DURATION OF RAPID LIGHT CURVES FOR DETERMINING THE PHOTOSYNTHETIC ACTIVITY OF MICROPHYTOBENTHOS BIOFILM IN SITU

Keywords: chlorophyll fluorescence, diatoms, intertidal mudflat, marine ecology, nonphotochemical quenching, photoacclimation

Abstract

This study is the first field work that examines the use of Rapid Light Curves (RLCs) for in situ assessment of microphytobenthos (MPB) photosynthetic activity. The advantages offered by RLC methodology respond primarily to in situ constraints concerning MPB ecology and behaviour. In particular, the rapidity of RLCs allows for a high number of replicates during emersion, while preventing disturbance of the measurement due to MPB cell migration. It is necessary to maintain this feature in order to thoroughly assess the large spatial and temporal variability of in situ MPB biomass and photosynthetic activity. Consequently, working with intertidal mudflat MPB biofilms dominated by epipelic (motile) species, we investigated the effect of 10-s and 30-s irradiance steps: two durations commonly used for building RLCs. We compared the performance of 10-s and 30-s RLCs in determining the in situ MPB photosynthetic activity without dark acclimation and in high irradiances (summer). Although a similar trend of MPB cell photoacclimation was observed using the two procedures, there were differences, revealed by lower values of α-10 s and rETR max -30 s compared with α-30 s and rETR max -10 s on some experimental days (49 and 34%, respectively). Discrepancies could mainly be explained by the unbalanced Q A redox state and the occurrence of photoprotective non-photochemical quenching (NPQ) present prior to RLC, which impacted the 10-s and 30-s RLCs differently. The initial slope (α) was strongly impacted by NPQ developed prior to RLCs. A correction performed a posteriori and based on the use of the maximum F m " obtained during the course of RLCs was not sufficient to remove the bias of pre-existing NPQ on the determination of α completely. Consequently, a good, direct and simultaneous determination of rETR max and α was virtually impossible in situ even when using RLCs. In contrast to published results from laboratory studies, the shorter irradiance steps (10 s) offers the better (i.e. the most reasonable) compromise for assessing MPB maximum photosynthetic capacity in situ using RLC.

Introduction

In shallow estuarine ecosystems, the benthic community of microalgae and cyanobacteria, designated as microphytobenthos (MPB), is a basic component that plays a major role in primary production, nutrient recycling and support of the trophic web [START_REF] Underwood | Primary production by phytoplankton and microphytobenthos in estuaries[END_REF][START_REF] Lefebvre | Seasonal dynamics of trophic relationships among co-occurring suspension-feeders in two shellfish culture dominated ecosystems[END_REF]. The activity of MPB is highly dynamic and is characterized by large spatial and temporal variability [START_REF] Herlory | Microstructure of microphytobenthic biofilm and its spatio-temporal dynamics in an intertidal mudflat (Aiguillon Bay, France)[END_REF][START_REF] Haubois | Spatiotemporal structure of the epipelic diatom assemblage from an intertidal mudflat in Marennes-Oleron Bay, France[END_REF][START_REF] Jesus | Spatial dynamics of microphytobenthos determined by PAM fluorescence[END_REF]. In order to improve our understanding of the underlying processes driving MPB ecology [START_REF] Underwood | Observations in ecology: you can"t make progresses without understanding the patterns[END_REF], one of the main challenges for evaluating in situ MPB features, such as biomass and photosynthetic activity, is to find a way to cope simultaneously with these variations, which occur at very different scales.

Pulse Amplitude Modulation (PAM) chlorophyll a fluorescence has been increasingly applied to MPB in situ to perform such measurements (Perkins et al., 2010a). A special fluorescence methodology, the rapid light response curves (RLCs), has recently been applied to MPB biofilms to assess their photosynthetic activity (e.g. Serodio et al., 2005a). RLCs use short (≤ 30 s) steps of increasing irradiance to determine the relative electron transport rate (rETR) according to irradiance (E) (i.e. rETR/E curves), and to estimate photosynthetic parameters that provide information on the photosynthetic activity, the photoacclimation state of MPB and their driven processes (Serodio et al., 2005a;[START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF][START_REF] Herlory | Methodology of light response curves: application of chlorophyll fluorescence to microphytobenthic biofilms[END_REF]Perkins et al., 2010b). The protocols sustaining RLCs proved to be particularly appropriate for MPB over traditional steady state light curves (SSLCs), for which a steadystate is usually reached at each irradiance step [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]Perkins et al., 2010a).

RLCs were primarily used on MPB to limit the effect of upward and downward cell migrations in the sediment during the time course of SSLC development (Perkins et al., 2010a). Later, RLCs became an essential tool to provide a view of the current status of the photosynthetic apparatus, almost independently of immediate variability resulting from cell migration and irradiance changes, two points of special importance in MPB in situ investigations (Perkins et al., 2010a). RLCs have been successfully used to study the variability of MPB photosynthetic activity as a function of environmental cues and species assemblage (Serodio et al., 2005a;[START_REF] Underwood | Patterns in microphytobenthic primary productivity: Species-specific variation in migratory rhythms and photosynthetic efficiency in mixed-species biofilms[END_REF]. This variability reflects the photoacclimation state (long-term light history) [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF] coupled with migration strategy [START_REF] Mouget | Migration or photoacclimation to prevent high irradiance and UV-B damage in marine microphytobenthic communities[END_REF][START_REF] Jesus | Adaptations of microphytobenthos assemblages to sediment type and tidal position[END_REF]Perkins et al., 2010b) and the dynamics of fast photosynthesis regulation via dissipation of excess energy by the non-photochemical quenching (NPQ) mechanism (photoprotection and short-term light history) (Serodio et al., 2005b;[START_REF] Serodio | Photosynthetic activity, photoprotection and photoinhibition in intertidal microphytobenthos as studied in situ using variable chlorophyll fluorescence[END_REF][START_REF] Van Leeuwe | Photoacclimation in microphytobenthos and the role of xanthophyll pigments[END_REF].

Several methodological studies have recently been conducted to improve RLC protocols used on MPB biofilms or benthic diatoms. These include (1) the comparison with SSLCs in the laboratory [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]Serodio et al., 2006a;[START_REF] Herlory | Methodology of light response curves: application of chlorophyll fluorescence to microphytobenthic biofilms[END_REF], (2) the assessment of light history and NPQ effects on the RLC pattern [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF][START_REF] Serodio | Photosynthetic activity, photoprotection and photoinhibition in intertidal microphytobenthos as studied in situ using variable chlorophyll fluorescence[END_REF], (3) the effect of changing the duration and/or the up-or down-increment of irradiance steps on the determination of photosynthetic parameters (Serodio et al., 2005a;[START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]. [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF] identified that both the light history of the cells prior to RLC and the light dose accumulation during RLC strongly influence the kinetics of fluorescence emission. They concluded that RLCs with 30 s irradiance steps were better suited for dark-acclimated samples than were RLCs with 10-s irradiance steps. Moreover, dark acclimation before RLCs may induce cell migration in situ [START_REF] Jesus | Chlorophyll fluorescence as a proxy for microphytobenthic biomass: alternatives to the current methodology[END_REF].

According to Perkins et al. (2010a) dark acclimation could also modify the photoacclimation state of the cells and should be avoided in order to standardise methodologies. Serodio et al. (2005a) showed that when the cells were light-acclimated (constant irradiance of 150 µmol photons m -2 s -1 ), there was no difference in the initial slope of the RLC () whatever the irradiance step (10 s, 20 s or 40 s) and a small increase of the maximum relative electron transport rate (rETRmax) was only observed when using 40 s. However, these studies were conducted ex situ with non-natural irradiances on either sediment cores containing active MPB (Serodio et al., 2005a), reconstituted biofilm [START_REF] Herlory | Methodology of light response curves: application of chlorophyll fluorescence to microphytobenthic biofilms[END_REF] or algal suspensions [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF].

The aim of our work was to make in situ comparisons between different durations of RLC irradiance steps on mudflat MPB biofilms and to weigh up their impact on the subsequent determination of photosynthetic parameters. The rationale behind this study is that in situ conditions of tide, irradiance and temperature, which are difficult to reproduce in the laboratory, affect the recent light history of MPB cells and their photoacclimation state, thus putting into question the validity of conclusions on irradiance steps of RLCs from studies performed ex situ [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]. For the purpose of the present study, 10-s and 30-s RLCs were performed on light-acclimated biofilms in the field. Irradiance steps of 10 s and 30 s are the most commonly used durations for RLCs in MPB in situ studies (e.g. [START_REF] Mouget | Migration or photoacclimation to prevent high irradiance and UV-B damage in marine microphytobenthic communities[END_REF][START_REF] Serodio | Photosynthetic activity, photoprotection and photoinhibition in intertidal microphytobenthos as studied in situ using variable chlorophyll fluorescence[END_REF]Perkins et al., 2010b). The experiments were run within 1 hour before the end of the diurnal emersion period on several consecutive days in summer to provide changes in irradiance, light dose and light history of MPB cells.

Material & Methods

Study site and measurement of environmental conditions

The study was completed on the Brouage intertidal mudflat located in Marennes-Oléron bay on the French Atlantic coast (45°54"50""N, 01°05"25""W) during six consecutive days in 2008 between the 13 and the 18 July. Sediment type is fine mud with a mean grain size < 10 µm [START_REF] Haubois | Spatiotemporal structure of the epipelic diatom assemblage from an intertidal mudflat in Marennes-Oleron Bay, France[END_REF]. In July, the MPB at the site is usually composed of around a dozen species of epipelic diatoms, largely dominated by three species: Navicula phyllepta (about 45% of the relative abundance), Amphora coffeaeformis (20%) and N. digitoradiata (10%) [START_REF] Haubois | Spatiotemporal structure of the epipelic diatom assemblage from an intertidal mudflat in Marennes-Oleron Bay, France[END_REF]. Irradiance and temperature and were measured and recorded every 5 min using a Li-1000 data recorder coupled to a Li-192 quantum sensor and a temperature probe (LI-COR, Lincoln, USA). Environmental conditions are presented in Table 1. RLCs were performed at the end of the diurnal emersion period (1 hour before immersion) so that the time was different from one day to another (semi-diurnal tide).

Rapid light response curves (RLCs)

Variable chlorophyll a fluorescence was measured using a Pulse Amplitude Modulation (PAM) fluorimeter including a PAM-control unit and a WATER-EDF-universal emitter-detector unit (Walz, Effeltrich, Germany). This apparatus is equipped with a modulated blue light (LEDs with maximum emission at 450 nm), which serves as the same light source for the measuring beam, and the actinic and saturating lights. Irradiances were calibrated against a quantum sensor (LI-COR Li 190) at a distance of 2 mm. The PAM fiber optic probe was placed carefully at 2 mm from the MPB biofilms, using a 4 cm diameter ring to isolate the sample from natural irradiance. It was maintained during the RLCs using a burette holder fixed on a base buried in the first 20 cm of sediment. Replicates were chosen randomly within an area of one square metre. An estimation of the effective quantum yield of PSII (F/F m ") was measured few seconds after positioning the probe [START_REF] Serodio | Photosynthetic activity, photoprotection and photoinhibition in intertidal microphytobenthos as studied in situ using variable chlorophyll fluorescence[END_REF], by applying a saturating flash of 800 ms at around 3800 µmol photons m -2 s -1 . RLCs were then constructed by exposing the MPB biofilms to eight steps of increasing irradiance without prior dark acclimation (around 70, 100, 150, 220, 310, 430, 710, 1050 µmol photons m -2 s -1 ).

The duration of the irradiance steps was 10 s or 30 s. Each experimental day, three replicates of each treatment were randomly performed in turn. The relative Electron Transport Rate (rETR) was calculated at each level of irradiance, as the product of the effective quantum yield of PSII and the delivered irradiance: rETR=F/F m " x E. The maximum quantum efficiency (F v5 /F m5 ) was conveniently estimated after 5 min dark acclimation at the end of the 10-s RLCs, to balance for the overall elapsed time of each RLC, longer for 30-s than for 10-s irradiance steps. For each experimental day, the whole process, which included six RLCs plus three 5 min dark acclimation periods and corresponding manipulation lasted between 32 and 50 min. Kinetics of NPQ development or relaxation during the RLC were calculated as suggested by Serodio et al. (2005b): NPQ= (F m " m -F m ")/F m " where F m " m is the maximum F m " value higher than the F m that is measured during the RLCs. In order to compare the kinetics of NPQ during both 10-s and 30-s RLCs, values were standardized between 0 and 1 by normalizing to F m",m (hereafter named "relative NPQ", rNPQ).

Statistics

The light response of rETR (rETR/E) curves were constructed using the model of [START_REF] Eilers | A model for the relationship between light intensity and the rate of photosynthesis in phytoplankton[END_REF]. This model allowed us to estimate the maximum rETR (rETR max ), which is the asymptote of the curve; the maximum light use efficiency (), which is the slope at the beginning of the curve; and the light saturation coefficient (E K ), calculated as (rETR max / ). Curve fitting was achieved using the downhill simplex method of the Nelder-Mead model, and standard deviation of parameters was estimated by an asymptotic method. All fittings were tested by analysis of variance (P < 0.001), residues being tested for normality and homogeneity of variance, and parameter significance by Student"s t-test (P < 0.05).

Comparisons of RLCs and photosynthetic parameters were performed using the method of [START_REF] Ratkowski | Non linear regression modeling. A unified practical approach[END_REF] for non-linear models. All the curve fitting processes and associated statistics were coded under MATLAB R2008b.

The rNPQ light response curves (rNPQ/E curves) were constructed using a combination of two exponential models varying between 0 and 1:

) d E c exp( 1 1 ) b E a exp( 1 1 1 rNPQ       
where a and c are the rates of rNPQ decrease and increase, respectively, and b and d are the half saturation parameters of the decreasing and increasing part of the curve, respectively.

Curve fitting was also done using the downhill simplex method of the Nelder-Mead model and comparisons of rNPQ/E curves and parameters were performed here as well (see above).

Results

Irradiance, effective quantum yield of PSII and maximum quantum efficiency of MPB biofilm

The semi-diurnal tidal regime that the mudflat was exposed to during the period of study (from the 13 to 18 July 2009) provided different conditions for both irradiance and the cumulative light dose (Table 1). The duration of the diurnal emersion was around 4.5 h. The mean irradiance increased with consecutive days, with the sampling time gradually getting closer to noon. Consequently, the temperature at the sub-surface of the sediment increased from 15°C to more than 23°C. The cumulative light dose also increased during the first three days and then stabilized. Irradiance just prior to RLCs was high, varying from 1447 to 1886 µmol photons m -2 s -1 over the six-day survey period and the mean irradiance during the whole diurnal emersion varied from 661 to 1584 µmol photons m -2 s -1 . The effective quantum yield of PSII (F/F m ") drastically decreased with increasing irradiance. The maximum quantum efficiency (F v5 /F m5 ) followed the same pattern during the first three days before stabilizing around 0.45 during the second half of the monitoring.

Rapid light response curves (RLCs) and photosynthetic parameters

Significant differences were found between the rETR/E curves with either 10-s or 30-s irradiance step (Ratkowski test for non linear models; P < 0.001), with the exception of 14 July (P > 0.069) (Fig. 1). For 13 and 18 July, the maximum relative electron transport rate (rETR max ) was significantly lower for 30-s RLCs (Fig. 2A) while the maximum light use efficiency () was no different. For 15, 16 and 17 July,  was significantly lower for 10-s RLCs (Fig 2B ), while there was no difference for rETR max whatever the treatment. Only on the first day (13 July) was a marked down-regulation observed in the rETR/E curve. For both treatments, rETR max increased with irradiance on the first three days and decreased during the second half of the monitoring, as the irradiance reached a maximum (see Table 1). The parameter  decreased by 60-65% with increasing irradiance and stabilized during the last three days. There was no significant linear relationship between  and rETR max whatever the treatment (P > 0.05). E K followed the opposite pattern to  (Fig. 2C).

Photoprotective non-photochemical quenching, NPQ

The maximum NPQ (NPQ max ) displayed by MPB during the course of each RLC was significantly higher when the duration of the irradiance steps was 30 s compared with 10 s (ANCOVA, P < 0.05; Table 1). For both treatments, NPQ max decreased with increasing irradiance prior to the RLCs (P < 0.001, n = 48, Fig. 3). In order to directly compare the kinetics of NPQ development or relaxation during both 10-s and 30-s RLCs, we calculated the relative NPQ (rNPQ) after normalization to the maximum F m " (F m " m ) displayed during each RLC (see Fig. 3). The rNPQ displayed two different patterns and changed gradually from one to another depending on whether the treatment was 10 s or 30 s (Fig. 3). On 13 July, the development of the rNPQ followed a classical shape for both treatments i.e., a sigmoidal increase in the development of NPQ with no relaxation phase for the lowest irradiance of the RLCs (Lavaud et al., 2002b;[START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Evolution, regulation and ecophysiology[END_REF][START_REF] Cruz | Relationship of rapid light curves of variable fluorescence to photoacclimation and non-photochemical quenching in a benthic diatom[END_REF]. The minimum rNPQ (rNPQ min ) was observed at the lowest and developed exponentially from about 250 µmol photons m -2 s -1 to reach a maximum (rNPQ max ) at the highest irradiance of the RLCs. No relaxation occurred at the lowest irradiance of the RLCs. On 14 July, while the trend of rNPQ development was the same for the 10 s treatment, even though it was faster than the previous day, it was different for the 30 s-RLC following a U-shaped pattern as a function of irradiance. Indeed, rNPQ was already developed prior to the RLC, it then relaxed to a minimum up to 400 µmol photons m -2 s -1 to develop once more, finally reaching a maximum value at the highest irradiance. This pattern of difference between 10 s-and 30 s-RLCs is inferred from the fact that F m " m was not reached at the same irradiance step during the RLC due to a different cumulative light dose resulting from 10-s or 30-s duration. During the following days, the two treatments showed the same U-shaped pattern with NPQ relaxation in the first part of the RLC higher than the NPQ development that occurred in the second part (i.e. for higher irradiance). The irradiance at which rNPQ min was observed increased with E K from a day to another (see Fig. 2C) until a plateau was reached. In contrast to 30-s RLC on the 14 July and for both treatments on 13 July, rNPQ max was observed at the lowest irradiance whatever the treatment. Finally, the relaxation of rNPQ at the beginning of RLCs increased with lower irradiances for the 30 s-RLCs.

Discussion

Physiological basis for the differences in photosynthetic parameters determined under different RLC irradiance step durations

The Rapid Light Curve (RLC) methodology applied to microphytobenthos (MPB) is based on a paradox: RLCs aim to determine the photoacclimation state of microalgal cells, but they also generate a dynamic phase of regulation/acclimation during the RLC, due to the cumulative actinic light dose, which is function of the duration of the irradiance step increments. On the one hand, short-term changes in regulation/acclimation states induced by RLC methodology can be visualized, but on the other hand, they are by nature limited in comparison with those occurring in steady state light curves (SSLCs). Although the general trends of photoacclimation were similar whatever the duration of the irradiance step increments (10 s or 30 s), duration clearly influenced the pattern of RLCs and the absolute values of photosynthetic parameters extracted. The differences between the 10-s and 30-s RLCs were observed throughout the six days of monitoring, independently of the light dose the MPB was exposed to prior to the RLCs. According to [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF], the light history of the cells prior to RLC, which governs their photoacclimation capacity, and the cumulative light dose during the RLC are key factors for explaining the discrepancy in the determination of photosynthetic parameters when the duration of irradiance steps differs. The cumulative light dose effect is based on two fast photo-chemical and photo-protective processes: the reoxidation of Q A and the development of NPQ. [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF] proposed that the primary misleading source for the determination of photosynthetic parameters of RLCs with 10-s light steps is the dynamic and incomplete reoxidation of Q A from the irradiance increment n to n+1 and so on. It generates the non steady-state accumulation of reduced electron acceptors down to the PSII during the course of the RLC and this has a growing impact on the higher irradiance increments. It ultimately leads to the underestimation of rETR max and potentially α in the case of very slow Q A reoxidation. This is especially true when the electron transfer between Q A and Q B is slowed down because of environmental stresses (light or nutrient stress, herbicide exposure, etc.) [START_REF] Materna | First induced plastid genome mutations in an alga with secondary plastids: psbA mutations in the diatom Phaeodactylum tricornutum (Bacillariophyceae) reveal consequences on the regulation of photosynthesis[END_REF]. This proposal was based on different studies performed on benthic diatoms acclimated to low irradiance or darkness for various periods prior to RLC measurements [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF][START_REF] Cruz | Relationship of rapid light curves of variable fluorescence to photoacclimation and non-photochemical quenching in a benthic diatom[END_REF]. However, the differences in rETR max determined after 10 s and 30 s (and also for longer durations) are lower and are even nearly eliminated when the cells have been previously acclimated to higher continuous irradiance (Serodio et al., 2005b;[START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]. We effectively did not observe any significantly lower values of rETRmax for the 10-s RLCs compared to the 30-s RLCs in MPB acclimated to high and variable irradiances in situ. This can be explained by a modification of the stochiometry PSII : Cytochrome b6/f : PSI, leading to an increase in the number of electron acceptors down to the PSII and a higher PSI cyclic electron activity, which would accelerate the reoxidation of Q A [START_REF] Kotakis | Evidence for active cyclic electron flow in twig chlorenchyma in the presence of an extremely deficient linear electron transport activity[END_REF]. The second possibility could be an increase in the extent of the PSII electron cycle. This mechanism deviates electrons from their linear course down to Q B and decreases the probability of Q A over-reduction (Lavaud et al., 2002a;[START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Evolution, regulation and ecophysiology[END_REF]. These two features were recently reported in PSII mutants of diatoms showing an impaired ETR [START_REF] Materna | First induced plastid genome mutations in an alga with secondary plastids: psbA mutations in the diatom Phaeodactylum tricornutum (Bacillariophyceae) reveal consequences on the regulation of photosynthesis[END_REF], supporting the hypothesis it could also occur in situ in natural MBP biofilm composed of benthic diatoms exposed to very high irradiances and showing a rETR decrease (see Fig. 2A).

The cumulative light dose during the RLC itself can lead to the fast development of photoprotective NPQ triggered by the build-up of the transthylakoid ΔpH and the xanthophyll cycle (so-called qE) (Serodio et al., 2005b;[START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Evolution, regulation and ecophysiology[END_REF]. NPQ can influence the pattern of RLCs when the irradiance steps are equal to or longer than 30 s [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]. The quenching of chlorophyll fluorescence generated by the NPQ mechanism disturbs the acquisition of the effective quantum yield of PSII (ΔF/F m ") [START_REF] Herlory | Methodology of light response curves: application of chlorophyll fluorescence to microphytobenthic biofilms[END_REF] and the subsequent calculation of rETR, leading to its underestimation [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]. This is especially true for diatoms, which can develop large NPQ (Lavaud et al., 2002b;[START_REF] Lavaud | Fast regulation of photosynthesis in diatoms: Evolution, regulation and ecophysiology[END_REF][START_REF] Serodio | Photosynthetic activity, photoprotection and photoinhibition in intertidal microphytobenthos as studied in situ using variable chlorophyll fluorescence[END_REF] as their potential increases with the irradiance [START_REF] Schumann | Investigation of the quenching efficiency of diatoxanthin in cells of Phaeodactylum tricornutum (Bacillariophyceae) with different pool sizes of xanthophyll cycle pigments[END_REF][START_REF] Cruz | Relationship of rapid light curves of variable fluorescence to photoacclimation and non-photochemical quenching in a benthic diatom[END_REF]. Such a situation was observed during the second half of our monitoring, when the irradiance was the highest. NPQ was high, the acquisition of ΔF/F m " was obviously disturbed as it was very low, while F v5 /F m5 remained stable (see Table 1) and rETR max was ultimately lower for 30 s-RLCs (18 th of July). rETR max determined with 30 s-RLCs was also underestimated when such conditions did not occur (the first day). In this case, rETR max was reduced, probably because of the fluorescence quenching generated by excess light photoinhibitory damage (so-called qI). The decrease of rETR max was observed for both 10-s and 30-s increments, and was more accentuated for 30 s-RLCs because of the higher cumulative light dose.

NPQ also strongly influences the determination of α (Serodio et al., 2006a;[START_REF] Cruz | Relationship of rapid light curves of variable fluorescence to photoacclimation and non-photochemical quenching in a benthic diatom[END_REF] by potentially generating its underestimation when the length of the RLC irradiance steps is too short (i.e. 10 s compared to 30 s) [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF]. This is crucial when diatoms are exposed to excess irradiance resulting in a sustained NPQ, as during our monitoring (apart from the first day); so that when the RLC starts, rNPQ is already at its maximum. For an accurate determination of α, NPQ needs to be relaxed. In diatoms, the relaxation of NPQ in the dark can last a long time because of the slow reversion of the xanthophyll cycle operation [START_REF] Lavaud | Photoprotection capacity differs among diatoms: Possible consequences on the spatial distribution of diatoms related to fluctuations in the underwater light climate[END_REF]Perkins et al., 2010b). It is accelerated by switching the cells to low irradiance (Serodio et al., 2005a;[START_REF] Goss | The importance of a highly active and delta pH-regulated diatoxanthin epoxidase for the regulation of the PSII antenna function in diadinoxanthin cycle containing algae[END_REF]. In our conditions, NPQ relaxed during the first low irradiance steps of RLCs; the longer the irradiance step, the faster the relaxation. Thus, NPQ relaxation was thus always faster for the 30 s-RLCs and α was systematically lower for the 10-s RLCs. Results differed on the last day since NPQ max was similar for both 10-s and 30-s increments and the NPQ relaxation at the beginning of RLCs did not show significant difference. The very high irradiances observed on the last day probably resulted in a preponderant behavioural photoacclimation strategy (i.e. cell micro-cycling and downward migrations), in contrast to physiological photoacclimation (i.e. NPQ) (Serodio et al., 2006b;[START_REF] Mouget | Migration or photoacclimation to prevent high irradiance and UV-B damage in marine microphytobenthic communities[END_REF]Perkins et al., 2010b). Downward cell migration has been shown to be an essential behaviour that MPB cells use to cope with high and photoinhibitory irradiances during emersion [START_REF] Mouget | Migration or photoacclimation to prevent high irradiance and UV-B damage in marine microphytobenthic communities[END_REF]Perkins et al., 2010b).

To our best knowledge, no studies have investigated the effect of MPB cell micro-migration during RLCs on the determination of photosynthetic parameters. Indeed, only a few of them were conducted on sediment containing migratory MPB (Serodio et al., 2005a;[START_REF] Herlory | Methodology of light response curves: application of chlorophyll fluorescence to microphytobenthic biofilms[END_REF]. Notably, [START_REF] Underwood | Patterns in microphytobenthic primary productivity: Species-specific variation in migratory rhythms and photosynthetic efficiency in mixed-species biofilms[END_REF] observed that specific diversity changes due to migration could occur in the MPB community at high irradiances of the rETR/E curves when using a 3 min irradiance step. Here, the longest RLCs lasted 4 min (8 steps of 30 s). In these conditions, and although RLCs were primarily used to prevent MPB cell migration, it cannot be totally ruled out. Cell migration could explain the differences observed between 10-s and 30-s treatments for the highest irradiances of the RLCs on 13 and 18 July. On 13 July particularly, it is probable that early in the morning the MPB community was enriched in low light-acclimated cells that were not able to cope with the 30-s step of highest irradiances.

They therefore possibly migrated down, which generated a decrease in rETR.

Comparison of RLCs with different duration of irradiance steps due to the constraints of field work

In our study, the high irradiance that the MPB was exposed to leads to opposite effects of the Q A redox state and extent of NPQ on the determination of photosynthetic parameters.

Additionally, the hierarchy of the impact of these two processes changed depending on the duration of the irradiance step increments of RLCs (Fig. 4). In summary, for 10-s duration, the unbalanced redox state of Q A from an irradiance step to another is the most influential in diatoms acclimated to low irradiance (as generally found in laboratory conditions, Perkins et al., 2006[START_REF] Herlory | Methodology of light response curves: application of chlorophyll fluorescence to microphytobenthic biofilms[END_REF]. It is important to underline that, independently from any field constraints for any sequential light curves (whatever the technique used: oxymetry, 14 C incorporation or modulated fluorimetry), the same processes (redox changes at Q A and NPQ relaxation/development) happen and could also be considered, with the exception of their dynamic phase. If RLCs are used to perform light curves, NPQ influence becomes significant under high irradiance (as generally found in situ) because of its development prior to the RLC, and of the balance between its relaxation during the first initial low irradiance steps, and its development during the highest irradiance steps of the RLC. For 10-s RLCs, pre-existing NPQ, present prior to the RLC is crucial, as it cannot be fully relaxed during the first irradiance steps because they are too short. For 30-s RLCs, the most influential process is the NPQ that develops during RLC as the relaxation of NPQ present prior to RLC and the reoxidation of Q A occur. The influence of NPQ is accentuated by the acclimation of the diatoms to higher light doses, as for in situ conditions (a noticeable effect is observed only with 50-60 s treatment in laboratory conditions, [START_REF] Perkins | Light response curve methodology and possible implications in the application of chlorophyll fluorescence to benthic diatoms[END_REF][START_REF] Herlory | Methodology of light response curves: application of chlorophyll fluorescence to microphytobenthic biofilms[END_REF]. In these conditions, NPQ present prior to RLC is better relaxed because of the longer duration of initial low irradiance steps, which ultimately leads to a better estimation of α for 30 s, in contrast to 10-s increments. If the light dose is moderate (i.e., in winter/spring or on a cloudy day) the balance between the effects of change in Q A redox state and NPQ relaxation/development on the determination of rETRmax and α would be modified, but its dynamics certainly depend on the photoacclimation state of the cells.

Two strategies could be proposed to remove these drawbacks inherent to RLC methodology. From a purely technical point of view, an initial non-academic approach would be to perform a first 10-s RLC, followed by a short (5 min) low irradiance or far-red exposure [START_REF] Jesus | Chlorophyll fluorescence as a proxy for microphytobenthic biomass: alternatives to the current methodology[END_REF], then by a 10-s RLC including only the lowest irradiance. The first RLC would provide a good estimation of rETR max , while the second would provide estimation of α after relaxation of NPQ. Nevertheless, two major criticisms can be made to this unconventional protocol. Its length, plus the low/far-red light period, would most probably generate the downward migration of MPB, the RLC methodology is supposed to counteract.

Additionally, it would reduce the possibility of performing a maximum of replicates during the emersion period. As an alternative, we checked the possibility of improving the determination of α both for 10-s and 30-s RLC a posteriori. In diatoms, low irradiance is a more efficient way to relax NPQ than dark acclimation (Serodio et al., 2005a;[START_REF] Goss | The importance of a highly active and delta pH-regulated diatoxanthin epoxidase for the regulation of the PSII antenna function in diadinoxanthin cycle containing algae[END_REF]. Using this characteristic, we hypothesize that F m " m values measured during RLCs after the initial relaxation of NPQ were the closest to F m . This hypothesis is strengthened by the fact that F m " m was usually reached for two consecutive light levels, in most cases for 30 s RLCs (but for 18 July, Fig 3). We then replaced the lower values of F m " recorded for lower values of irradiance by the value of F m " m in each RLC. The initial slope of the RLC was then re-estimated. Values of the parameter α were improved by this approach, which allowed us to partially counteract the impact of NPQ (Fig. 5). The parameter α increased by 43 to 169% and by 37 to 74% for 10 s and 30 s, respectively. Moreover, this correction was neutral for RLCs without a preliminary phase of NPQ relaxation (13 July, and 14 July for 10 s only). However, this correction was not as efficient for 10 s as for 30 s and most probably the influence of preexisting NPQ present prior to the RLC was not fully removed for 30-s RLCs. For this reason, regarding irradiance step duration for RLC methodology, although a shorter irradiance step is not the ultimate objective for MPB studies in situ, it is the best compromise, especially when field constraints and MPB behaviour are taken into account. RLCs were run with either 10-s or 30-s light step increments. The curves were constructed using the [START_REF] Eilers | A model for the relationship between light intensity and the rate of photosynthesis in phytoplankton[END_REF] model. Significant differences between the treatments were determined using the method of [START_REF] Ratkowski | Non linear regression modeling. A unified practical approach[END_REF] for non linear models (P values are indicated). Values are mean ± SD of three independent measurements Fig. 2. Photosynthetic parameters extracted from rETR/E curves (Fig. 1). Significant differences between the treatments were determined using the method of [START_REF] Ratkowski | Non linear regression modeling. A unified practical approach[END_REF] for non linear model (n.s. non significant * P<0.05 **P<0.01 ***P<0.001) Fig. 3. Relative non-photochemical quenching of fluorescence (rNPQ) developed during each RLC. On some occasions, symbols of the two treatments are surimposed. NPQ values were normalized to the maximum NPQ observed (see Fig. 3). Values are mean ± SD of three independent measurements. The curves were constructed using the equation given in the Material and Methods, statistics section. Significant differences between the treatments were determined using the method of [START_REF] Ratkowski | Non linear regression modeling. A unified practical approach[END_REF]

Figure captions

Table 1

Temperature, light environment and photosynthetic capacity of the microphytobenthos biofilm on the Brouage mudflat, Marennes-Oléron bay, July 2009. The duration of the emersion was around 4.5 h (semi-diurnal tide). Measurement time indicates the beginning and the end of the six RLCs measurements (three replicates of 10-s and 30-s light-step RLCs randomly displayed) plus the three 5 min dark acclimations (for more details see materials and methods section). RLCs were performed at the end of the diurnal emersion period (1 hour before the immersion) so that the time was different from a day to another. Values are mean ± SD of three independent measurements.

Fig. 1 .

 1 Fig. 1. Rapid light response curves (RLCs) of the relative electron transport rate (rETR) of in

Fig. 4 .

 4 Fig. 4. Diagram of the influence of the duration of RLC irradiance steps, the light acclimation

Fig. 5 .

 5 Fig. 5. Re-estimation of α (maximum light use efficiency) using a posteriori corrections of

Acclimation to: Low light / Dark -Continuous Acclimation to: High light -Variable / Continuous

  F/F m ", effective quantum yield of PSII; F v5 /F m5 , maximum quantum efficiency measured after 5 min dark acclimation; RLC, rapid light curve; NPQmax, maximum non-photochemical quenching of chlorophyll fluorescence (NPQ max ) developed (with an asterix) or relaxed during each RLC.

		Figure 4
			10 s irradiance steps		30 s irradiance steps
		Q A reoxidation "unbalanced"	No NPQ development prior to and	Q A reoxidation "balanced"	Slow relaxation of initial NPQ during RLC	Low to moderate NPQ development during RLC	No NPQ development prior to RLC	Relaxation of initial NPQ nearly complete during RLC	F/F m "	F v5 /F m5	NPQmax	NPQmax
			during RLC
		rETR max Underestim.			α Underestim.	proxy for Valuable		α Underestim.			(10s)	(30s)
						rETR max
		E K Underestim.			E K Overestim.
	13	07:14/07:55	15.3 ± 0.1	8.1		661.1 (0-1512)	1447.2 (1360-1512)	0.58 ± 0.04 0.63 ± 0.01 0.45±0.11* 1.60±0.18*
	14	08:03/08:53	18.3 ± 0.4	15.8		1009.4 (0-1597)	1548.2 (1516-1575)	0.46 ± 0.10 0.53 ± 0.05 0.44±0.07* 0.53±0.06*
	15	09:28/10:00	21.6 ± 0.1	22.1		1361.0 (647-1745)	1703.6 (1674-1735)	0.31 ± 0.04 0.49 ± 0.04 0.19±0.06	0.52±0.25
	16	10:08/10:58	23.5 ± 0.2	25.6		1499.7 (977-1850)	1807.6 (1760-1838)	0.14 ± 0.08 0.43 ± 0.06 0.15±0.07	0.41±0.06
	17	10:59/11:37	21.6 ± 0.2	25.2		1457.2 (636-1885)	1797.2 (1552-1881)	0.33 ± 0.24 0.43 ± 0.02 0.18±0.01	0.71±0.07
	18	11:40/12:17	22.2 ± 0.1	28.4		1583.8 (443-1888)	1886.5 (1884-1886)	0.21 ± 0.06 0.47 ± 0.04 0.13±0.02	0.29±0.18
		T, temperature; PPFD, 29

Acclimation to: Low light / Dark -Continuous Acclimation to: High light -Variable / Continuous

	Small NPQ	Q A reoxidation	Large NPQ
	development	balanced	development
	during RLC		during RLC
	rETR max		rETR max
	Underestim.		Underestim.
	E K	E K	E K
	Underestim.	Overestim.	Underestim.

Acknowledgments

This work was supported by the French ANR (National Research Agency) through the VASIREMI project "Trophic significance of microbial biofilms in tidal flats" (contract ANR-06-BLAN-0393-01). The authors are grateful to the scientific community of VASIREMI with special thanks to Christine Dupuy, Brenda Waqueteux and Katell Guizien for their support during this study. We are embedded to Dany Prevostat (Aeroglisseurs Services) for technical support during the field survey. Finally, we thank two anonymous reviewers for their constructive comments.


Benoit Daniel

Isabel Fernandez

Pablo Mira
email: pablo.mira@upct.es

Î T Beno

Isabel Fern Daniel

Pablo Ández

Mira

THE GAUSS MAP OF SURFACES IN PSL 2 (R)

Keywords: 2010 Mathematics Subject Classification. Primary: 53C42, 53C30. Secondary: 53A10, 53C43 Homogeneous Riemannian manifold, constant mean curvature surface, harmonic map

Introduction

An important property in the theory of constant mean curvature (CMC) surfaces is that the Gauss map (i.e., the unit normal vector viewed as a map into the 2sphere S 2 ) of a CMC surface in R 3 is a harmonic map into S 2 . If the surface in R 3 is minimal, the Gauss map is holomorphic.

When we consider an immersed oriented surface Σ in a Riemannian threemanifold M , its unit normal vector is a map N : Σ → UM that takes values in the five-dimensional unit tangent bundle UM of M . It is then unclear how to define for Σ a Gauss map into S 2 that provides relevant information about the surface. The construction of such a Gauss map will depend on the geometric properties of M , and in general the Gauss map will not be harmonic for CMC surfaces in M . Nonetheless, for some ambient spaces M such a Gauss map has been constructed, and has provided key information about the global properties of certain classes of surfaces in M .

An illustration of this fact is the case of CMC surfaces in hyperbolic 3-space H 3 . There is a well known way to define a "hyperbolic Gauss map" for surfaces in H 3 [START_REF] Epstein | The hyperbolic Gauss map and quasiconformal reflections[END_REF][START_REF] Bryant | Surfaces of mean curvature one in hyperbolic space[END_REF] taking values in the asymptotic boundary ∂ ∞ H 3 ≃ C where C = C ∪ {∞} is the Riemann sphere. R. Bryant [START_REF] Bryant | Surfaces of mean curvature one in hyperbolic space[END_REF] proved that this hyperbolic Gauss map is holomorphic for CMC 1 surfaces in H 3 and he obtained a Weierstrass-type representation formula for these surfaces; these techniques were then developed by M. Umehara and K. Yamada [START_REF] Umehara | Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space[END_REF] and many other authors, leading to very important improvements in the study of these surfaces. See also [START_REF] Gálvez | Flat surfaces in the hyperbolic 3-space[END_REF][START_REF] Espinar | Hypersurfaces in H n+1 and conformally invariant equations: the generalized Christoffel and Nirenberg problems[END_REF] for other applications of the hyperbolic Gauss map.

Recently, much attention was drawn to surfaces in simply connected homogeneous 3-manifolds. The homogeneous spaces diffeomorphic to R 3 with a 4dimensional isometry group constitute a two-parameter family; they will be denoted by E 3 (κ, τ ) where κ 0 and (κ, τ ) = (0, 0). When τ = 0 we get the product space H 2 (κ) × R; when κ = 0 we obtain the 3-dimensional Heisenberg group Nil 3 endowed with a left-invariant metric; otherwise we obtain the universal cover of the Lie group PSL 2 (R) endowed with a left-invariant metric; we will denote it by PSL 2 (R). Thus, we see that the metrics of H 2 × R and Nil 3 arise as limits of certain left invariant metrics on PSL 2 (R).

The second and third authors [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] defined a "hyperbolic Gauss map" taking values in the hyperbolic plane H 2 for surfaces in H 2 × R having a regular projection on H 2 ; they moreover proved that for CMC 1/2 surfaces this Gauss map is harmonic, and they obtained a Weierstrass-type representation formula for these surfaces. Then, in Nil 3 , the first author [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF] studied a Gauss map defined using the Lie group structure and obtained obtained a Weierstrass-type representation formula for minimal surfaces in Nil 3 . These two Gauss maps have quite different definitions and properties. Moreover, the use of these Gauss maps has led to important discoveries in their respective theories, such as solutions to Bernstein type problems, classification of complete multigraphs or half-space theorems [START_REF] Fernández | Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space[END_REF][START_REF] Daniel | Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group[END_REF].

Let us indicate that minimal surfaces in R 3 and Nil 3 , CMC 1 surfaces in H 3 and CMC 1/2 surfaces in H 2 ×R all have critical constant mean curvature. That is, their mean curvature equals the value c such that there exist compact CMC H surfaces respectively in R 3 , Nil 3 , H 3 , H 2 × R if and only if |H| > c. By the same definition, the value c for the critical mean curvature in PSL 2 (R) = E 3 (κ, τ ) for κ < 0 and τ = 0, is c = √ -κ/2. Moreover, CMC surfaces with critical mean curvature in H 2 × R, Nil 3 and PSL 2 (R) are related by a Lawson-type correspondence [START_REF] Daniel | Isometric immersions into 3-dimensional homogeneous manifolds[END_REF].

The aims of this paper are:

• to define geometrically a smooth Gauss map for smooth surfaces in PSL 2 (R) taking values in a 2-sphere, in a way that does not depend on a choice of model and such that the Gauss map is "compatible" with ambient isometries (in a sense that will be made precise), • to try to unify the definitions of the Gauss map of surfaces in H 2 × R, PSL 2 (R) and Nil 3 , • to prove that the Gauss maps of CMC local graphs with critical mean curvature in PSL 2 (R) are harmonic maps into H 2 (local graphs will be defined in Section 2).

Let us first eliminate two approaches that could seem natural to solve this problem.

The first approach would be to define a left-invariant Gauss map using the Lie group structure, as it is implicitely done in R 3 and as it was done in Nil 3 [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF] (see also [START_REF] Daniel | Existence and uniqueness of constant mean curvature spheres in Sol 3[END_REF]). Isometries coming from the Lie group structure in H 2 × R and PSL 2 (R) do not have a geometric characterization (see Remark 3.10) as in R 3 and Nil 3 (where they are "translations"). Moreover, the homogeneous manifold PSL 2 (R) is isometric to another Lie group with a left-invariant metric (see Theorem 2.14 and Corollary 3.19 in [START_REF] Meeks | Constant mean curvature surfaces in metric Lie groups[END_REF] for a discussion of these Lie group structures), so defining a left-invariant Gauss map would depend on the choice of a Lie group structure. These left-invariant Gauss maps do not relate well to isometries. One can compute that, for surfaces with critical CMC, they satisfy some second order elliptic partial differential equations that are not harmonic map equations.

The second approach would be to define a Gauss map taking values in the asymptotic boundary of the ambient manifold, as it is again implicitly done in R 3 and as it was done in H 3 . However, even in H 2 × R this is not suitable: indeed, H 2 × R admits no differentiable Hadamard compactification [START_REF] Kloeckner | Symmetric spaces of higher rank do not admit differentiable compactifications[END_REF], and hence the Gauss map of a smooth surface would not necessarily be smooth.

The paper is organised as follows. In Section 2, we introduce some preliminary material about the manifolds E 3 (κ, τ ). In Section 3, we prove the existence and the uniqueness of a Gauss map satisfying some geometric conditions; this Gauss map reaches the first two of the abovementioned aims of the paper. In Section 4, we gather some preliminary computations related to conformal immersions and their Gauss map. Section 5 is devoted to the Gauss map of surfaces with critical CMC: we prove that it is harmonic into H 2 for local graphs and we obtain a Weierstrass-type representation theorem; hence this Gauss map reaches the third of the abovementioned aims. We also relate the Gauss map of the surface with the Gauss map of its sister minimal surface in Nil 3 . In Section 6 we give the expression of the Gauss map using the Lorentzian model for H 2 .

2.

The manifolds E 3 (κ, τ )

The simply connected homogeneous Riemannian 3-manifolds with a 4-dimensional isometry group constitute, together with Euclidean 3-space and round 3-spheres, a 2-parameter family (E 3 (κ, τ )) (κ,τ )∈R 2 , such that there exists a Riemannian fibration π : E 3 (κ, τ ) → M 2 (κ) with bundle curvature τ , where M 2 (κ) is the simply connected surface of curvature κ. The fibration is a product fibration if and only if τ = 0. We refer to [START_REF] Daniel | Isometric immersions into 3-dimensional homogeneous manifolds[END_REF] and references therein for more details.

There are different types of manifolds according to the values of κ and τ . In this paper we will only consider the case where κ 0.

The manifold E 3 (κ, τ ) is diffeomorphic to R 3 and isometric

• to Euclidean space R 3 when (κ, τ ) = (0, 0); in this case the isometry group has dimension six, and the fibration π is not unique, • to the 3-dimensional Heisenberg group endowed with a left-invariant metric when κ = 0 and τ = 0; we will denote it by Nil 3 (τ ) (up to dilations, all these metrics are isometric), • to H 2 (κ) × R when κ < 0 and τ = 0, where H 2 (κ) is the hyperbolic plane of constant curvature κ, • to the universal cover of PSL 2 (R) endowed with certain left-invariant metrics when κ < 0 and τ = 0; we will denote it by PSL 2 (R) (up to dilations, all these metrics are isometric to one such that κ -4τ 2 = -1).

A vector v is said to be vertical if dπ(v) = 0 and horizontal if it is orthogonal to vertical vectors. We let ξ be a unit vertical field in E 3 (κ, τ ) (the field ξ is unique up to multiplication by -1). We say that a vector v is upwards pointing (respectively, downwards pointing) if v, ξ > 0 (respectively, v, ξ < 0). We set

c := √ -κ 2 .
This constant is called the critical mean curvature because there exist compact CMC H surfaces in E if and only if |H| > c. When κ < 0, horocylinders, i.e. inverse images by π of horocycles of H 2 (κ), have constant mean curvature c. When κ = 0, vertical planes, i.e. inverse images by π of straight lines of R 2 , are minimal, and hence have constant mean curvature c. In particular, any E 3 (κ, τ ) with κ 0 can be foliated by topological planes of critical CMC, all of them congruent to each other.

A surface is said to be nowhere vertical (or a local graph) if ξ is nowhere tangent to it, that is, if the restriction of π to the surface is a local diffeomorphism (in H 2 (κ) × R this means that the surface has a regular projection on H 2 (κ)); it is said to be an entire graph if the restriction of π to the surface is a global diffeomorphism onto M 2 (κ). The angle function of an oriented surface Σ is the function N, ξ where N is the unit normal vector field to Σ. Hence, a surface is nowhere vertical if and only if its angle function does not vanish. CMC local graphs, and in particular entire graphs, play an important role among CMC surfaces; see for instance [START_REF] Fernández | Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space[END_REF][START_REF] Hauswirth | Infinite boundary value problems for constant mean curvature graphs in H 2 × R and S 2 × R[END_REF][START_REF] Hauswirth | On complete mean curvature 1 2 surfaces in H 2 ×R[END_REF][START_REF] Daniel | Half-space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group[END_REF][START_REF] Mazet | The Dirichlet problem for the minimal surface equation, with possible infinite boundary data, over domains in a Riemannian surface[END_REF][START_REF] Manzano | Parabolic stable surfaces with constant mean curvature[END_REF][START_REF] Mazet | The half space property for cmc 1/2 graphs in E(-1, τ )[END_REF].

For ρ > 0 we set

D(ρ) = {z ∈ C; |z| < ρ}.
We also set D = D(1).

We use the following model for the hyperbolic plane of constant curvature κ:

H 2 (κ) = D 2 √ -κ
endowed with the metric given in canonical coordinates (x 1 , x 2 ) by

Λ 2 (dx 2 1 + dx 2 2 ) where Λ = 1 1 + κ 4 (x 2 1 + x 2 2 ) = 1 1 -c 2 |ζ| 2 with ζ = x 1 + ix 2 .
The model we use for

E 3 (κ, τ ) is D 2 √ -κ × R when κ < 0 or R 3 when κ = 0, endowed with the metric given in canonical coordinates (x 1 , x 2 , x 3 ) by Λ 2 (dx 2 1 + dx 2 2 ) + (τ Λ(x 2 dx 1 -x 1 dx 2 ) + dx 3 ) 2 . In this model we have π(x 1 , x 2 , x 3 ) = (x 1 , x 2 ).
We consider the following orthonormal frame:

(2.1) V 1 = 1 Λ ∂ ∂x 1 -τ x 2 ∂ ∂x 3 , V 2 = 1 Λ ∂ ∂x 2 + τ x 1 ∂ ∂x 3 , V 3 = ∂ ∂x 3 = ξ.
The fields V 1 and V 2 are the horizontal lifts in E 3 (κ, τ ) by the fibration π of the fields

1 Λ ∂ ∂x1 and 1 Λ ∂ ∂x2 in M 2 (κ).

Definition of the Gauss map

In this section we set κ < 0.

We will define a "geometric" Gauss map for surfaces in E 3 (κ, τ ) that coincides with the hyperbolic Gauss map [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] for graphs in H 2 (κ) × R when τ = 0 and such that, by taking a limit when κ → 0, we get the Gauss map defined for surfaces in Nil 3 using the Lie group structure studied in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF].

For simplicity we set E = E 3 (κ, τ ). We denote by Isom 0 (E) the connected component of the identity in the isometry group of E. All isometries in Isom 0 (E) leave the field ξ invariant, since the sectional curvature of the plane ξ ⊥ is a strict minimum (see [START_REF] Daniel | Isometric immersions into 3-dimensional homogeneous manifolds[END_REF] for more details); as a consequence, a fiber is mapped to another fiber. An isometry f ∈ Isom 0 (E) induces via π an orientation-preserving isometry f of H 2 (κ), i.e., f • π = π • f ; we will say that f is the horizontal part of f . Moreover, the field ξ is a Killing field; the isometries it generates are called vertical translations and their horizontal part is the identity.

Let

SU 1,1 = α β β ᾱ ; (α, β) ∈ C 2 , |α| 2 -|β| 2 = 1 , PSU 1,1 = SU 1,1 /{±I}.
We let S denote the Riemann sphere C = C ∪ {∞} together with a marked oriented circle E in C. We call E the equator of S. Then S \ E has two connected components: we will call northern hemisphere and denote by S + the one whose oriented boundary is E, and we will call southern hemisphere and denote by S -the one whose oriented boundary is E with the opposite orientation.

The connected group that acts naturally on S is the group of (orientationpreserving) conformal diffeomorphisms of C that leave E invariant and preserve the orientation of E. This group is PSU 1,1 ≃ PSL 2 (R), which is also the group of orientation-preserving isometries of H 2 (κ). Hence to each M ∈ SU 1,1 we associate a conformal diffeomorphism ψ M : S → S as above and an orientation-preserving isometry ϕ M : H 2 (κ) → H 2 (κ), so that

ψ M N = ψ M • ψ N , ϕ M N = ϕ M • ϕ N .
In this way we obtain a unique conformal diffeomorphism σ :

H 2 (κ) ∪ ∂ ∞ H 2 (κ) → S + ∪ E such that σ • ψ M = ϕ M • σ for all M .
Without loss of generality we can assume that E is the circle |z| = 1 oriented counter clockwise, the northern hemisphere is

S + = {|z| < 1},
the southern hemisphere is

S -= {|z| > 1} ∪ {∞},
and, for

M = α β β ᾱ ∈ SU 1,1 , ψ M and ϕ M are defined by ψ M (z) := αz + β βz + ᾱ , ϕ M (ζ) := 1 c αcζ + β βcζ + ᾱ . In this way, we have σ(z) = z/c.
Inspired by the properties of the "hyperbolic Gauss map" of surfaces in hyperbolic 3-space H 3 , we seek a Gauss map for oriented surfaces in E with values in S having the following properties:

• two surfaces passing through the same point x have the same Gauss map at x if and only if they have the same unit normal vector at x, • applying an isometry f of E belonging to the connected component of the identity switches the Gauss map g of the surface to ψ M •g, where M ∈ SU 1,1 corresponds to the horizontal part of f , • the Gauss map at a point lies on the equator E (respectively, in the northern hemisphere) if and only if the unit normal vector at that point is horizontal (respectively, upwards pointing), The first property implies that the Gauss map of an oriented surface Σ factorizes as g = Π • N for some map Π : UE → S, where UE denotes the unit tangent bundle to E and N : Σ → UE the unit normal vector to Σ. Theorem 3.1. There exists a unique map

Π : UE → S (x, Z) → Π x (Z) such that (a) for any f ∈ Isom 0 (E), if M ∈ SU 1,1 is such that the horizontal part of f is ϕ M , then Π • df = ψ M • Π, (b) for any (x, Z) ∈ UE, Π x (Z) ∈ E (respectively Π x (Z) ∈ S + , Π x (Z) ∈ S -)
if and only if Z is horizontal (respectively, upwards pointing, downwards pointing), (c) for any point x ∈ E and any horizontal vector

Z ∈ U x E, Π x (Z) = σ(p)
where p denotes the endpoint of the oriented geodesic of H 2 (κ) passing through π(x) with dπ(Z) as tangent vector, (d) for any point x ∈ E, the map Π x : U x E → S is a conformal diffeomorphism. Moreover the map Π is analytic, and its expression in the frame

(V 1 , V 2 , V 3 ) is given by (3.1) Π x (Z) = Z 1 + iZ 2 + cζ(1 + Z 3 ) c ζ(Z 1 + iZ 2 ) + 1 + Z 3 where ζ = x 1 + ix 2 and Z = Z 1 V 1 + Z 2 V 2 + Z 3 V 3 .
Proof. We first notice that proving existence and uniqueness of Π is equivalent to proving that there exists a unique map Π : UE → S satisfying property (a) and properties (b), (c) and (d) at one point, for instance at the origin O = (0, 0, 0) ∈ E.

We first prove the uniqueness of Π O . At the origin the frame (V 1 , V 2 , V 3 ) is just the coordinate frame. We let s : U O E → S denote the stereographic projection with respect to the South Pole, i.e., if

Z = Z 1 V 1 + Z 2 V 2 + Z 3 V 3 ∈ U O E, then s(Z) = Z 1 + iZ 2 1 + Z 3 .
So, by property (d) there exists a conformal diffeomorphism h : S → S such that

Π O = h • s.
By property (b), h(S + ) = S + . Moreover, property (a) holds for all isometries f ∈ Isom 0 (E) preserving O. Applying this property to the vector Z = V 3 , we obtain that for all rotations R around 0 ∈ S we have h(0) = R(h(0)), which implies that h is a rotation around 0 ∈ S. Finally, using property (c) we conclude that h is the identity, and hence Π O = s.

Observe that it is easy to check that Π O = s satisfies (b), (c), (d) at x = O and also (a) for all isometries f ∈ Isom 0 (E) preserving O.

We now prove existence and uniqueness of Π, as well as the announced expression (3.1) for Π.

We consider a point y = (y 1 , y 2 , y 3 ) ∈ E and an isometry f ∈ Isom 0 (E) such that

f (y) = O. To f is associated a matrix M = α β β ᾱ ∈ SU 1,1 such that β = -αcw where w = y 1 + iy 2 . Let Z = Z 1 V 1 (y) + Z 2 V 2 (y) + Z 3 V 3 (y) ∈ U y E and set df (Z) = Y 1 V 1 (0) + Y 2 V 2 (0) + Y 3 V 3 (0). Since isometries in Isom 0 (E) preserve the field ξ, we have df (Z), ξ(O) = Z, ξ(y) , i.e., Y 3 = X 3 . On the other hand, since ϕ M is holomorphic, using complex notation in H 2 (κ) we have dϕ M = ϕ ′ M (dx 1 + idx 2 )
. Consequently, since Λ(0) = 1 and dπ(df (Z)) = dϕ M (dπ(Z)), we get

Y 1 + iY 2 = ϕ ′ M (w) Λ(w) (Z 1 + iZ 2 ) = α ᾱ (Z 1 + iZ 2 ). From this we deduce that Π O (df (Z)) = Y 1 + iY 2 1 + Y 3 = α ᾱ Z 1 + iZ 2 1 + Z 3 .
Then by property (a) we have

Π y (Z) = ψ -1 M (Π O (df (Z))) = ᾱΠ O (df (Z)) -β -βΠ O (df (Z)) + α = Z 1 + iZ 2 + cw(1 + Z 3 ) c w(Z 1 + iZ 2 ) + 1 + Z 3 .
Let us observe that this expression does not depend on the choice of the isometry f ; this is because Π O satisfies (a) for all isometries in Isom 0 (E) preserving O. This gives the uniqueness of Π as well as the announced expression, from which we also deduce analyticity. Conversely, by construction this map satisfies (a), (b), (c) and (d).

Definition 3.2. Let Σ be an oriented surface in E. Then the Gauss map of Σ is the map Π • N : Σ → S where N is the unit normal vector to Σ. Remark 3.3. Though we have used a specific model for E to prove Theorem 3.1, its statement about existence and uniqueness of Π : UE → S satisfying conditions (a)-(d) does not involve a model for E but only needs the choice of an isomorphism between the group of orientation-preserving isometries of H 2 (κ) and the group of orientation-preserving conformal diffeomorphisms of S that leave E invariant and preserve the orientation of E. In particular, Definition 3.2 is also independent of the model chosen for E.

If we consider the specific model for E we are working with in this paper, we get the following expression for the Gauss map.

Corollary 3.4. If N = N 1 V 1 + N 2 V 2 + N 3 V 3 , then the Gauss map of Σ is (3.2) g = N 1 + iN 2 + cζ(1 + N 3 ) c ζ(N 1 + iN 2 ) + 1 + N 3 with ζ = x 1 + ix 2 .
Remark 3.5. In the case of H 2 × R, the hyperbolic Gauss map defined in [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] for CMC 1/2 graphs can be written as Π • N with Π satisfying conditions (a)-(d) in Theorem 3.1, so it coincides with g.

Remark 3.6. The formula given in Corollary 3.4 extends for κ = 0, and in this case we get the Gauss map studied in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF]. Hence formula (3.2) provides a unified treatment for Gauss maps in all manifolds E 3 (κ, τ ) with κ 0. Remark 3.9. Let r : E → E be the map defined by r(x 1 , x 2 , x 3 ) = (x 1 , -x 2 , -x 3 ). This map is an isometry of E (but does not lie in Isom 0 (E)); it is the rotation of angle π around the x 1 -axis, which is a horizontal geodesic. Let x ∈ E and

X ∈ U x E. Then Π r(x) (dr(X)) = 1 Π x (X)
.

Indeed, this follows from (3.1) with (ζ, X 1 , X 2 , X 3 ) replaced by ( ζ, X 1 , -X 2 , -X 3 ).
Remark 3.10. The homogeneous manifold PSL 2 (R) is isometric to the universal cover of UH 2 (κ) endowed with a Sasaki metric (see [START_REF] Daniel | Isometric immersions into 3-dimensional homogeneous manifolds[END_REF] for details). Via this identification, the Riemannian fibration π is the canonical projection onto H 2 (κ), and isometries of PSL 2 (R) coming from the Lie group structure (i.e., left-multiplications) are of the form (f, df ) where f is a direct isometry of H 2 (κ). For example, among all screw motions around the x 3 -axis in PSL 2 (R), those coming from the Lie group structure are exactly those whose pitch has a particular value that depends on κ and τ . Vertical translations do not come from the Lie group structure.

Preliminary calculations

This section is devoted to preliminary equations for conformal immersions in E = E 3 (κ, τ ) where κ 0 and their Gauss map. 4.1. Expression of the connection of E. First we compute the connection ∇ of

E in the frame (V 1 , V 2 , V 3 ). We have [V 1 , V 2 ] = - κ 2 x 2 V 1 + κ 2 x 1 V 2 + 2τ V 3 , [V 1 , V 3 ] = [V 2 , V 3 ] = 0,
and so

(4.1) ∇V1 V 1 = κ 2 x 2 V 2 , ∇V2 V 1 = -κ 2 x 1 V 2 -τ V 3 , ∇V3 V 1 = -τ V 2 , ∇V1 V 2 = -κ 2 x 2 V 1 + τ V 3 , ∇V2 V 2 = κ 2 x 1 V 1 , ∇V3 V 2 = τ V 1 , ∇V1 V 3 = -τ V 2 , ∇V2 V 3 = τ V 1 , ∇V3 V 3 = 0.
4.2. Conformal immersions. Let X = (x 1 , x 2 , x 3 ) : Σ → E be a conformal immersion from a Riemann surface Σ into E. We shall denote by N : Σ → UE its unit normal. If we fix a conformal coordinate z = u + iv in Σ, then we have

X z , X z = λ 2 > 0, X z , X z = 0,
where λ is the conformal factor of the metric with respecto to z. Moreover, we will denote the coordinates of X z and N in the frame (V 1 , V 2 , V 3 ) by

X z =   A 1 A 2 A 3   , N =   N 1 N 2 N 3   .
The function N 3 is the angle function (see Section 2). The usual Hopf differential of X, i.e., the (2, 0) part of its complexified second fundamental form, is defined as

P dz 2 = N, ∇Xz X z dz 2 .
From the definitions, we have the basic algebraic relations (4.2)

             |A 1 | 2 + |A 2 | 2 + |A 3 | 2 = λ 2 , A 2 1 + A 2 2 + A 2 3 = 0, N 2 1 + N 2 2 + N 2 3 = 1, A 1 N 1 + A 2 N 2 + A 3 N 3 = 0.
A classical computation proves that the Gauss-Weingarten equations of the immersion read as

             ∇Xz X z = λ z λ X z + P N, ∇Xz X z = λH 2 N, ∇Xz X z = λ z λ X z + P N,      ∇Xz N = -HX z - 2P λ X z , ∇Xz N = - 2 P λ X z -HX z ,
where H is the mean curvature function of X.

Using (4.1) in these equations we get

(4.3)              A 1z = λH 2 N 1 + κ 2 x 2 Ā1 A 2 - κ 2 x 1 |A 2 | 2 -τ ( Ā2 A 3 + A 2 Ā3 ), A 2z = λH 2 N 2 + κ 2 x 1 A 1 Ā2 - κ 2 x 2 |A 1 | 2 + τ ( Ā1 A 3 + A 1 Ā3 ), A 3z = λH 2 N 3 + τ (A 1 Ā2 -Ā1 A 2 ). (4.4) N 3z = -HA 3 - 2P λ Ā3 + τ (A 2 N 1 -A 1 N 2 ).
Moreover, the fact that X z × X z = i λ 2 N implies that

N 1 = - 2i λ (A 2 Ā3 -A 3 Ā2 ), N 2 = - 2i λ (A 3 Ā1 -A 1 Ā3 ), N 3 = - 2i λ (A 1 Ā2 -A 2 Ā1 ). 4.
3. An auxiliary map. We now introduce the auxiliary map

G := N 1 + iN 2 1 + N 3 , so that (4.5) N = 1 1 + |G| 2   2 Re G 2 Im G 1 -|G| 2   .
Remark 4.1. This map G does not have a geometric meaning since it depends on the choice of the frame (V 1 , V 2 , V 3 ); however |G| has a geometric meaning, since it depends only on the angle function of the immersion. In particular, X defines a local graph with upwards pointing normal if and only if |G| < 1.

Let us set

η := 2 X z , ξ = 2A 3 . A straightforward computation from (4.2) proves that η/ Ḡ extends smoothly at points where G = 0 and that (4.6)

A 1 = - 1 -Ḡ2 4 Ḡ η, A 2 = i 1 + Ḡ2 4 Ḡ η, A 3 = η 2 and thereby (4.7) λ = (1 + |G| 2 ) 2 |η| 2 4|G| 2 .
Then,

denoting ζ = x 1 + ix 2 : Σ → C, (4.3) becomes (4.8) ( Ḡ2 + 1) η Ḡz 4 Ḡ2 + 1 4 Ḡ - 1 Ḡ η z = (1 + |G| 2 ) |η| 2 4|G| 2 H Re(G) -κ |η| 2 32|G| 2 (1 + Ḡ2 )(ζ + G 2 ζ) + iτ |η| 2 8 1 G - 1 Ḡ + G -Ḡ , (4.9) i( Ḡ2 -1) η Ḡz 4 Ḡ2 + i 4 Ḡ + 1 Ḡ η z = (1 + |G| 2 ) |η| 2 4|G| 2 H Im(G) + iκ |η| 2 32|G| 2 (1 -Ḡ2 )(ζ + G 2 ζ) + τ |η| 2 8 - 1 G - 1 Ḡ + G + Ḡ , (4.10) 1 2 η z = (H + iτ ) |η| 2 8|G| 2 (1 -|G| 4 ).
Reporting (4.10) into (4.8) + i (4.9) gives

G Ḡz 2η = H 8 (1 + |G| 2 ) 2 + iτ 8 (1 -|G| 2 ) 2 - κ 16 Ḡ(ζ + G 2 ζ), i.e. (4.11) η = 4 ḠG z U (G, ζ)
where

U (G, ζ) = H(1 + |G| 2 ) 2 -iτ (1 -|G| 2 ) 2 + 2c 2 G( ζ + Ḡ2 ζ).

4.4.

Preliminary formulas for the Gauss map. Using (4.6) and (2.1) we first have (4.12)

ζ z = A 1 + iA 2 Λ(ζ) = -(1 -c 2 |ζ| 2 ) η 2 Ḡ , (4.13
) ζz = A 1 -iA 2 Λ(ζ) = (1 -c 2 |ζ| 2 ) Ḡη 2 = -Ḡ2 ζ z .
By Corollary 3.4, the Gauss map g and the auxiliary map G are related by

(4.14) g = G + cζ c ζG + 1 , G = g -cζ -c ζg + 1
, from where we also get

(4.15) 1 -|G| 2 = (1 -c 2 |ζ| 2 )(1 -|g| 2 ) (1 -c ζg)(1 -cζ ḡ) . (4.16) ζ + Ḡ2 ζ = ( ζ + ḡ2 ζ)(1 + c 2 |ζ| 2 ) -4c|ζ| 2 ḡ (1 -cζ ḡ) 2 .
4.5. Preliminary formulas for the derivatives of the Gauss map. Differentiating the left equation in (4.14) we get

g z g = G z + cζ z G + cζ -c ζz G + ζG z c ζG + 1 = 1 -c 2 |ζ| 2 (G + cζ)(c ζG + 1) G z + c ζ z G + cζ -c G ζz c ζG + 1 .
Using (4.12), (4.13) and (4.11) we obtain

g z g = 1 -c 2 |ζ| 2 (G + cζ)(c ζG + 1) ηU (G, ζ) 4 Ḡ -c(1 -c 2 |ζ| 2 ) η 2 Ḡ(G + cζ) + |G| 2 η 2(c ζG + 1) = (1 -c 2 |ζ| 2 )η 4 Ḡ(G + cζ)(c ζG + 1) (U (G, ζ) -2c(c ζG + 1) -2cG Ḡ2 (G + cζ)) = (1 -c 2 |ζ| 2 )η 4 Ḡ(G + cζ)(c ζG + 1) V (G) with (4.17) V (G) = (H -c)(1 + |G| 2 ) 2 -(c + iτ )(1 -|G| 2 ) 2 .
On the other hand, differentiating the right equation in (4.14) and using (4.12) and (4.13) we get

(4.18) Ḡz Ḡ = 1 -c 2 |ζ| 2 (ḡ -c ζ)(1 -cζ ḡ) ḡz -cηḡ + c 2 η 2 ( ζ + ḡ2 ζ) .

CMC surfaces with critical mean curvature

We recall that there exist compact CMC H surfaces in E if and only if |H| > c. Indeed, there exist CMC H spheres if |H| > c (see for instance [START_REF] Abresch | A Hopf differential for constant mean curvature surfaces in S 2 × R and H 2 × R[END_REF][START_REF] Torralbo | Rotationally invariant constant mean curvature surfaces in homogeneous 3manifolds[END_REF][START_REF] Figueroa | Invariant surfaces of the Heisenberg groups[END_REF]), and since horocylinders (vertical planes when κ = 0) have mean curvature c, by the maximum principle there cannot exist compact CMC H surfaces if |H| c. CMC surfaces with mean curvature c are called CMC surfaces with critical mean curvature.

In this section we focus on CMC surfaces with critical mean curvature, and we assume that they are oriented by their mean curvature vector when κ < 0, i.e., we set H = c. The results of these section generalize those of [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF] (which correspond to the case where κ = 0 and where τ has been normalized to 1/2).

Harmonicity of the Gauss map.

Theorem 5.1. Let X = (x 1 , x 2 , x 3 ) : Σ → E be a CMC immersion with critical mean curvature. Assume that X is nowhere vertical and with upwards pointing unit normal vector. Let g : Σ → D be its Gauss map (see Remark 3.8). Then, g is nowhere antiholomorphic, i.e., g z = 0 at every point for any local conformal parameter z on Σ, and g verifies the second order elliptic equation

(5.1) (1 -|g| 2 )g z z + 2ḡg z ḡz = 0.
In other words, g is harmonic in D endowed with the hyperbolic metric (of curvature -1) 4|dw| 2 /(1 -|w| 2 ) 2 . Moreover, the immersion X = (x 1 , x 2 , x 3 ) : Σ → E can be recovered in terms of the Gauss map g by means of the representation formula

(5.2)                ζ z = 2 c + iτ (1 -cζ ḡ) 2 (1 -|g| 2 ) 2 g z , ζ z = - 2 c -iτ (g -cζ) 2 (1 -|g| 2 ) 2 ḡz , (x 3 ) z = - 2 c + iτ (ḡ -c ζ)(1 -cζ ḡ) (1 -c 2 |ζ| 2 )(1 -|g| 2 ) 2 g z + iτ 2 ζ ζz -ζζ z 1 -c 2 |ζ| 2 , where ζ = x 1 + ix 2 .
Conversely, assume that a map g : Σ → D from a simply connected Riemann surface Σ verifies (5.1) and is nowhere antiholomorphic. Then the map X : Σ → E given by the representation formula (5.2) is a conformal CMC immersion with critical mean curvature whose Gauss map is g.

Proof. Since H = c, V (G) defined in (4.17) simplifies and we obtain, using (4.15) and the second equation in (4.14), (5.3)

g z = - c + iτ 4 (1 -c 2 |ζ| 2 )(1 -|g| 2 ) 2 (ḡ -c ζ)(1 -cζ ḡ) η.
Differentiating (5.3), we get

g z z g z = η z η -c 2 ζζ z + ζ ζz 1 -c 2 |ζ| 2 -2 ḡg z + gḡ z 1 -|g| 2 - ḡz -c ζz ḡ -c ζ + c ḡζ z + ζ ḡz 1 -cζ ḡ = (c + iτ ) 1 -|G| 4 4|G| 2 η + c 2 (ζ -ζG 2 ) η 2G -2 ḡg z 1 -|g| 2 -2 gḡ z 1 -|g| 2 -c 1 -c 2 |ζ| 2 ḡ -c ζ η 2G + c (1 -c 2 |ζ| 2 )ḡ 1 -cζ ḡ Gη 2 + -1 -c 2 |ζ| 2 + 2cζ ḡ (ḡ -c ζ)(1 -cζ ḡ)
ḡz .

We compute

B 1 := (c + iτ ) 1 -|G| 4 4|G| 2 η = (c + iτ )(1 -c 2 |ζ| 2 )(1 -|g| 2 ) (1 -c ζg)(1 -cζ ḡ) + (g -cζ)(ḡ -c ζ) 4(g -cζ)(ḡ -c ζ)(1 -c ζg)(1 -cζ ḡ) η, B 2 := c 2 (ζ -ζG 2 ) η 2G = c 2 (ζ -ζg 2 )(1 -c 2 |ζ| 2 ) 2(g -cζ)(1 -c ζg) η, B 3 := -c 1 -c 2 |ζ| 2 ḡ -c ζ η 2G + c (1 -c 2 |ζ| 2 )ḡ 1 -cζ ḡ Gη 2 = c(1 -c 2 |ζ| 2 ) cḡ(ζ -ζg 2 )(1 -c 2 |ζ| 2 ) + c 2 (ζ 2 ḡ2 -ζ2 g 2 ) + 2cg( ζ -ζ ḡ2 ) + |g| 4 -1 2(g -cζ)(ḡ -c ζ)(1 -c ζg)(1 -cζ ḡ) η,
and so

B 1 +B 2 +B 3 = -(c-iτ )(1-c 2 |ζ| 2 )(1-|g| 2 ) (1 -c ζg)(1 -cζ ḡ) + (g -cζ)(ḡ -c ζ) 4(g -cζ)(ḡ -c ζ)(1 -c ζg)(1 -cζ ḡ) η.
Moreover, we have

-2 gḡ z 1 -|g| 2 + -1 -c 2 |ζ| 2 + 2cζ ḡ (ḡ -c ζ)(1 -cζ ḡ) ḡz = - (1 -c ζg)(1 -cζ ḡ) + (g -cζ)(ḡ -c ζ) (1 -|g| 2 )(ḡ -c ζ)(1 -cζ ḡ) ḡz ,
and we conclude from (5.3) that

g z z g z = -2 ḡg z 1 -|g| 2 ,
which means that g : Σ → D is harmonic for the hyperbolic metric.

Combining (4.12), (4.13), (4.14) and ( 5.3) gives the first order differential system satisfied by ζ in terms of g and g z :

(5.4)

       ζ z = 2 c + iτ (1 -cζ ḡ) 2 (1 -|g| 2 ) 2 g z , ζ z = - 2 c -iτ (g -cζ) 2 (1 -|g| 2 ) 2 ḡz .
Observe that (5.3) gives η in terms of g, g z and ζ:

(5.5) η = - 4 c + iτ (ḡ -c ζ)(1 -cζ ḡ) (1 -c 2 |ζ| 2 )(1 -|g| 2 ) 2 g z . Finally, since x 3z = A 3 + τ (x 1 A 2 -x 2 A 1 ) = A 3 + τ Λ(x 1 x 2z -x 2 x 1z ), we have (5.6) (x 3 ) z = η 2 + iτ 2 ζ ζz -ζζ z 1 -c 2 |ζ| 2 .
Moreover, if g z = 0 at some point, then these formulas show that ζ z = 0, ζ z = 0 and (x 3 ) z = 0 at this point, contradicting the fact that X is an immersion. Consequently, g is nowhere antiholomorphic. We observe that the conformal factor of X is given in terms of g as

λ = 4 |1 -c ζg| 2 + |g -cζ| 2 2 (c 2 + τ 2 )(1 -c 2 |ζ| 2 ) 2 (1 -|g| 2 ) 4 .
Conversely, assume that a map g : Σ → D from a simply connected Riemann surface Σ verifies (5.1) and is nowhere antiholomorphic. A long but straightforward computation shows that if z denotes a complex parameter on Σ, then

2 c + iτ (1 -cζ ḡ) 2 (1 -|g| 2 ) 2 g z z = - 2 c -iτ (g -cζ) 2 (1 -|g| 2 ) 2 ḡz z .
By the Frobenius theorem, given

p 1 + ip 2 ∈ D(1/c) and z 0 ∈ Σ, there is a unique solution ζ : Σ → C to (5.4) such that ζ(z 0 ) = p 1 + ip 2 . Let us check now that |ζ| < 1/c on Σ. Let φ := 1 -c 2 |ζ| 2 . Then, from (5.4) we get (5.7)
φ z = -2c 2 ( ζ -ḡ2 ζ) (c + iτ )(1 -|g| 2 ) 2 g z φ.
Assume that |ζ| < 1/c does not hold in Σ (observe that |ζ(z 0 )| < 1/c). Then there exists a regular path γ(t) : [0, 1] → Σ such that (φ • γ)(0) > 0, (φ • γ)(1) = 0 and (φ • γ) ′ (t) < 0 for all t ∈ [0, 1). As γ[0, 1] is compact, we get from (5.7) that

- (φ • γ) ′ (t) (φ • γ)(t) C
for every t ∈ [0, 1) and some constant

C > 0. Denote f (t) = log((φ•γ)(t))-log((φ• γ)(0)). Then f (0) = 0 and f ′ (t) -C for every t ∈ [0, 1). Thus f (t) -Ct, which contradicts that lim t→1 -f (t) = -∞. So, |ζ| < 1/c on Σ.
Once here, a computation shows that, if we write the third equation in (5.2) as (x 3 ) z =: A, then A z = Āz . Again by the Frobenius theorem there exists x 3 : Σ → R such that (x 3 ) z = A, unique once we fix an initial condition x 3 (z 0 ) = p 3 .

Therefore, we get a unique map

X = (x 1 , x 2 , x 3 ) : Σ → E with X(z 0 ) = (p 1 , p 2 , p 3 ) that verifies system (5.2), where ζ = x 1 + ix 2 .
To complete the proof of the converse, it remains to check that X is a conformal immersion with critical mean curvature c and Gauss map g. This is a long but standard computation. The idea is, starting with (5.2), to show that if

X z = A 1 V 1 + A 2 V 2 + A 3 V 3 , then A 1 , A 2 are
given by (4.12), (4.13) and A 3 =: 2η is given by (5.5). From here we can check that X is a conformal immersion with Gauss map g, and a computation using (4.17) gives that X has critical constant mean curvature c. We omit the details. Remark 5.2. If X : Σ → E is a CMC immersion with critical mean curvature possibly with vertical points, then its Gauss map g : Σ → C still satisfies (5.1), and it is nowhere antiholomorphic at points where |g| = 1. Indeed, the whole proof can be extended to this case.

However, conversely, if Σ is simply connected and g : Σ → C satisfies (5.1) without assuming that |g| < 1, then the representation formula (5.2) only locally defines a possibly branched immersion away from points where |g| = 1 (see Remark 4.2 in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF]).

Remark 5.3. When κ < 0, if X is a nowhere vertical CMC immersion with critical mean curvature, it may not be possible to orient it so that its unit normal is upwards pointing, since the orientation is chosen so that the mean curvature is positive. However it is not a loss of generality to assume that the unit normal is upwards pointing. Indeed, assume that X has a downwards pointing unit normal and let r the rotation of angle π around the x 1 -axis (see Remark 3.9). This map r is an isometry of E, and hence r • X is a nowhere vertical CMC immersion with critical mean curvature c and upwards pointing normal. Then Theorem 5.1 applies to r • X. Also, if g denotes the Gauss map of X, then the Gauss map of r • X is 1/g. Remark 5.4. Theorem 5.1 implies that, when κ < 0 (i.e., in H 2 ×R and PSL 2 (R)), there generically exists a two-parameter family of non-congruent CMC immersions with critical mean curvature sharing the same Gauss map. Indeed, in Theorem 5.1, for a given z 0 ∈ Σ one can prescribe ζ(z 0 ) and x 3 (z 0 ); changing x 3 (z 0 ) simply corresponds to a vertical translation, but changing ζ(z 0 ) gives a new non-congruent immersion unless the Gauss map has some symmetry, by property (a) in Theorem 3.1. In H 2 × R this was noticed in Remark 5 in [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF]. When κ = 0 (i.e., in Nil 3 ) this is not the case anymore (see Section 6 in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF]). 5.2. The Hopf differential. One can associate two natural holomorphic quadratic differentials to the immersion X:

• the Hopf differential Q(g)dz 2 of g (in the sense of harmonic maps into D endowed with the hyperbolic metric):

Q(g) := 4g z ḡz (1 -|g| 2 ) 2 ,
• the Abresch-Rosenberg differential Φdz 2 of X [1, 2] (see also [START_REF] Fernández | A characterization of constant mean curvature surfaces in homogeneous 3-manifolds[END_REF] for its expression):

Φ := 2(H + iτ )P -(κ -4τ 2 ) η 2 4 = 2(c + iτ )P + (c 2 + τ 2 )η 2 .
Proposition 5.5. We have

Q(g) = -Φ.
Proof. By (5.3) we have

Q(g) = -(c + iτ ) (1 -c 2 |ζ| 2 ) (ḡ -c ζ)(1 -cζ ḡ) ηḡ z .
We now compute Φ. First we compute P , starting from the identity (4.4) and using (4.5), (4.6), (4.7) and (4.11):

P = λ η 2 ḠG z + G Ḡz (1 + |G| 2 ) 2 -(H -iτ ) η 2 = η 2|G| 2 U (G, ζ)η 4 + G Ḡz -(H -iτ )(1 + |G| 2 ) 2 η 2 8|G| 2 = iτ 2 η 2 + c 2 η 2 4 Ḡ ( ζ + Ḡ2 ζ) + Ḡz η 2 Ḡ .
Then, using (4.18) and (4.16), we get

P = iτ 2 η 2 + c 2 η 2 2 ζ + ḡ2 ζ (ḡ -c ζ)(1 -cζ ḡ) - cḡη 2 2 1 + c 2 |ζ| 2 (ḡ -c ζ)(1 -cζ ḡ) + η 2 1 -c 2 |ζ| 2 (ḡ -c ζ)(1 -cζ ḡ) ḡz = - c -iτ 2 η 2 + η 2 1 -c 2 |ζ| 2 (ḡ -c ζ)(1 -cζ ḡ)
ḡz .

Finally we get Φ = -Q(g).

5.3.

The Gauss map of the sister minimal surface in Nil 3 (τ ). Let X : Σ → E 3 (κ, τ ) be a CMC immersion with critical mean curvature. Assume that Σ is simply connected. Then, by the Lawson-type correspondence in [START_REF] Daniel | Isometric immersions into 3-dimensional homogeneous manifolds[END_REF], there exists a minimal immersion X : Σ → Nil 3 (τ ) = E 3 (0, τ ) where τ 2 = τ 2 + c 2 that is isometric to X. Moreover, the immersions X and X have the same angle function (see Section 2) and their associated Abresch-Rosenberg differentials Φ and Φ are related by Φ = e -2iθ Φ, where τ + ic = e iθ τ (see for instance the end of Section 3 in [START_REF] Gálvez | The Bonnet problem for surfaces in homogeneous 3-manifolds[END_REF] and the expression of the Abresch-Rosenberg differential given in Section 5.2).

A pair of such isometric surfaces are called sister surfaces. The minimal sister surface in Nil 3 (τ ) of a CMC surface in E 3 (κ, τ ) with critical mean curvature is unique up to direct isometries of Nil 3 (τ ).

If g : Σ → D is a harmonic map for the hyperbolic metric on D, we consider the form µ(g)|dz| 2 (which is independent of the choice of the conformal parameter z) where

µ(g) := 4(|g z | 2 + |g z | 2 ) (1 -|g| 2 ) 2 .
Two harmonic maps g 1 and g 2 are said to be associate if there exists a real constant θ such that Q(g 2 )dz 2 = e -2iθ Q(g 1 )dz 2 and µ(g 2 )|dz| 2 = µ(g 1 )|dz| 2 .

We now consider a conformal immersion X : Σ → E 3 (κ, τ ) from a simply connected Riemann surface Σ into E 3 (κ, τ ) such that X(Σ) is a local graph with positive angle function. We let X : Σ → Nil 3 (τ ) be its sister minimal immersion. Let g be the Gauss map of X and ĝ the Gauss map of X (which is unique only up to rotations around 0 ∈ D, see Section 6 in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF]).

Proposition 5.6. The Gauss maps g and ĝ are associate and Q(g) = e -2iθ Q(ĝ) with θ defined by (5.8) τ + ic = e iθ τ .

Proof. We keep for the immersion X the same notation as in the previous sections. By Proposition 5.5 we have Q(g) = -Φ, and similarly we have Q(ĝ) = -Φ where Φ is the Abresch-Rosenberg differential of X. Since the differentials Φ and Φ satisfy Φ = e -2iθ Φ, we get Q(g) = e -2iθ Q(ĝ). in Nil 3 . Consequently, by Theorem 3.1 in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF], X(Σ) is an entire graph in Nil 3 . By Corollary 13 in [START_REF] Fernández | Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space[END_REF] (which uses a result from [START_REF] Cheng | Maximal space-like hypersurfaces in the Lorentz-Minkowski spaces[END_REF]), the metric ν 2 ds 2 is complete, and so by Lemma 9 in [START_REF] Fernández | Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space[END_REF] the surface X(Σ) is an entire graph in E 3 (κ, τ ). This shows that (2) implies (1) and completes the proof.

Example 5.9 (Surface with conformal Gauss map). We seek a local graph with critical CMC whose Gauss map g : Σ → D is conformal, i.e., such that the Hopf differential of g vanishes identically. Then ĝ is also conformal, so the surface will be the sister surface of that of Example 8.1 in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF]. Up to a reparametrization we may assume that Σ = D and that ĝ(z) = z. Then up to an isometry of E 3 (κ, τ ) we can assume that g(z) = z.

Then by (5.12) we get

ζ = e 2iθ -1 c z e 2iθ |z| 2 -1 .
Using (5.10), (5.11) and (5.8) we get

η = 4i τ + ic z (1 -|z| 2 ) 2 . Next, we compute that ζ z = 1 -e 2iθ c(e 2iθ |z| 2 -1) 2 , ζz = (1 -e -2iθ )e -2iθ c z2 (e -2iθ |z| 2 -1) 2 , 1 -c 2 |ζ| 2 = (1 -|z| 2 ) 2 (e 2iθ |z| 2 -1)(e -2iθ |z| 2 -1)
.

Then we compute using (5.6) that

(x 3 ) z = 2i τ + ic - iτ (1 -e 2iθ )(1 -e -2iθ ) 2c 2 |z| 4 -2|z| 2 e -2iθ + 1 (e 2iθ |z| 2 -1)(e -2iθ |z| 2 -1) z (1 -|z| 2 ) 2 .
By (5.8) we have (1 -e 2iθ )(1 -e -2iθ ) = 4c 2 /(τ 2 + c 2 ), so we get

(x 3 ) z = 2 τ 2 + c 2 c(|z| 4 + 1) + 2(τ sin(2θ) -c cos(2θ))|z| 2 (1 -|z| 2 ) 2 (e 2iθ |z| 2 -1)(e -2iθ |z| 2 -1) z = 2c τ 2 + c 2 (1 + |z| 2 ) 2 z (1 -|z| 2 ) 2 (|z| 4 -2|z| 2 cos(2θ) + 1)
, and finally, up to a vertical translation,

x 3 = - τ c 2 arctan |z| 2 -cos(2θ) sin(2θ) + 2 c(1 -|z| 2 )
.

Hence we obtain a surface of revolution about the x 3 -axis; this surface is also an entire graph and x 3 → +∞ as |ζ| → 1/c. This surface is described in [START_REF] Torralbo | Rotationally invariant constant mean curvature surfaces in homogeneous 3manifolds[END_REF] (item 2 of Theorem 2 with 4H 2 + κ = 0 and E = 0). In H 2 (κ) × R (i.e., when τ = 0), this is the surface described on page 1172 in [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] (see also [START_REF] Abresch | A Hopf differential for constant mean curvature surfaces in S 2 × R and H 2 × R[END_REF]).

Example 5.10 (Surfaces with singular Gauss map). We seek complete local graphs with critical CMC whose Gauss map g : Σ → D is singular on an open set of Σ. Then g(Σ) and ĝ(Σ) are geodesics of D for the hyperbolic metric, and these surfaces will be sister surfaces of those of Example 8.2 in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF]. These minimal surfaces in Nil 3 constitute a one-parameter family of surfaces that are not isometric one to another, the parameter being the hyperbolic distance between ĝ(Σ) and the origin 0 ∈ D. Since sister surfaces are isometric, we get a one-parameter family of CMC surfaces in E 3 (κ, τ ) that are not isometric one to another. Since their sister minimal surfaces in Nil 3 are invariant by a one-parameter family of translations, a standard argument using the integrability equations for surfaces in E 3 (κ, τ ) [START_REF] Daniel | Isometric immersions into 3-dimensional homogeneous manifolds[END_REF] shows that these surfaces are invariant by a one-parameter family of isometries of E 3 (κ, τ ) (see Proposition 4.3 in [19]). Moreover, since their sister minimal surfaces in Nil 3 are entire graphs, it follows from Proposition 5.8 that these surfaces are also entire graphs. In H 2 (κ)×R (i.e., when τ = 0), these are the surfaces described in Proposition 18 in [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] (see also [START_REF] Earp | Parabolic and hyperbolic screw motion surfaces in H 2 × R[END_REF]).

The Gauss map in the Lorentzian model

In this section we assume that κ < 0 and we will show an alternative expression for the Gauss map of nowhere vertical surfaces when we consider the Lorentzian model for the hyperbolic space, that is, we now consider π : E → H 2 (κ), where H 2 (κ) = {p = (p 0 , p 1 , p 2 ) ∈ L 3 | p, p = 1/κ, p 0 > 0} ⊂ L 3 .

Here p 0 denotes the timelike coordinate in Lorentz space L 3 and p, p = -p 2 0 + p 2 1 + p 2 2 . This construction is inspired by the one given in [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] for surfaces in H 2 × R. In particular, it agrees with that one when τ = 0. Theorem 6.1. Let X : Σ → E be a nowhere vertical surface oriented so that its unit normal vector field N points upwards. Define X * = π • X : Σ → H 2 (κ) ⊂ L 3 , N * = dπ(N ) : Σ → L 3 and ν = N, ξ .

Then the map g : Σ → H 2 = H 2 (-1) given by g = 1 ν (2cX * + N * ) .

agrees with the Gauss map of X as defined in Corollary 3.4.

Proof. We first observe that g actually takes values in H 2 . Indeed, since N * takes values in T X * (•) H 2 (κ) = X * (•) ⊥ ⊂ L 3 , we have N * , X * = 0. Also, since π is a Riemannian submersion, we have N, N = N * , N * + N, ξ 2 , from where we get N * , N * = 1-ν 2 , and so g, g = -1. Finally, since N points upwards, ν is positive, and therefore g, X * = -1/(2cν) < 0, implying that the timelike coordinate of g is positive.

Let F : D → H 2 be the natural isometry given by Let us prove that Π = F • Π, where Π is the map given in Theorem 3.1. Since Π satisfies hypothesis (a) in abovementioned theorem, following the ideas of the proof of this result, it suffices to check that both maps agree at the origin O = (( 1 2c , 0, 0), 0) ∈ E.

F (w) = 1
Since Z * = d O π(Z) ∈ T ( 1 2c ,0,0) H 2 = ( 1 2c , 0, 0) ⊥ ⊂ L 3 , then its timelike coordinate vanishes, and so ΠO (Z) = 1 Z 3 (1, Z 1 , Z 2 ).

On the other hand, since Π O (Z) = 1 1+Z3 (Z 1 + iZ 2 ), it is straightforward to check that ΠO (Z) = F (Π O (Z)). Remark 6.2. It is possible then to follow the computations in [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] and find an alternative Weierstrass-type representation for surfaces of critical CMC in E in terms of the Gauss map, using this model. This method uses the fact that a nowhere antiholomorphic harmonic map into H 2 is (locally) the Gauss map of a CMC 1/2 surface in L 3 [START_REF] Akutagawa | The Gauss map and spacelike surfaces with prescribed mean curvature in Minkowski 3-space[END_REF]. In this setting, one first needs to integrate a differential system to obtain η = 2 X z , ξ and then X * is obtained from η and the harmonic map without integration. This gives a representation formula equivalent to (5.2) but that differs significantly from it. Remark 6.3. The condition for being the Gauss map in H 2 × R that we obtain in the present paper (being nowhere antiholomorphic) is different from the one obtained in [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] (the existence of Weierstrass data). This difference comes from the fact that in [START_REF] Fernández | Harmonic maps and constant mean curvature surfaces in H 2 × R[END_REF] the orientation induced by the Riemann surface Σ is not assumed to coincide with the orientation induced by the unit normal for which the angle function of the surface is positive. In the present paper, we always assume that the orientation of Σ is coherent with the choice of the unit normal of the immersion.

Example 3 . 7 .

 37 Let γ be a curve of constant curvature k in H 2 (κ) and let Σ = π -1 (γ) ⊂ E. Then, by property (b) of Theorem 3.

1 -|w| 2 1 +

 11 |w| 2 , 2Re(w), 2Im(w) and Π : UE → H 2 the map so that g= Π • N , that is, for x ∈ E and Z ∈ U x E, Π(x, Z) = Πx (Z) = 1 Z 3 (2cx * + Z * ),where x * = π(x) ∈ H 2 (κ), Z * = d x π(Z) and Z 3 = Z, ξ .

The first author was partially supported by the ANR-11-IS01-0002 grant. The second author was partially supported by MCYT-FEDER research project MTM2010-19821 and Junta de Andalucía Grant P09-FQM-5088. The third author was partially supported by MICINN-FEDER, Grant No. MTM2010-19821, and the Programme in Support of Excellence Groups of Murcia, reference 04540/GERM/06.

We now prove that µ(g) = µ(ĝ). Since sister surfaces have the same angle function, it follows from (4.5) and the analogous formula in Nil 3 (τ ) (see Definition 3.1 in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF]) that (5.9)

Denote by (ĝ, η) the Weierstrass data of X (see Defintion 3.2 in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF]). By equation [START_REF] Daniel | Existence and uniqueness of constant mean curvature spheres in Sol 3[END_REF] in [START_REF] Daniel | The Gauss map of minimal surfaces in the Heisenberg group[END_REF] (or by (4.11) with (η, H, τ, c) replaced by (η, 0, τ , 0)) we have

On the other hand we have (see for instance the end of Section 3 in [START_REF] Gálvez | The Bonnet problem for surfaces in homogeneous 3-manifolds[END_REF]) (5.11) η = e -iθ η.

So using (5.3) we get

and so, using (5.8), (5.9) and (4.15), (5.12)

which implies, using (5.9) and (4.14),

(5.13)

(5.13) and the fact that |Q(g)| = |Q(ĝ)| imply that (5.14)

Finally, we deduce from (5.13) and (5.14) that µ(g) = µ(ĝ). This proves that g and ĝ are associate.

Remark 5.7. Formula (5.12) gives the expression of ζ in terms of g and ĝ without the need to integrate a differential system, contrarily to the representation formula (5.2). We will use (5.12) to explicitely compute Example 5.9.

We recall the following result from [START_REF] Daniel | Constant mean curvature surfaces in homogeneous manifolds[END_REF] (see Corollaries 4.6.3 and 4.6.4 there; see also Theorem 6.10 in [START_REF] Fernández | Constant mean curvature surfaces in 3-dimensional Thurston geometries[END_REF]).

Proposition 5.8. The following conditions are equivalent for a surface X(Σ) of critical CMC in E 3 (κ, τ ):

(1) X(Σ) is an entire graph, (2) X(Σ) is a complete local graph. In particular, the sister correspondence preserves entire graphs of critical CMC (see Section 2 for the definition).

Proof. That (1) implies ( 2) is trivial. Conversely, let X(Σ) be a complete local graph in E 3 (κ, τ ), which we will assume to be simply connected by passing to its universal covering if necessary. Let ds 2 denote the metric induced by X on Σ and ν the angle function of X(Σ). Let X(Σ) denote the sister surface of X in Nil 3 . Then, ds 2 and ν coincide with the metric induced by X on Σ, and with the angle function of X(Σ), respectively. Then, in particular, X(Σ) is a complete local graph