostatic interactions. It has been assumed that hydrogen bonds between the amide functions are formed indirectly from interactions with chloride counterions and water molecules (Figure 1). The nature of counterions could thus play a predominant role in the gelation mechanism. We choose to work with two halide counterions (Cl and F -) and also to compare with a divalent counterion (SO 2-4 ).

Counterion eects on the macroscopic properties

The counterion specic eect on ionene-based gels was rst investigated on macroscopic properties. For all counterions studied and for all concentrations investigated, modied ionene hydrogels exhibit a viscoelastic behaviour. In the linear domain, the elastic modulus G' is always a decade higher than the viscous modulus G and they are both independent of the frequency as shown in the frequency sweep in Supporting information Figure S1 (A), a typical feature of a gel. As a function of strain, the systems exhibit a linear viscoelastic plateau at low strains (less than 1% in strain) for which the modied ionene gels have an elastic behaviour (see the strain sweep and the evolution of stress as a function of strain in Supporting information Figure S1 (B)-(C)). Above a threshold stress (from 3% to 25% in strain), the modulus G' decreases sharply with deformation and becomes lower than the modulus G: the gel is destructured and ows.

The evolution of the elastic modulus G' as a function of the 6-X modied ionene concentration (C 6-X ) in the linear domain is shown in Figure 2 (A). C 6-X is in moles of repeat unit presented in Figure 1 (B), each repeat unit containing two charges. We ensured that dialysis had no inuence on the rheological properties of the system by performing rheology measurements on 6-Cl hydrogel with non-dialysed and dialysed 6-Cl (see the evolution of G' as a function of concentration with non-dialysed and dialysed 6-Cl in Supporting information Figure S2). The elastic modulus values overlap well for both systems. The rst remarkable dierence between the studied counterions is the critical gelation concentration (CGC) of the corresponding gels. The CGC is dierent for each counterion and evolves as follow: CGC (6-F) = 0.035 M (20 g/L) > CGC (6-Cl) = 0.017 M (10 g/L) > CGC (6-SO 4 ) = 0.007 M (5 g/L). Moreover, the values of G' are much higher for 6-SO 4 than for 6-Cl and 6-F at the same concentration. At 0.035 M for example, the order of magnitude of G' is ∼ 5000 Pa for 6-SO 4 , ∼ 400 Pa for 6-Cl and only ∼ 1 Pa for 6-F. However, from 0.04 M, 6-SO 4 no longer forms a gel and precipitates, as a result of too strong attractive chain-chain interactions.

For all three counterions, the evolution of G' as a function of the 6-X concentration is described by a power law, with a dierent characteristic exponent for each counterion. Thus the disappearance of this peak is not due to a change in the concentration regime but due to a specic eect of the counterions. In the case of polarizable monovalent counterions, it is suggested that the absence of the peak is due to partial "condensation" of counterions onto the ionene backbone, leading to screening of repulsions between the chains. For the divalent counterions SO 2-4 and C 2 O 2-4 , the absence of a polyelectrolyte peak could be due to a bridging eect of counterions between the ionene chains, 37 a phenomena already observed for others charged systems.

3739

The structure of the 6-Cl modied ionene solution and gel are dierent from the one of conventional ionene solutions as already described in a previous paper.

13 Figure 3 (A) displays the scattering intensity of a 6-Cl solution at 70°C and gel at room temperature after the heating-cooling process at 0.051 M. In the middle-q region, the scattering intensity evolves as I ∼ q -2 , a typical feature for polymer systems, be it for solutions or gels. [START_REF] Horkay | Structural investigations of a neutralized polyelectrolyte gel and an associating neutral hydrogel[END_REF] At low q region, an upturn in the scattering intensity is observed which is associated with large-scale heterogeneities, very often seen for polyelectrolytes and gels.

41

Figure 3: (A) Scattering intensity I(q) as a function of the wave vector q and (B) Kratky plots (q 2 I(q) as a function of the wave vector q) for the 6-Cl hydrogel (RT) and the 6-Cl solution (70°C). (C) Scheme illustrating the two characteristic lengths of the gel network.

(D) Scattering intensity I(q) as a function of the wave vector q by SAXS and SANS for the The scattering curve of the 6-Cl gel in Figure 3 (A) presents a broad peak around 0.03 Å -1

and a shoulder around 0.1 Å -1 . Regarding the former, it is not a well dened polyelectrolyte peak but on the other hand, the scattered intensity cannot be fully described by an Ornstein-Zernike (OZ) function (I = I 0 1+(qΞ) 2 , Ξ being the correlation length of the system) [START_REF] Ornstein | Accidental Deviations of Density and Opalescence at the Critical Point of a Single Substance[END_REF] which is traditionally used to describe the typical behaviour for uncharged polymer solutions and gels. 40,43,44 In other words, in this polyelectrolyte gel the electrostatic chain-chain repulsions compete with attractions between the "aromatic" blocks in the chains, leading to a "weak" polyelectrolyte behaviour. As the counterions are involved in both the electrostatic chainchain repulsion (through screening) and in the attractive interactions (see again Figure 1 (C)), the system is indeed very sensitive to the counterion nature.

Regarding the shoulder around 0.1 Å -1 , this contribution is already present in the 6- Cl solution at high temperature, as is seen in the scattering curve in Figure 3 (A). This observation leads to the description of the scattering intensity of the gel in terms of two distinct contributions:

45 one due to thermal concentration uctuations corresponding to the scattering intensity of the polymer solution, and an excess scattering intensity due to heterogeneities introduced by cross-linking. From this two contributions, we can extract two characteristic correlation lengths related to the polymer network in the gel. The Kratky representation displayed in Figure 3 (B) reveals in the best way the values of these correlation lengths. The rst peak in the low q region, around 0.03 Å -1 characterises the distance between the cross-linking nodes. The mesh-size of the gel due to these nodes is thus estimated as Ξ = 2π q * = 21 nm, q * being the value of the maximum of the peak. The second peak around 0.1 Å -1 characterises the chain-chain correlation due to thermal concentration uctuations already present in solution before gelation. [START_REF] Stieger | Small-angle neutron scattering study of structural changes in temperature sensitive microgel colloids[END_REF] This correlation length is estimated as ξ = 6

nm. These two correlation lengths are depicted schematically in Figure 3 (C).

For this study we aimed to take advantage of both small angle X-rays and neutron scattering techniques to describe the structure of the hydrogel as a function of the nature of counterions. The scattering curves from SAXS and SANS shown in Figure 3 (D) for a 6-Cl gel display very similar proles (curves in arbitrary y-units). When plotting the Kratky plots q 2 I(q) of the SAXS and SANS curves (Figure 3 (E) and Figure 3 (F)), it can be seen that the two correlation peaks present dierent relative intensities. Indeed, the rst correlation peak corresponding to the correlation length Ξ is much more intense in SAXS curve while the second correlation peak is better visible in SANS curve. The mesh-size of the gel Ξ is located at similar q values in SAXS and SANS curves whereas the correlation length ξ is located at lower q values by SAXS compared to SANS. This feature has already been observed in polyelectrolyte solutions and is attributed to dierences in the intra-molecular interactions that can be seen between SAXS and SANS due to the dierent contrasts.

47 In small angle scattering, the scattered intensity I(q) by polyelectrolytes is composed of both the signal of the chains and the signal of the counterions as follows:

I(q) = (∆ρ) 2 m S mm (q) + (∆ρ) 2 c S cc (q) + 2(∆ρ) m (∆ρ) c S cm (q) (1)
Where S mm (q), S cc (q) and S cm (q) are respectively the monomer/monomer, counterion/counterion and counterion/monomer partial scattering functions.

37,47

6im , the (∆ρ) X (∆ρ) 6im values and the contrast values of the dierent counterions (∆ρ) 2 X in heavy water (for SANS) and in light water (for SAXS).

The respective contributions of the dierent partial scattering functions are weighted by the contrasts of the monomers and the counterions. Thus, from the contrast values it is possible to estimate the relative strength of each contribution. The relative proportions of the contrasts to the total scattering intensity for the 6-Cl gel are given in Table 2. The scattering intensity in SAXS thus comes mainly from the contrast of the chloride counter ions with water (at 75%) while in SANS the scattered intensity comes mainly from the monomer/monomer and monomer/counter ion correlations. In SAXS, the peak corresponding to the gel mesh size Ξ is much more intense than the one corresponding to the inter-chain distance ξ. This points clearly towards the chloride counterions being heavily involved in the formation of the mesh of the gel. The correlation length ξ mainly involves monomer/monomer and monomer/counterion correlations.

The SAXS and SANS scattering intensities of the modied 6-F ionene hydrogel (0.054 M) are plotted Figure 4 (A)-(C). With uoride counterions, three dierent correlation peaks are now observed: a rst one around 0.027 Å -1 , a second one around 0.058 Å -1 and a third one around 0.11 Å -1 . The third correlation peak observed in the gel is attributed to the inter-chain correlation length ξ, estimated to be around 6 nm because we observed the same correlation length in the corresponding high-temperature solution (the scattering intensity and the Kratky plot of the high-temperature solution are presented in Figure S3). Thus the rst two correlation peaks are assigned to two correlation lengths Ξ 1 and Ξ 2 related to the gel mesh of 23 nm and 11 nm respectively. Comparing the SAXS and SANS Kratky plots of the same gel (Figure 4 (B)-(C)), the second correlation peak corresponding to Ξ 2 is more pronounced in SAXS than in SANS. According to Table 2, in the same way as for the chloride counterions, the scattered intensity in SAXS comes mainly from the contrast of the uoride counterions with water (at 67%). In SANS, the scattered intensity comes mainly from the monomer/monomer and monomer/counterion correlations. Thus, the correlation length Ξ 2 extracted from the second correlation peak is related to the F --F -correlation within the gel. techniques. This is due to the fact that the relative proportions of the contrasts to the total scattering intensity are similar in SAXS and SANS (see Table 2). In the same way as for the 6-Cl gel, two correlation peaks can again be distinguished. The rst peak at 0.026 Å -1

is attributed to the correlation length Ξ 1 estimated at 23 nm. A second peak at 0.07 Å -1 corresponding to a distance of 9 nm is also observed. It is dicult to determine whether this correlation peak corresponds to a distance associated with the gel mesh or to an inter-chain distance as we will see later on. We will refer to this distance as Ξ 2 in the following.

We now look at the evolution of the structure of the modied ionene gels as a function of the polyelectrolyte concentration, see Figure 5 (A) to Figure 5 (H). We observe for the 6-Cl gels that the rst peak evolves towards larger q values, i.e. the gel mesh Ξ decreases when increasing the concentration of modied ionene (Figures 5 (A)-(B) and 5 (G)). The length ξ evolves with a power law as ξ ∼ c -0.45 , which corresponds to the typical concentra-tion behaviour of a polyelectrolyte solution in the semi-diluted regime [START_REF] De Gennes | Remarks on polyelectrolyte conformation[END_REF] (ξ ∼ c -0.5 in that case) (Figure 5 (H)). The two correlation peaks become increasingly intense with increasing concentration. For 6-F gels, the longest correlation length Ξ 1 also decreases with increasing concentration as seen in Figures 5 (C)-(D) and (G). A weaker dependence is seen for Ξ 2 .

What is especially remarkable is that the second correlation peak related to Ξ 2 becomes increasingly more intense, as the polyelectrolyte concentration increases. The intensity evolution of the two peaks corresponding to Ξ 1 and Ξ 2 eliminates the hypothesis that these two peaks are overtones of each other. The closest picture we propose is the presence of two types of domains in the gel, corresponding to the two dierent Ξ. The domains with a denser mesh become dominant as the 6-F concentration increases. The length ξ follows the same trend as for the 6-Cl gels. Overall, the length Ξ 1 (6-F) is larger than the length Ξ (6-Cl) while the length Ξ 2 (6-F) is smaller than that of 6-Cl as depicted in Figure 5 (G). When concentrating the system, the smaller distance Ξ 2 within the 6-F gel mesh overrides the larger characteristic length Ξ 1 . The emergence of this smaller distance could explain the rapid evolution of the elastic modulus (dashed line in Figure 2 (A)) of the 6-F gel compared to 6-Cl (G' ∼ c 4 ) once the CGC of 6-F is reached, and up to 0.07 M (Figure 2 (A)). A second denser network is then present, observed only for the gel with the uoride counterion. Beyond this critical point, the evolution of G' as a function of concentration becomes similar to that observed for the other two counterions, with G' ∼ c 4.5 . At the same time, we notice that the peak corresponding to Ξ 2 (6-F) starts to shift towards larger values of q. The Ξ 2 length then decreases from 11 nm to 9.5 nm. The beginning of a behaviour similar to the one of 6-Cl is then observed, i.e. a decrease of Ξ with the increase of the modied ionene concentration. To conrm this, it will be necessary to measure the scattering intensity of 6-F gels at higher concentrations of observed. When NaF salt is added to the 0.09 M 6-F gel (Figure 6 (D)), the correlation length Ξ 2 is observed to decrease from 9.5 nm to 8 nm. Moreover, the higher the amount of salt, the more the three correlation peaks lose intensity. The addition of salt therefore has an eect on all correlation lengths of the system. As expected, the increase of the ionic strength leads to charge screening. The loss of intensity of the correlation peaks is a result of the decrease of electrostatic repulsions between the polyelectrolyte chains. The decrease in these repulsions favours the associations between the chains, cross-linking can occur more frequently and a denser mesh is observed (smaller mesh size), especially in the case of the 6-Cl gel.

The above structural changes induced by the addition of salt are reected also in the macroscopic properties of the hydrogels. Frequency sweeps (from 1 to 100 rad/s) of hydrogels upon addition of salts show the same behavior, i.e. a fully at G'-G prole, as those for pure hydrogels. Is it thus possible to compare the G' values when adding salts. It is seen from Figure 7 that it leads to an increase in the elastic modulus G'. Thus, the addition of salt allows the formation of a denser network by decreasing the electrostatic repulsions between the modied ionene chains and thus favouring the attractive interactions between the chains.

For the modied 6-Cl ionene gel, the G' increases up to 0.075 M in salt. From 0.1 M in NaCl, the system precipitates. For 6-F, systems with NaF additions at concentrations above 0.1 M have not been studied, but it is reasonable to assume that the system also precipitates above a certain salt concentration. Indeed, addition of salt screens the charges and when the attraction between the chains is too strong, the ionene precipitates. The F ions are more strongly hydrated than the Cl ions which explains why, compared to the 6-F gel, the 6-Cl gel strengthens over a narrower range of salt concentrations prior to precipitation.

Conclusion

Ionene-based hydrogels are formed through hydrogen bonding, π-π stacking interactions and cation-π interactions with the quaternary ammonium N + charged sites of the modied ionene chains and anion-π interactions with the counterions. These attractive interactions counterbalance the electrostatic repulsions between chains inherent in the presence of charged groups. This behaviour is clearly visible in the SAXS and SANS scattering curves. Indeed, the correlation peak corresponding to the characteristic distance of the gel mesh cannot be completely described either by an Ornstein Zernike function characteristic of a neutral polymer, or by a polyelectrolyte peak. The behaviour of the modied ionene chain thus results in an intermediate behaviour between that of a polyelectrolyte and that of a neutral polymer. A strong counterion specic eect exists within modied ionene hydrogels. In the case of halide counterions, gelling is less ecient when the counterions are strongly hydrated (F -), i.e. when the ions are better dissociated from polyelectrolyte chains with a more extended counterion cloud around the chain. This extends the concentration range where chain-chain electrostatic repulsions dominate and delays the gel formation, for which attractive chain-chain interaction are needed in order to form cross-linking nodes. The critical gelation concentration is thus higher than with a more weakly hydrated halide counterion (Cl -). Once the critical gelation concentration is reached, a very rapid increase in the elastic modulus of the gel is observed with the uoride counterion. In SAXS, two distances are observed characterising the mesh of 6-F gel: a larger distance than the one characterising the mesh of the 6-Cl gel and a smaller one. As the two peaks in SAXS corresponding to the two distances are not overtones (intensity evolution with concentration is inconsistent with this), two types of gel domains seem to co-exist (one related to Ξ 1 and one to Ξ 2 ). The domains with a denser mesh become dominant with the increase of the 6-F concentration in the gel and this correlates with a fast increase of the elastic modulus of the gel (the dashed line for 6-F as compared to G' ∼ c 4 for 6-Cl). Once the elastic modulus reaches values similar to those of 6-Cl, the evolution of the elastic modulus of 6-F with concentration becomes similar to that of 6-Cl. In the case of divalent counterions (SO 2-4 ), the critical gelation concentration is lower than for monovalent ions. In addition, the elastic modulus of the gel is higher than for gels with monovalent counterions at the same concentration. The presence of divalent counterions decreases electrostatic repulsions and favours the association of chains, probably by a bridging eect. On the other hand, there is a critical concentration above which the gel precipitates as a result of too strong attractive interactions between chains. The network structure of the 6-SO 4 gel seems to have two dierent characteristic lengths. The concentration dependence of the second (shorter) length suggests the formation of larger structures (as opposed to a densication of a mesh) in this particular system. Overall, in the reported gels, the balance between electrostatic repulsion and attractive interactions (hydrogen bonding, π-π stacking interactions and (N + )-π cation interactions) is necessary to create cross-linking nodes and at the same time allow hydrated regions to be retained. If the electrostatic repulsion is too strong, the gel cannot form, while if the attractive interactions are too strong, the chains aggregate and the system precipitates. In conclusion, ionenebased hydrogels present a family of gels with very ion-sensitive structural and rheological properties. At this stage, the permeability of these gels to (charged) molecular and colloidal species is a promising eld of study and could pave the way to potential applications for controlled release of active substances.

Figure 1 :

 1 Figure 1: (A) Structure of a linear conventional ionene, y and z representing the number of methylene groups linking the charged quaternary ammonium centers. X represents the counterions. (B) Structure and typical lengths of the 6-X modied ionene (6 representing the number of carbons between two charged quaternary ammonium) composed of charged and aromatic blocks (in green and orange respectively). X represents the counterions. (C) Interchain physical interactions necessary for gel formation in the case of modied ionenes: hydrogen bonding, π-π stacking, (N + )-π and anion-π interactions, as well as indirect hydrogen bonding via counterions and water molecules.

  have been partially described in the literature byMisawa et al.,

10 Figure 2 :

 102 Figure 2 (A). The exponent of 2.6 is close to the predicted value of 2.67 correponding to a

6 -

 6 Cl hydrogel at 6-Cl 0.051 M. The peak marked by a star on the gure corresponds to a parasitic reection arising from the Kapton polymer used as a window material. (E) and (F) Kratky plots by SAXS and SANS for the 6-Cl modied ionene hydrogel at 6-Cl 0.051 M.

Figure 4 :

 4 Figure 4: (A) Scattering intensity as a function of q and (B)-(C) Kratky plots (q 2 I(q) as a function of the wave vector q) by SAXS and SANS respectively for the 6-F modied ionene hydrogel at 0.054 M. (D) Scattering intensity as a function of q and (E)-(F) Kratky plots by SAXS and SANS respectively for the 6-SO 4 modied ionene hydrogel at 0.01 M. The peak marked by a star on the gures corresponds to a parasitic reection arising from the Kapton polymer used as a window material.

Figure 5 :

 5 Figure 5: (A)-(F) Scattering intensity and Kratky plots (q 2 I(q) as a function of the wave vector q) for modied ionene hydrogels at dierent modied ionene concentrations by SAXS for Cl and F counterions and by SANS for SO 2-4 counterion. The curves (B), (D) and (F) are shifted along the y-axis for a better visualization. (G)-(H) Evolution of the gel mesh size Ξ and the inter-chain distance ξ as a function of modied ionene concentration for the three counterions studied. The peak marked by a star on the gures corresponds to a parasitic reection arising from the Kapton polymer used as a window material.

Figure 6 :

 6 Figure 6: Scattering intensity and Kratky plots from SAXS measurements (q 2 I(q) as a function of the wave vector q) for (A)-(B) 6-Cl and (C)-(D) 6-F modied ionene hydrogels upon addition of salt (NaCl and NaF respectively). (E)-(F) evolution of Ξ and Ξ 2 for 6-Cl and 6-F modied ionene hydrogels upon addition of salt (NaCl and NaF respectively). The curves in (B) and (D) are shifted along the y-axis for a better visualization. The peak marked by a star on the gures corresponds to a parasitic reection arising from the Kapton polymer used as a window material.

Figure 7 :

 7 Figure 7: Evolution of the storage modulus G' of the hydrogel as a function of the concentration of (A) NaCl for 6-Cl modied ionene gel and (B) NaF for 6-F modied ionene gel.

Table 1

 1

gives the contrast values of the modied ionene monomer (∆ρ)

Table 1

 1

	: Values of the contrast (∆ρ	2 ) (cm	-4 ) for the 6 modied ionene monomer and the
	several counterions as well as the (∆ρ) X (∆ρ) 6im values with H 2 O and D 2 O in SAXS and
	SANS respectively.
	SAXS (solvent: H 2 O)	(∆ρ) 2 6im 3.8 10 19	(∆ρ) 2 Cl -1.45 10 21	(∆ρ) 2 F -7.45 10 20	(∆ρ) 2 SO 2-4 7.2 10 19	(∆ρ) Cl -(∆ρ) 6im 2.3 10 20	(∆ρ) F -(∆ρ) 6im 1.7 10 20	(∆ρ) 6im 5.2 10 19 (∆ρ) SO 2-4
	SANS (solvent: D 2 O)	2.5 10 21	1.4 10 21	1.3 10 21	2.2 10 21	1.9 10 21	1.8 10 21	2.35 10 21

Table 2 :

 2 Relative proportions of contrast to total scattering intensity in SAXS and SANS for the 6-Cl, 6-F and 6-SO 4 modied ionene hydrogels.

	6-Cl SAXS (solvent: H 2 O)	%(∆ρ) 2 6im 2%	%(∆ρ) 2 Cl -75%	%(∆ρ) Cl -(∆ρ) 6im 23%
	SANS (solvent: D 2 O)	33%	18%	49%
	6-F SAXS (solvent: H 2 O)	%(∆ρ) 2 6im 3%	%(∆ρ) 2 F -67%	%(∆ρ) F -(∆ρ) 6im 30%
	SANS (solvent: D 2 O)	33%	18%	49%
	6-SO 4 SAXS (solvent: H 2 O)	%(∆ρ) 2 6im 18%	%(∆ρ) 2 SO 2-4 34%	%(∆ρ) SO 2-4 44% (∆ρ) 6im
	SANS (solvent: D 2 O)	27%	24%	49%

Acknowledgement

The authors thank the SOLEIL synchrotron for awarding the beamtime on beamline SWING.

We would like also to acknowledge the Laboratoire Léon Brillouin for awarding the beamtime on PAXY beamline. The authors thank the Institut Laue-Langevin for the beamtime allocation on fast access (DOI: 10.5291/ILL-DATA.[86516]) for the SANS measurements. The authors thank Laurent Michot, Véronique Peyre and Jérôme Combet for fruitful discussions.

modied ionene.

The case of 6-SO 4 gels needs to be considered apart. The SO 4 -gels show a signicant departure from the q -2 dependence in scattering intensity for the two highest concentrations compared to the gels with halide counterions, see Figure 5 (F). The strong attraction between the chains, via the bridging eect of divalent counterions (the "egg-box" model), probably leads to the formation of larger structures (contrary to a densication of a mesh). This is consistent with the concentration dependence of the second, shorter characteristic length Ξ 2 , which increases with increasing concentration, unlike all other characteristic lengths presented here. Note that the larger length, Ξ 1 , is constant between 25 nm and 23 nm.

The "egg-box" model evokes the formation of linear, lament structures, however no q -1 dependence emerges in the spectra. 49 We do not have the scattering spectra of the SO 4 solutions, which would to some extent help with the assignment of the peaks and thus we do not attempt a more detailed tting at this stage. Lastly, the strong chain-chain attraction in the case of SO 4 gels is also reected in the appearance of syneresis when the gel is left to rest after several weeks. It is the only gel type presented here, for which syneresis is observed.

Eect of salinity

In parallel to the nature of the counterions, additional salts dramatically change the conformation of polyelectrolytes leading to the alteration of the gelation behaviour. To study the eect of salt, we form the gels in salt solutions of the same anion as their counterion by heating at 80

• C and then by cooling down the system to room temperature. In 6-SO 4 gels, when 0.01 M of Na 2 SO 4 salt is added, the system already precipitates, so we perform this study only for 6-Cl and 6-F gels. Figure 6 shows the scattering intensities of 6-Cl gels at 0.068 M and 6-F gels at 0.09 M upon addition of salt (NaCl and NaF respectively), and the evolution of Ξ (6-Cl) and Ξ 2 (6-F) distances in this case. For the 0.068 M 6-Cl gel, a shift in the rst correlation peak, i.e. a decrease of the gel mesh size when salt is added to the system, is observed in Figure 6 (B). In addition, a disappearance of the second correlation peak is

Declaration of Competing Interest

The authors declare that they have no known competing nancial interests or personal relationships that could have appeared to inuence the work reported in this paper.

Supporting Information Available

Supporting information. Evolution of the storage modulus and the viscous modulus as a function of the oscillation frequency and strain for the 6-X hydrogels for all counterions studied, evolution of the stress as a function of strain for the 6-X hydrogels, evolution of the storage modulus as a function of 6-Cl concentration for dialysed and non-dialysed polyelectrolyte, scattering intensity and Kratky plots for 6-F solution (80°C) and for the 6-F hydrogel (room temperature, RT).


Séverine Fontaine
email: severine.fontaine@anses.fr

Laëtitia Caddoux

Benoit Barrès

First report of the kdr pyrethroid resistance mutation in a French population of the English grain aphid, Sitobion avenae

Keywords: insecticide resistance monitoring, cereals, aphids

The lack of recent insecticide innovations and the withdrawal of efficient, but ecotoxic molecules (e.g. neonicotinoids) may lead to a shift to older insecticidal active substances, particularly pyrethroids. The evolution of resistance to this family of insecticides has long been recognized, including in many aphid species such as the grain aphid Sitobion avenae. The target-site resistance mechanism involved, named kdr (knock-down resistance), was first identified in the United Kingdom in 2014. In this study, we screened for the presence of this mutation in 25 French populations of S. avenae sampled between 2017 and 2021 on various cultivated cereals. Aphids were genotyped individually for the kdr mutation using PCR-RFLP or partial sequencing of the voltage-gated sodium channel gene. Of the 400 S. avenae individuals tested, one carried a heterozygous kdr mutation. This S. avenae kdr mutant was sampled in Northern France, the French region closest to the United Kingdom. The low frequency of the detected mutation and the geographical proximity to the UK suggest that it arose from a

Manuscript

Click here to view linked References migration event and not an independent convergent mutation event, although this hypothesis remains to be confirmed. Our study also reveals that partial sequencing of the voltage-gated sodium-channel gene can be used for another cereal aphid species, Rhopalosiphum padi.

INTRODUCTION

The English grain aphid, Sitobion avenae (Fabricius, 1775) is one of the major cereal aphid species in the temperate areas of the Northern Hemisphere. This aphid can infest a wide range of cultivated or wild cereals (wheat, barley, oat, maize, rice, cocksfoot grass, etc.). It causes direct damage by sucking sap from the host, which can lead to senescence of the flag leaf and reduced grain weight [START_REF] Wratten | The nature of the effects of the aphids Sitobion avenae and Metopolophium dirhodum on the growth of wheat[END_REF]). This species is also a vector of viruses, such as barley yellow dwarf virus (BYDV), which causes a reduction in yield and grain quality [START_REF] Edwards | Effect of Barley yellow dwarf virus infection on yield and malting quality of barley[END_REF][START_REF] Gray | Aphid transmission of barley yellow dwarf virus: acquisition access periods and virus concentration requirements[END_REF]. Controlling this pest generally involves the use of insecticides in conventional agriculture. Repeated application of pesticides exerts selection pressure on pests, which can lead to the evolution of resistance within the species with -in the long run -the risk of losing treatment effectiveness [START_REF] Hawkins | The evolutionary origins of pesticide resistance[END_REF]. S. avenea has great evolutionary potential, as demonstrated by the multiple types of resistance that have evolved against several active substances, including target-site resistance mechanisms [START_REF] Foster | A mutation (L1014F) in the voltage-gated sodium channel of the grain aphid, Sitobion avenae, is associated with resistance to pyrethroid insecticides[END_REF]) and non-target site resistance mechanisms (Zang et al. 2017[START_REF] Gong | Field evolved resistance to pyrethroids, neonicotinoids, organophosphates and macrolides in Rhopalosiphum padi (Linnaeus) and Sitobion avenae (Fabricius) from China[END_REF]. Some French populations of S. avenea can reproduce both sexually and asexually [START_REF] Simon | Reproductive mode and population genetic structure of the cereal aphid Sitobion avenae studied using phenotypic and microsatellite markers[END_REF][START_REF] Dedryver | Coexistence in space and time of sexual and asexual populations of the cereal aphid Sitobion avenae[END_REF], thereby allowing genetic recombination and rapid dispersal of competitive clones by asexual multiplication. Although other active substances have since been developed, pyrethroids remain one of the main insecticide families used to control aphid populations on cereal crops in France. In Europe, pyrethroid-resistant individuals of S. avenae were first reported in 2014 on populations collected in 2011 in England and Scotland [START_REF] Foster | A mutation (L1014F) in the voltage-gated sodium channel of the grain aphid, Sitobion avenae, is associated with resistance to pyrethroid insecticides[END_REF]. The resistance mechanism involves a target-site mutation, resulting in the aminoacid substitution of a leucine by a phenylalanine on codon 1014 (L1014F) of the voltage-gated sodium channel, the pyrethroid target. This mutation, named kdr (knock-down resistance), is one of the most common pyrethroid-resistance mechanisms in insects [START_REF] Dong | Molecular biology of insect sodium channels and pyrethroid resistance[END_REF].

Several other mutations affecting the voltage-gated sodium channel have also been described, especially in aphid species. They are generally located at position 918 and they can be found with or without kdr [START_REF] Eleftherianos | Characterization of the M918T sodium channel gene mutation associated with strong resistance to pyrethroid insecticides in the peach-potato aphid, Myzus persicae (Sulzer)[END_REF][START_REF] Fontaine | Uncommon associations in target resistance among French populations of Myzus persicae from oilseed rape crops[END_REF][START_REF] Panini | Presence and impact of allelic variations of two alternative s-kdr mutations, M918T and M918L, in the voltage-gated sodium channel of the green peach aphid Myzus persicae[END_REF][START_REF] Carletto | Insecticide resistance traits differ among and within host races in Aphis gossypii[END_REF]. Mutations affecting this codon are called skdr (super knock-down resistance), because they most often lead to very high level of resistance to pyrethroids [START_REF] Eleftherianos | Characterization of the M918T sodium channel gene mutation associated with strong resistance to pyrethroid insecticides in the peach-potato aphid, Myzus persicae (Sulzer)[END_REF][START_REF] Fontaine | Uncommon associations in target resistance among French populations of Myzus persicae from oilseed rape crops[END_REF].

Considering the pyrethroid spray failures caused by a resistant S. avenae clone in the UK [START_REF] Foster | A mutation (L1014F) in the voltage-gated sodium channel of the grain aphid, Sitobion avenae, is associated with resistance to pyrethroid insecticides[END_REF]) and the wide use of pyrethroids on cereals in France, we screened for the presence of kdr between 2017 and 2021, as part of the French system for monitoring the adverse effects of plant protection products.

MATERIALS AND METHODS

Aphid sampling

Field samples were done from 2017 to 2021 by harvesting 30 to 40 randomly distributed cereal plants infested with S. avenae. Live aphids were collected with a brush and individually placed in 2 mL Eppendorf tubes with a sterile stainless-steel bead of 4.76 mm diameter. Aphids were stored at -80°C until RNA extraction.

RNA Purification

Samples were ground in a TissueLyser II (Qiagen, Hilden, Deutschland) with two cycles of 2 min at 20 Hz after adding 100 µL of buffer RA1 with 2% TCEP provided in the Nucleospin RNA Mini kit (Macherey Nagel, Düren, Deutchland). RNA was then purified according to the manufacturer's instructions. Final elution was done in 10 µL of RNase-free water. Purified RNA was stored at -80°C until use in a reverse-transcription polymerase chain reaction (RT-PCR).

Pyrethroid resistance genotyping PCR-RFLP genotyping for kdr mutation

From 2017 to 2018, the presence of the kdr mutation in the voltage-gated sodium-channel gene was determined using the RT-PCR method described in [START_REF] Foster | A mutation (L1014F) in the voltage-gated sodium channel of the grain aphid, Sitobion avenae, is associated with resistance to pyrethroid insecticides[END_REF]. This method requires reverse transcription of RNA into DNA, followed by PCR with the Sa1 and Sa3 primers.

RT-PCR was performed with the Superscript One Step RT-PCR system with Platinum Taq kit (Invitrogen, city, CA, USA). The reaction mix contained 1x reaction buffer, 400 µM of each primer and 0.25 µL of RT/Platinum Taq mix. PCR products were digested with the enzyme BssSαI (New England Biolabs, Ipswich, United Kingdom) at 37°C for at least 4 h. This enzyme cuts the sequence '5'-CACGAG-3', which can differentiate the wild-type sequence (a single restriction site) from the mutated sequence (no restriction site).

Partial sequencing of the voltage-gated sodium channel

In addition to the PCR-RFLP method, the voltage-gated sodium-channel mRNA was partially sequenced in 2019-2021 samples. This method has the advantage of including codon 918 and codon 1014 of voltage-gated sodium channels, both known to be implicated in pyrethroid resistance, i.e. the skdr and kdr mutations, respectively. The sequencing approach also provides the opportunity to investigate the few atypical digestion profiles that were obtained due to incorrect aphid species identification during sampling (see below). RT-PCR products were directly sequenced by Genewiz (South Plainfield, NJ, USA) or Genoscreen (Lille, France) using the Sanger method with the forward primer.

RESULTS

A total of 25 populations (from 1 to 9 per year) were sampled in small grain cereals (24) and in a maize field (1) (Figure 1). Field sampling was carried out mostly in conventional fields (18 vs. 7 from trial fields). The objective was to collect 30 aphids in each sampled field, but this goal was not always reached due to the low infestation rate of some samples. In total 482 out of 501 aphids sampled (mean per sample = 20 +/-10.75 -Supplementary Data 1) were successfully analyzed. Over the four years of survey, almost two thirds of aphids were sampled in 2019 and 2021. The amplification failure rate of the analyzed aphids was low (3.6%) and related to low-quality RNA extracts.

During the two first years of the survey, resistance genotyping was performed using the PCR-RFLP method on 103 aphids; 98 of them showed a digestion profile with two fragments (one of 74 bp and one of 397 bp) corresponding to susceptible individuals. Five aphids showed atypical band profiles consisting of two bands at approximately 230 bp and 240 bp. These PCR fragments were sequenced. A BLAST analysis showed that the highest identity score (98.97%) of these fragments matched with the Rhopalosiphum padi sodium channel 1 mRNA sequence (accession number: MN972451.1). The result of the Blast analysis as well as the frequent cooccurrence of R. padi and S. avenae on straw cereals in France make us think that an error of identification took place during the sampling in the field. Interestingly, this result reveals that the primers used in this study can also amplify the voltage-gated sodium channel from RNA extracted from R. padi. In these five aphids, no mutations were identified at codon 918 or 1014, known to be implicated in pyrethroid resistance. The sequence was deposited in GenBank (accession number OM585630).

From 2019 to 2021, to avoid genotyping failure due to incorrect species identification, RT-PCR was directly followed by sequencing. The analysis revealed that one aphid had been wrongly identified and belonged to R. padi. Among 385 S. avenae successfully analyzed during the last three years of the survey, only one aphid was heterozygous for the kdr mutation (L1014F) (Figure 2). This S. avenae individual was sampled in 2020 in a straw cereal crop field, located in the located in one of the northernmost plots in this study (Figure 3). None of the analyzed aphids showed a mutation at codon 918. The partial the voltage-gated sodium-channel gene sequences of a wild and mutant clone were deposited in GenBank (accession numbers OM585631 and OM585632 for the wild and kdr-mutated sequence, respectively).

DISCUSSION

This is the first report of S. avenae with a kdr mutation in France. Using bioassays, heterozygous kdr shows resistance level that is 30-40 times greater than that of the homozygous wild-type and is associated with reports of pyrethroid spray failure [START_REF] Foster | A mutation (L1014F) in the voltage-gated sodium channel of the grain aphid, Sitobion avenae, is associated with resistance to pyrethroid insecticides[END_REF]. In the UK, only heterozygous aphids have been detected and all of them have belonged to a unique clonal lineage (Sa3). In French S. avenae populations, the kdr mutation appears to be present at a very low frequency, which contrasts strongly with the situation observed in the UK, where more than 50 % S. avenae collected were resistant [START_REF] Walsh | In-field prevalence of resistant grain aphid Sitobion avenae (fabricius)[END_REF]). We could not determine whether the clone identified in France is identical to that detected in the UK. Given the geographical proximity between the UK and Northern France where the resistant aphid was detected, a migratory event (natural or related to human activity) is the most likely origin of the mutation. However, we cannot completely rule out independent emergence of the kdr mutation. Genotyping neutral markers is required to determine whether the resistant French aphid is related to the Sa3 clone. Our study was conducted on RNA and not on DNA, precluding DNA analysis. Recent studies have reported a fitness trade-off in S. avenae clones with kdr, such as lower population growth and lower individual relative growth as well as a greater rate of parasitization by Aphidius ervi [START_REF] Jackson | Grain aphids (Sitobion avenae) with knockdown resistance (kdr) to insecticide exhibit fitness trade-offs, including increased vulnerability to the natural enemy Aphidius ervi[END_REF]) and later onset of male forms [START_REF] Walsh | Evaluating the impact of pyrethroid insecticide resistance on reproductive fitness in Sitobion avenae[END_REF]. Although the very low kdr allele frequency in our survey (0.2%) can be suggestive of fitness trade-offs, this finding must be interpreted with caution. The situation in the UK has shown how a resistant clone can spread rapidly across a large area due to selection pressure in the presence of pyrethroids.

Interestingly, the primers published by [START_REF] Foster | A mutation (L1014F) in the voltage-gated sodium channel of the grain aphid, Sitobion avenae, is associated with resistance to pyrethroid insecticides[END_REF] and used in this study were able to amplify the sodium channel from RNA of another aphid species living on cereal crops. Cereals are often co-infected with different aphid species at different stages of development, which can complicate aphid identification. Adult aphids can be easily identified, but species determination can be more difficult at the larval stage. Therefore, some identification errors (1%) can occur during aphid sampling of infested cereals. Both aphid species experience comparable insecticide treatments, and pyrethroid resistance involving voltage-gated sodium-channel mutations has been recently reported in Chinese populations of R. padi [START_REF] Wang | Super-kdr mutation M918L and multiple cytochrome P450s associated with the resistance of Rhopalosiphum padi to pyrethroid[END_REF]. Therefore, the availability of a tool for the characterization of the kdr mutation compatible with S. avenae and R. padi, two aphids widespread on cereal crops, offers the advantage of detecting resistance-related genotypes even if the exact identification of the species is uncertain. This tool could be easily used in the monitoring framework planned to early detection and following of pyrethroid resistance involving target site modification specially, in countries where the withdrawal of neonicotinoids has occurred or where cases of resistance are suspected in these species of cereals.

Figure 1 :

 1 Figure 1: Map of fields sampled from 2017 to 2021. The pie charts indicate the proportion of

Figure 2 :

 2 Figure 2: Chromatographs of wild type and kdr heterozygous S. avenae sequences around the

Figure 3 :

 3 Figure 3: Map of the distribution of kdr, the target site of pyrethroid resistance, in populations

  Figure(s) Click here to access/download;Figure(s);fig1Host.pdf


Samuel Bronstein

Graham Andrew Smith

A geometric description of the set of almost-fuchsian representations

We prove that the set of proper minimal disks in the real hyperbolic 3-space whose principal curvatures belong in a closed set of (-1, 1) is parametrized by a convex set of Hopf differentials. We give a quantitative estimate of this convex set in terms of sup norms of the quadratic differentials. Applying this result to disks equivariant under some surface group action, we get a parametrization of almost-fuchsian representations by a fiberwise convex open set in the cotangent bundle to Teichmüller space.

Proposition 3.3. Let φ be a holomorphic quadratic differential on the disk. Then there is at most one bounded solution to (3.4)

In this paper we study the geometry of the space of complete almost-fuchsian minimal disks in H 3 . In the special case of disks equivariant under the action of some cocompact surface group in H 3 , this naturally gives a description of the moduli space of almost-fuchsian representations by a fiberwise convex subset of the cotangent bundle of Teichmüller space. The same discussion yields a description of minimal almost-fuchsian disks invariant under some fixed group Γ < PSL(2, R).

Let Σ g be a closed hyperbolic surface of genus g ≥ 2 and let Γ = π 1 (Σ g ) denote its fundamental group. In her seminal work on minimal surfaces in hyperbolic 3-space [Uhl83], K. Uhlenbeck defines the notion of almost-fuchsian representation: it is a discrete and faithful representation ρ of Γ into PSL(2, C) such that there is a ρ-equivariant immersion f : H 2 → H 3 with minimal image and principal curvatures in (-1, 1). Uhlenbeck showed that those representations are always convex-cocompact, and that they form an open neighborhood of the fuchsian locus inside Hom(Γ, PSL(2, C)). In H 3 , convex-cocompact representations are quasi-fuchsian representations, hence the denomination of almost-fuchsian for this stronger notion.

Considering the minimal surface problem in H 3 , almost-fuchsian representations are of special interest because they admit a unique invariant minimal disk with small principal curvatures. This is not the case for arbitrary convex-cocompact representations, see for instance [START_REF] Huang | Counting minimal surfaces in quasi-fuchsian threemanifolds[END_REF].

Date: January 3, 2023. 1 Krasnov-Schlenker [START_REF] Krasnov | Minimal surfaces and particles in 3manifolds[END_REF] have extended the notion to almost-fuchsian immersions or almost-fuchsian disks, not necessarily invariant under a surface group. Seppi [START_REF] Seppi | Minimal discs in hyperbolic space bounded by a quasicircle at infinity[END_REF] used it to show that quasi-circles with quasisymmetry constant close to 1 bound minimal disks.

We will work with the following notion of almost-fuchsian immersion.

Definition 1.1. An immersion f : D → H 3 is said to be almost-fuchsian if it is proper, conformal with minimal image, and has principal curvatures in (-1 + ǫ, 1 -ǫ), for some ǫ > 0.

A subdisk of H 3 is said to be an almost-fuchsian subdisk if it is the image of an almostfuchsian immersion.

Let Q(D) denote the space of holomorphic quadratic differentials over D. The main result we will prove in this paper is the following: Theorem 1.2. Let C ⊆ Q denote the set of Hopf differentials of almost fuchsian immersions. Then

(1) 2 disks with same Hopf differentials are identified by an isometry of H 3 .

(2) C is a bounded, open and convex subset of Q, and

(3) C contains all holomorphic quadratic differentials such that |φ| ∞ ≤ 1 2 . (4) For any φ ∈ C, |φ| ∞ ≤ 1.
Krasnov-Schlenker [START_REF] Krasnov | Minimal surfaces and particles in 3manifolds[END_REF] already consider this parametrization for the case of holomorphic quadratic differentials invariant under a closed surface subgroup of PSL(2, C). In that case, it was already known that the set is open and star-shaped around 0, see for instance [START_REF] Samuel Trautwein | The hyperkähler metric on the almost-fuchsian moduli space[END_REF].

The almost-fuchsian moduli space is an example of moduli space endowed with a hyperkähler structure, see [START_REF] Simon K Donaldson | Moment maps in differential geometry[END_REF] and [START_REF] Wolf | Hyperkähler geometry and Teichmüller space[END_REF]. The hyperkähler structure actually coincides with the Feix-Kaleidin structure when considering C as a neighborhood of the fuchsian moduli space, [START_REF] Samuel Trautwein | The hyperkähler metric on the almost-fuchsian moduli space[END_REF].

This description of C has 2 main consequences. First, it extends the notion of almostfuchsian representation to any infinite discrete subgroup of PSL(2, R). Also, considering holomorphic quadratic differentials invariant under a surface group representation, we prove the following result.

Theorem 1.3. There exists a Finsler metric p on the Teichmüller space T g such that

(1.1) ∀φ ∈ T * T g , p(φ) < 1 if and only if φ ∈ C .
This Finsler metric is uniformly equivalent to the Finsler metric dual to the unit L ∞ balls under the identification of the cotangent bundle with the space of holomorphic quadratic differentials.

The authors thank Nicolas Tholozan for his insights and discussions on this topic.

introduction and notation

This section is a classical survey of the Gauss-Codazzi equation for almost-fuchsian disks. The results are well-known, we explain them here to make our notation clear. The reader comfortable with Gauss-Codazzi theory may go directly to the next section.

In the following, Σ is a connected, oriented surface, with χ(Σ) < 0. Let (M, h) is a complete, 3-dimensional hyperbolic manifold, and let f : Σ → M be a smooth immersion.

Page 2/13 Let g 0 denote the induced metric over S, let II denote the second fundamental form of f , and let A denote its shape operator. Recall that f is said to be minimal whenever its mean curvature H := Tr(A) vanishes. Let κ denote the intrinsic curvature of g 0 . The Gauss equation is

(2.1) κ = det g 0 (II) -1 .
where the determinant of the second fundamental form II is taken with respect to g 0 . The determinant of II is also the determinant of the shape operator A.

Definition 2.1 (Hopf differential). The Hopf differential φ of f is the (2, 0)-component of II. That is, φ is the symmetric bilinear form given by

(2.2) φ(ξ, ν) := 1 4 II(ξ, ν) -iII(Jξ, ν) -iII(ξ, Jν) -II(Jξ, Jν) .
In particular, when f is minimal we readily obtain:

(2.3) det g 0 (II) = -|φ| 2 g 0 . In the following, we will denote g a fixed hyperbolic metric on a Riemann surface, we call the pair (u, φ) Gauss-Codazzi data for the immersion, where e 2u g is the induced metric and φ is the Hopf differential of the corresponding immersion.

Lemma 2.2. Let (Σ, g 0 ) be a riemannian surface and let g 1 = e 2u g 0 be a metric conformal to g 0 . Let ∆ i and κ i denote respectively the laplacian and curvature of the metric g i for i ∈ {0, 1}. Then

κ 1 e 2u = κ 0 -∆ 0 u (2.4) κ 0 = (κ 1 + ∆ 1 u)e 2u .
(2.5) Proof. Denote Ric i the Ricci tensor of g i . The following equation in Besse's book, 1.159 d) [START_REF] Arthur | Einstein manifolds[END_REF], relates the Ricci curvatures of g 1 and g 0 :

(2.6) Ric 1 = Ric 0 -(∆ 0 u)g 0
The remaining equations follow from the fact that, for a riemannian surface (Σ, g i ), Ric g i = κ i g i . Note that Besse uses the analysts' laplacian, whose sign is the inverse of that of the geometer's laplacian.

Combining these equations yields the Gauss-Codazzi equation.

Proposition 2.3 (Gauss-Codazzi). Let f : Σ → (M, h) be a minimal immersion of a surface into a hyperbolic manifold. Let g 0 denote the induced metric and let g = e -2u g 0 denote the unique, complete hyperbolic metric in its conformal class. Then u satisfies

(2.7) ∆u = -1 + e 2u + e -2u |φ| 2 g . Proof. By lemma 2.2, (2.8)

κ 0 = (-1 -∆u)e -2u .
By Gauss equation, (2.9) κ 0 = -1 -e -4u |φ 0 | 2 g . The result follows upon combining these two equations.

Page 3/13

By the fundamental theorem of surface theory, for every pair (u, φ), where u : Σ → R and φ is a holomorphic quadratic differential on Σ satisfying the Gauss-Codazzi equation (2.7), there exists an immersion of Σ into H 3 having this Gauss-Codazzi data. As the surface considered is hyperbolic Σ identifies with the open disk D, and the immerssion is equivariant with regard to some representation ρ : π 1 Σ → Isom(H 3 ) Furthermore, this immersion is unique up to post composition by an isometry of H 3 .

We will require the following result of Li-Mochizuki [START_REF] Li | Complete solutions of toda equations and cyclic higgs bundles over non-compact surfaces[END_REF].

Proposition 2.4 (Sub-supersolution method). Let (Σ, g) be a riemannian surface, and let F be a smooth function defined on Σ × R. Consider the equation (E):

(2.10)

∆ g u = F (x, u)
Assume there are u -, u + : Σ → R satisfying:

u -< u + (2.11) ∆ g u -≥ F (x, u -) (2.12) ∆ g u + ≤ F (x, u + ) (2.13)
Then the equation (E) admits at least one solution u satisfying u -≤ u ≤ u + .

First remark that almost-fuchsian subsurfaces of H 3 are necessarily uniformized by the Poincaré disk, and that the conformal factor between the induced metric and the hyperbolic one is bounded: Lemma 2.5. Let M be a complete hyperbolic 3-manifold. Let f : Σ → M be an almostfuchsian immersion, then its induced metric g 0 is conformal to a complete hyperbolic metric g. Denote g 0 = e 2u g. Then u satisfies.

(2.14) - ln 2 2 ≤ u ≤ 0 .
Proof. Consider the prescribed curvature equation on the universal cover :

(2.15)

∆ g 0 v = e 2v -κ g 0 .
For any solution v, the corresponding metric e 2v h 0 will be hyperbolic. As f is almost fuchsian, the curvature κ h 0 satisfies -2 ≤ κ h 0 ≤ -1. Upon applying the sub-supersolution method (Prop 2.4) to (2.15) with subsolution v -= 0 and supersolution v + = ln 2 2 , we get a solution v satisfying

(2.16) 0 ≤ v ≤ ln 2 2 .
Since v is bounded, (Σ, e 2v g 0 ) is quasi-isometric to (Σ, g 0 ). As f is almost-fuchsian, it is proper. Thus g 0 is complete, and e 2v g 0 is also complete. By the Ahlfors-Schwarz-Pick lemma [Ahl38], the complete hyperbolic metric. is unique in its conformal class, hence g = e 2v g 0 is complete hyperbolic. Denote u = -v, we check:

(2.17)

- ln 2 2 ≤ u ≤ 0 ,
As asserted.

Remark. Page 4/13

• With the same proof, one can prove that this is a general property of complete pinched Hadamard disks, due to Bland and Kalka [START_REF] Bland | Complete metrics conformal to the hyperbolic disc[END_REF]. It is also stated in Benoist-Hulin [BH20], Prop. 3.1. : A complete pinched hadamard disk is uniformly quasiconformal to the complete hyperbolic disk. • In [AS20], Alvarez and Smith show a more general version of this: The conformal factor is actually bounded in C k+2,α .

Theorem 2.6. Let g be a complete hyperbolic metric on Σ, u : Σ → R a bounded continuous map, and φ a holomorphic quadratic differential on Σ satisfying

∆ g u = -1 + e 2u + e -2u
|φ| 2 , and (2.18)

sup z∈Σ e -2u(z) |φ(z)| < 1 . (2.19)
Then there is an almost-fuchsian immersion of Σ into a hyperbolic manifold such that e 2u g is the induced metric and φ is the (2, 0)-part of the second fundamental form. Conversely, for any almost-fuchsian immersion the associated data satisfies these relations.

Proof. By the fundamental theorem of surface theory, we get an immersion into a hyperbolic manifold. Since u is bounded, the induced metric is complete and it follows from (2.19) that the principal curvatures lie in (-1, 1). Hence f is almost-fuchsian.

Conversely, if f is almost-fuchsian, consider (u, φ) the associated Gauss-Codazzi data. Then Gauss-Codazzi is verified, the bound on the principal curvatures yields (2.19) and the boundedness of u follows from the previous lemma.

Note that almost-fuchsian subdisks of H 3 are always properly embedded and bound a quasicircle at infinity. Indeed, this readily follows from the original work [START_REF] Uhlenbeck | Closed minimal surfaces in hyperbolic 3-manifolds[END_REF] of Uhlenbeck and the argument of Theorem 1.3 of [START_REF] Huang | On almost-fuchsian manifolds[END_REF]. For the reader's convenience, we state and prove this result here.

Theorem 2.7. Let e : D → H 3 be an almost-fuchsian disk. Then the disk e(D) is properly embedded and bounds a quasi-circle at infinity. Furthermore, if k denotes the supremum of the principal curvatures of e, the quasisymmetry constant of the quasi-circle is bounded by where exp denotes the exponential map of H 3 . Let A denote the shape operator of f . The same computation as of lemma 2.1. of [START_REF] Huang | On almost-fuchsian manifolds[END_REF] gives the following expression of the metric induced by e :

(2.21)

h ν (x, t) := h t (x) ⊕ dt 2 ,
where, for all t,

(2.22) h t (•, •) = e 2u g((cosh(t)I 0 + sinh(t)A)•, (cosh(t)I 0 + sinh(t)A)•) .

Consider 0 < k < 1 such that the operator norm of A is less than k. Since |A| ≤ k < 1, h ν is complete and thus e is a diffeomorphism. In particular, e is properly embedded.

Let f : D → H 3 be an isometric parametrization of a totally geodesic plane, let v f denote its unit normal vector field, and define f :

D × R → H 3 by (2.23) f (x, t) = exp f (x) (tν f (x)) .
The metric induced by f is

(2.24) g(x, t) = cosh 2 (t)g ⊕ dt 2 .
It follows from (2.24) and (2.22) that α := e • f -1 is bilipschitz. It therefore extends (see Gehring [START_REF] Gehring | Rings and quasiconformal mappings in space[END_REF] or Mostow [START_REF] George D Mostow | Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms[END_REF]) to a quasiconformal map α from ∂ ∞ H 3 to itself. It follows that Γ := ∂ ∞ e(D) is a quasicircle. Note that, for all t, the restrictions of h 0 and g to the slice D × {t} are K-quasiconformal, with (2.25)

K = 1 + k 1 -k .
Letting t tend to +∞, it follows that α is K-quasiconformal. As the boundary of the disk is the image by α of an equator in S 2 , this completes the proof.

Proof of main theorem

As in this paper we are mainly interested in the description of conformal almost-fuchsian disks, let us define the space we want to describe: Definition 3.1. We denote AF the set of almost-fuchsian immersions D 2 → H 3 up to postcomposition by an isometry of H 3 .

The set AF, equipped with its quotient topology, can be identified with the set of almostfuchsian immersions with prescribed image and differential of the origin o. As such, it can be viewed as a subspace of the space of immersions from the disk into H 3 .

Remark. The space AF does not parametrize almost-fuchsian subdisks of H 3 . However, the set of almost-fuchsian subdisks up to isometry can be identified to the quotient of AF by a PSL(2, R) action by precomposition, which is not a properly discontinous action.

In order to prove our theorems in a noncompact setting, we will use the following version of Omori-Yau's theorem [START_REF] Omori | Isometric immersions of riemannian manifolds[END_REF][START_REF] Yau | Harmonic functions on complete riemannian manifolds[END_REF]. It can be found in Li-Mochizuki [START_REF] Li | Complete solutions of toda equations and cyclic higgs bundles over non-compact surfaces[END_REF], we reproduce it here for the reader's convenience.

Theorem 3.2 (Omori-Yau). Let (M, h) be a complete manifold with Ricci curvature bounded from below. Let u : M → R be a map bounded from above and let ε > 0. Then there is

x ∈ M such that u(x) ≥ sup u -ε , (3.1) |∇ h u(x)| ≤ ε , and (3.2) ∆ h u(x) ≤ ε . (3.3)
The first step is to show that the moduli space of almost-fuchsian disks injects into the space of bounded holomorphic quadratic differentials. ∆u = -1 + e 2u + e -2u |φ| 2 such that e -2u |φ| < 1 everywhere.

Proof. Suppose (u, φ) and (v, φ) are two bounded solutions of (3.4) such that e -2u |φ| < 1 and e -2v |φ| < 1. Then u -v is bounded and satisfies (3.5) ∆(u -v) = (e 2v -e -2v |φ| 2 )(e 2(u-v) -1) + |φ| 2 e -2v (e 2(u-v) + e 2(v-u) -2) .

Using the positivity of the second term of the right hand side and the almost-fuchsian property of v, one gets

(3.6) ∆(u -v) ≥ (e 2v -e -2v |φ| 2 )(u -v) .
Applying Omori-Yau's maximum principle gives u -v ≤ 0. Exchanging u and v, one has u = v, thus the corresponding almost-fuchsian disks are conjugated.

Remark. The same reasoning actually shows that if (e 2u g, φ) is almost-fuchsian, then any upper bounded v solution to the Gauss-Codazzi equation with quadratic differential φ must satisfy:

(3.7) v ≤ u .
In [START_REF] Huang | Bifurcation for minimal surface equation in hyperbolic 3-manifolds[END_REF], Huang, Lucia and Tarantello study the solutions of the Gauss-Codazzi equation on closed surfaces, of which the maximality of almost-fuchsian solutions is a consequence.

Moreover, we will show that the image of the almost-fuchsian locus in the space of bounded holomorphic quadratic differentials on the disk is actually open. Combining Proposition 3.3 with the fundamental theorem of surface theory, we get : Corollary 3.4. For f : D 2 → H 3 an immersion, denote φ f its Hopf differential. Then the map:

(3.8) Φ : AF → QD(D 2 ) f → φ f is injective.
Remark. The statement of this corollary in the equivariant case is the Lemma 2.11 of Krasnov-Schlenker [START_REF] Krasnov | Minimal surfaces and particles in 3manifolds[END_REF].

Proof. We already know that the Gauss-Codazzi data of an immersion characterizes it up to isometry. The prop (3.3) shows that if two almost-fuchsian immersions have same Hopf differentials, then they have the same Gauss-Codazzi data, hence the injectivity of our map. We will prove that the linearized operator of (3.9) L u : v → -∆v + 2(e 2u -e -2u |φ| 2 )v is an isomorphism between the Banach spaces C 2,α (D) and C 0,α (D), where here D is equipped with its Poincaré metric, hence, using the local inversion principle, we will be able to find a solution u uniformly close to u on a neighborhoof of φ, which then has to be almost-fuchsian.

At first, as u is bounded, L u trivially defines a bounded linear map from C 2,α (D) towards C 0,α (D). The almost-fuchsian condition implies

(3.11) δ 2 ≤ e 2u -e -2u |φ| 2 ≤ 1.
Hence L u is strictly elliptic. We now show that L u is injective. Indeed, choose v ∈ ker L u . Since v is bounded, Omori-Yau's maximum principle implies

(3.12) v ≤ 0 .
The same reasoning to -v shows that v = 0, hence L u is injective, as asserted.

It remains only to prove surjectivity. First consider a compact set K in D with smooth boundary. It is a standard consequence of elliptic theory and of the strict ellipticity of L u that L u is an isomorphism from C 2,α 0 (K) into C 0,α (K). Now choose u ∈ C 0,α (D). and a compact exhaustion (K n ) of D such that each K i has smooth boundary. For each i there is a unique

v i ∈ C 2,α 0 (K i ) such that L u v i = w1 K i .
We claim that the sequence (|v n | ∞ ) is bounded. Indeed, by the maximum principle, for all i,

(3.13) |u i | ∞ ≤ max{2δ, 1}|w1 K i | ∞ ≤ 2|w| ∞ .
Fix a radius r > 0 and x ∈ D. For i big enough such that B(x, 2r)

⊂ K i , |L u v i | C 0,α (B(x, 2r
)) ≤ |w| C 0,α , and (3.14)

|v i | C 0 (B x,2r ) ≤ 2|w| C 0 . (3.15)
Hence, by the Schauder estimates, there is a C > 0 depending only on r such that (3.16)

|v i | C 2,α (B(r,x)) ≤ C|w| C 0,α .
By the Arzela-Ascoli theorem, up to extracting a subsequence, we can assume there exists v ∞ ∈ C 2,α loc such that v n → v ∞ on every compact set in the C 2,β topology for β < α. Furthermore

|v ∞ | C 2,α ≤ C ′ sup x∈H 2 |v ∞ | C 2,α (B(r,x)) ≤ C ′ sup x∈H 2 lim sup i→∞ |v i | C 2,α (B(r,x)) ≤ C ′ C|w| C 0,α , so that v ∞ ∈ C 2,α (H 2 ). By continuity of L u , (3.17) L u v ∞ = w .
Thus L u is surjective. By the Closed Graph Theorem, L u is a linear isomorphism. It follows by the inverse function theorem that the image of the almost-fuchsian locus is open in the space of bounded quadratic differentials, as desired.

Equipped with this theorem, we will consider a perturbation φ t of φ in the image of the almost-fuchsian locus and we will consider the family (u t ) of bounded maps satisfying the Gauss-Codazzi equation. In particular, we will need the derivatives ut and üt to be bounded.

The following is actually a straightforward consequence of the implicit function theorem we used.

Proposition 3.6. Consider (e 2u , φ) an almost-fuchsian disk. Let f t be a C 0,α perturbation of |φ| 2 . Denote by (u t ) the smooth branch of solutions to :

(3.18) ∆u t = -1 + e 2ut + e -2ut f t
for t in a neighborhood of 0. Then ut and üt are bounded.

Proof.

Consider F : C 2,α (D) × [0, 1] → C 0,α (D) defined by F (v, t) = ∆v + 1 -e 2v -e -2v f t .
Thanks to the previous lemma, we know the local existence of a smooth G : [0, 1] → C 2,α (D) such that F (G(t), t) = 0. It follows by smoothness of G that ut , üt ∈ C 2,α (D) for all t, and the result follows.

We are now able to show that the almost-fuchsian locus retracts to the set of almostfuchsian immersions with zero quadratic differentials.

Theorem 3.7. Let φ ∈ ℑ(AF) and let ψ be a holomorphic quadratic differential on the disk satisfying

(3.19) ∀z ∈ D, |ψ(z)| ≤ |φ(z)| .
Then ψ ∈ ℑ(AF).

Proof. Let u : D → R be such that (u, φ) is the Gauss-Codazzi data of an almost-fuchsian disk. Let f t : D → [0, ∞[ be a smooth map such that f 0 = |ψ| 2 and f 1 = |φ| 2 . As |ψ| ≤ |φ|,
one can choose f so that the maps t → f t (z) are nondecreasing. We will show that ψ ∈ ℑ(AF), i.e. that there is a solution v to ∆v = -1 + e 2v + e -2v f 0 , and (3.20)

sup z∈D e -4v f 0 < 1 . (3.21)
To do so, consider for t ∈ [0, 1] the equation.

(3.22) ∆u t = -1 + e 2ut + e -2ut f t .

Let I ⊆ [0, 1] denote the set of all s such that, for all t ≥ s, this equation has a bounded solution u s satisfying |e -4us f s | ∞ < 1, for some ǫ > 0. By Theorem 2.6, 1 ∈ I. By Theorem 3.5 applied to a C 0,α perturbation, I is open. It thus remains only to show that I is closed. At first, consider t 0 = inf I and (t n ) ∈ I a monotonous sequence such that t n → t 0 . By Lemma 2.5, the sequence (u tn ) is uniformly bounded, from which it follows that (∆u tn ) is also uniformly bounded. By Schauder's estimates, the sequence (u tn ) is uniformly bounded in C 1,α on any compact set. Hence we may assume that u tn converges towards a map u t 0 ∈ C 2,α (H) in the C 2,β topology for all β < α. Trivially u t 0 solves (3.22).

It remains only to check that |e -4ut 0 f t 0 | ∞ < 1. Differentiating the above equation with respect to t yields, (3.23) ∆ ut = 2(e 2ut -e -2ut f t ) ut + e -2ut ḟt .

By Lemma 2.5, there exists B > 0 such that |u t | ∞ < B for all t. It follows by the Omori-Yau maximum principle that, over I, ut ≤ 0, so that, for all t ∈ I,

(3.24) e -4ut f t ∞ ≤ e -4u 1 f 1 ∞ < 1 .
Closedness of I now follows so that, by connectedness, 0 ∈ I. At t = 0, this means that (u 0 , ψ) is the Gauss-Codazzi data of an almost-fuchsian disk. We have shown that ψ ∈ ℑ(AF), as desired. Denote φ t = tφ for t ∈ [0, 1]. Then for any t ∈ [0, 1], there is a u t : Σ → R bounded such that (e 2ut g, φ t ) satisfies the Gauss-Codazzi equation. Furthermore, for all t ∈ [0, 1], (e 2ut g, φ t ) is almost-fuchsian.

Remark. For the equivariant case, this computation can already be found in a survey of Trautwein [START_REF] Samuel Trautwein | The hyperkähler metric on the almost-fuchsian moduli space[END_REF], step 3 of proof of theorem 5.13.

Corollary 3.9. The moduli space of almost-fuchsian disks up to isometry is contractible. The set of almost-fuchsian immersions retracts onto the set of geodesic immersions of disks in H 3 .

We continue with the proof of our main theorem: Theorem 3.10 (Convexity). Let (e 2u 0 g, φ 0 ) and (e 2u 1 g, φ 1 ) be two pairs corresponding to almost-fuchsian disks. For t ∈ [0, 1], denote φ t := (1 -t)φ 0 + tφ 1 . Then for any t, there is (e 2wt g, φ t ) satisfying the Gauss-Codazzi equation, and it corresponds to an almost-fuchsian disk.

Proof. First, we write down Gauss-Codazzi and we differentiate it twice in t ∆w t = -1 + e 2wt + e -2wt |φ t | 2 , (3.27) ∆ ẇt = 2 ẇt e 2wt -2 ẇt e -2wt |φ t | 2 + 2e -2wt g(φ t , φ 1 -φ 0 ) , and (3.28)

∆ ẅt = 2 ẅt (e 2wt -e -2wt |φ t | 2 ) + 4 ẇ2 t (e 2wt + e -2wt |φ t | 2 ) -8 ẇt e -2wt g(φ t , φ 1 -φ 0 ) + 2e -2wt |φ 1 -φ 0 | 2 . (3.29)
We first show that as long as (e 2wt g, φ t ) is almost-fuchsian, (3.30) ẅt + 4 ẇ2 t ≤ 0 . This is a consequence of the Omori-Yau maximum principle and the following standard formula.

(3.31) ∆( ẇ2 t ) = 2 ẇt ∆ ẇt + 2|∇ ẇt | 2 ≥ 2 ẇt ∆ ẇt . Then, using equations (3.28) and (3.29), we obtain

∆( ẅt + 4 ẇ2 t ) ≥ 2 ẅt (e 2wt -e -2wt |φ t | 2 ) + 20 ẇ2 t e 2wt -12 ẇ2 t e -2wt |φ t | 2 +8 ẇt e -2wt g(φ t , φ 1 -φ 0 ) + 2e -2wt |φ 1 -φ 0 | 2 . ( 3
-φ s,0 ) + 2|φ s,1 -φ s,0 | 2 .
= -4( ẅs,t + 4 ẇ2 s,t )|φ s,t | 2 + 2|4 ẇs,t φ s,t -(φ s,1 -φ s,0 )| 2 . This is nonnegative, and convexity follows. By convexity, e -4ws,t |φ s,t | 2 is smaller than the maximal value for t = 0 and t = 1, and we deduce that it is smaller than 1 -δ, for δ > 0 depending only on φ 0 and φ 1 . It now readily follows that S is closed, so that, by connectedness, 1 ∈ S. In other words, J 1 = [0, 1], and the result follows.

We also have the following quantitative estimate for this convex set. Fixing now a subgroup Γ of PSL(2, R), one can look at the subset AF Γ of elements of AF equivariant under some representation of Γ. The uniqueness properties shown imply that this set can be identified with a subset of Hom(Γ, PSL(2, C))/PSL(2, C). In the case of a closed surface group, this subset are known as almost-fuchsian representations considered by Uhlenbeck [START_REF] Uhlenbeck | Closed minimal surfaces in hyperbolic 3-manifolds[END_REF].

Page 11/13 Fix a hyperbolic surface Σ g,n , and consider Γ its fundamental group. Considering now holomorphic quadratic differentials equivariant under the action of Γ, identifying holomorphic quadratic differentials with the cotangent bundle to Teichmüller space of genus (g, n), we get a smooth family of open bounded sets in each leaf. As these sets are isotropic (if (e 2u g, φ) is an almost-fuchsian disk, so is (e 2u g, zφ) for any z of modulus one), it induces a Finsler metric on Teichmüller space, which is quasi-isometric to the Finsler metric dual to the L ∞ unit balls in T * T g . Corollary 3.12. For g ≥ 2 and T g the Teichmüller space of genus g, there is a Finsler metric p on T g such that:

(3.40) p(φ) < 1 ⇐⇒ φ ∈ ℑ(AF) This Finsler metric is quasi-isometric to the Finsler metric dual to the L ∞ unit balls in T * T g .

Remark. It is interesting to compare this Finsler metric to the metrics dual to L p -unit balls in T * T g . For p = 2, one gets Weil-Petersson's metric, see Wolpert [START_REF] Scott | The weil-petersson metric geometry[END_REF]. For p = 1, one gets the Teichmüller metric, see Papadopoulos-Su [START_REF] Papadopoulos | On the finsler structure of teichmüller's metric and thurston's metric[END_REF].

Remark. With this discussion, we have a description of the fibration map from AF Γ → T g,n , which is simply the conformal class of the induced metric on Σ g,n .

  1+k 1-k . Proof. Let g be the Poincaré metric of D. Let g 0 = e 2u g denote the induced metric of e on the disk D. Consider ν the unit normal vector field, and define e by (2.20) e : D × R → H 3 , and (x, t) → exp x (tν x ) .

Theorem 3. 5 .

 5 The image of the almost-fuchsian locus is open in the space of bounded holomorphic quadratic differentials. Proof. Consider φ in ℑ(AF), u and 0 < δ < 1 satisfying ∆u = -1 + e 2u + e -2u |φ| 2 and (3.9) sup z∈D e -2u |φ| ≤ 1 -δ . (3.10)

Corollary 3. 8 .

 8 Let φ ∈ ℑ(AF) and let u : D → R be the associated bounded map such that ∆u = -1 + e 2u + e -2u |φ| 2 , and (3.25) sup z∈D e -2u |φ| ≤ 1 -δ . (3.26)

  Theorem 3.11. The convex set ℑ(AF) satisfies: (AF) ⊂ {φ : sup z∈D |φ(z)| < 1} Proof. Take φ with |φ| ∞ ≤ 1 2 . Thenln 2 2 and 0 are respectively sub and supersolutions of (3.39) ∆u = -1 + e 2u + e -2u |φ| 2 . Hence by Proposition 2.4, we have a bounded solution satisfying Gauss-Codazzi, and necessarily e -2u |φ| ≤ 2|φ| ∞ < 1. Hence φ ∈ ℑ(AF). Conversely, if φ ∈ ℑ(AF), u the solution to Gauss-Codazzi satisfies u ≤ 0, and |φ| ∞ ≤ |e -2u φ| ∞ < 1.

  )(e 2wt -e -2wt |φ t | 2 ) + 12 ẇ2 t (e 2wt -e -2wt |φ t | 2 ) .For all s, let J s denote the set of all t ∈ [0, 1] for which there exists a function w s,t making (e 2ws,t g, φ s,t ) into an almost Fuchsian pair. Let S denote the set of all s ∈ [0, 1] for which J s equals the interval [0, 1]. Trivially S contains 0. By Theorem 3.5, S is open. We now show that S is closed. First, if s 0 ∈ S, the equicontinuity of all functions (u s,t ) for s ∈ S, t ∈ [0, 1] allows us to get a limiting family of functions which are solutions to Gauss-Codazzi equation. We claim that, for all s, the function t → e -4ws,t |φ s,t | 2 is convex over J s . Indeed,

	As 8A 2 + 8AB + 2B 2 = 2(2A + B) 2 is nonnegative, we obtain
	(3.34)	∆( ẅt + 4 ẇ2 t ) ≥ 2( ẅt + 4 ẇ2
	Hence, provided the pair is almost-fuchsian, we get
	(3.35)	∆( ẅt + 4 ẇ2 t ) ≥ 2( ẅt + 4 ẇ2 t )(e 2wt -e -2wt |φ t | 2 ) .
	Thus we deduce that
	(3.36)	ẅt + 4 ẇ2 t ≤ 0 .
	We have shown that as long as (e 2ut g, φ t ) is almost-fuchsian, ẅt + 4 ẇ2 t ≤ 0, as asserted.
	Consider now the family of quadratic holomorphic differentials
	(3.37)	φ s,t := s(1 -t)φ 0 + stφ 1
		e 4ws,t ∂ 2
	.32)
	Judiciously rearranging the terms, we obtain
	(3.33)	∆( ẅt + 4 ẇ2

t ) ≥ 2( ẅt + 4 ẇ2 t )(e 2wt -e -2wt |φ t | 2 ) + 12 ẇ2 t (e 2wt -e -2wt |φ t | 2 ) +8 ẇ2 t e -2wt |φ t | 2 + 8e -2wt g( ẇt φ t , φ 1 -φ 0 ) + 2e -2wt |φ 1 -φ 0 | 2 . t

t (e -4ws,t |φ s,t | 2 ) = -4( ẅs,t + 4 ẇ2 s,t )|φ s,t | 2 + 32 ẇ2 s,t |φ s,t | 2 -16 ẇs,t g(φ s,t , φ s,1