Thibault Falque
email: falque@cril.fr

Bertrand Mazure
email: mazure@cril.fr

Karim Tabia
email: tabia@cril.fr

Predicting Off-Block Delays: A Case Study at Paris -Charles de Gaulle International Airport

Keywords: Machine Learning, Application, Real Time Delay

Punctuality is a sensitive issue in large airports and hubs for passenger experience and for controlling operational costs. This paper presents a real and challenging problem of predicting and explaining flight off-block delays. We study the case of the international airport Paris Charles de Gaulle (Paris-CDG) starting from the specificities of this problem at Paris-CDG until the proposal of modelings then solutions and the analysis of the results on real data covering an entire year of activity. The proof of concept provided in this paper allows us to believe that the proposed approach could help improving the management of delays and reduce the impact of the resulting consequences.

INTRODUCTION

In the context of an airport, there are several typical problems of artificial intelligence, such as planning, optimization, simulation, and prediction. Indeed, many air transport problems exist where decision support systems are used while integrating artificial intelligence components.

Before the COVID-19 health crisis, the International Air Transport Association (IATA) forecasts showed that passengers would double by 2036, reaching 7.8 billion. The COVID-19 pandemic has slowed air traffic considerably, especially in 2020 and early 2021, but competitive pressure is always present, even in reduced activity. In recent months, air traffic has picked up in several world regions. In France, Paris-Charles de Gaule airport (Paris-CDG) Air France's main hub will see its passenger numbers increase by 35 to 40 million to reach 100 million by 2036. This will require around 400 additional aircraft movements (take-offs and landings) per day. The development project for Paris-CDG airport does not include any extension or new runway. It is, therefore, essential to improve the following:

• passenger flows by offering a simplified, fluid, and personalized route;

• the punctuality of flights by anticipating delays as far as possible;

• aircraft movements with optimized and adaptive planning of resources such as aircraft parking lots, check-in counters, baggage chutes, etc.

The problem of flight delays, for example (landing and take-off delays), does not only have immediate financial consequences. These delays can also cause a chain effect, other delays, and problems that affect the delays and rankings of airlines and airports.

In [START_REF] Xu | Multifactor Model for Predicting Delays at U.S. Airports[END_REF], the authors point out that about 84% of delays are generated by the airports. The problem of delay prediction is studied in a few works, but it mainly concerns landing or taxiing delays. The delay at departure from the parking lot (the delay between the scheduled time and the actual time at which an aircraft leaves its parking position or gate) depends on several factors. Many of these are specific to each airport (such as the amount of traffic and resources, passenger processes, weather conditions, air traffic control, Etc.). At Paris-CDG, there is a takeoff almost every minute, and the slightest delay can have a cascading effect on the takeoff schedule, which can take several hours to be resolved and return to a normal situation. Predicting and forecasting delays in real-time will allow us to anticipate management by providing delay management plans and adjustments to established schedules such as gateway, check-in, and baggage chutes.

One of the specificities of our work is that we focused on very fluctuating data due to the COVID-19 pandemic. The main contributions of the paper are the study of the existing situation at Paris-CDG, the needs, and then the identification of three main tasks real-time prediction of parking delays, forecasting, and explicability of predictions. Concerning the modeling, we have identified five categories of data for our problem: flight data, data on the progress of passenger processes (security, boarding, Etc.), weather data, current delay status data, Etc. Finally, we have conducted an empirical study to perform the data, feature, and model selection, and we provide an overview of the results obtained.

FLIGHT DELAYS: STATE OF THE ART

The problem of aircraft delays is a well-known problem. Different models among random forests, support vector machines and logistic regression have been studied [START_REF] Natarajan | A Novel Approach: Airline Delay Prediction Using Machine Learning[END_REF] to predict whether a flight will be delayed. A logistic regression model to predict a class of departing flights is proposed in [START_REF] Nigam | Cloud based flight delay prediction using logistic regression[END_REF]. In [START_REF] Venkatesh | Iterative machine and deep learning approach for aviation delay prediction[END_REF], the authors study arrival delays and propose different approaches to predict whether a specific flight will be delayed. In [START_REF] Ibrahem | A Comparative Analysis of Models for Predicting Airline Arrival Delays[END_REF] the authors compared different machinelearning approaches (random forest, logistic regression, Bayesian naive classifier, and decision trees) for delay prediction on arrival. In [START_REF] Tang | Airline Flight Delay Prediction Using Machine Learning Models[END_REF], a comparison of seven binary models is performed. In [START_REF] Yi | Flight Delay Classification Prediction Based on Stacking Algorithm[END_REF], the authors have proposed several stacked approaches for the Boston Logan International Airport flight dataset from January to December 2019.

We also find different studies on the impact of different factors on flight delays. For example, the authors of [START_REF] Wang | Flight connections and their impacts on delay propagation[END_REF] studied the impact of flight connections on delay. In [START_REF] Markovic | A statistical study of the weather impact on punctual-ity at Frankfurt Airport[END_REF], a statistical study on the impact of weather at Frankfurt airport is proposed. In (Yogita [START_REF] Borse | Flight Delay Prediction System[END_REF], the authors focused on weather data as the main feature to predict the delay class. In [START_REF] Esmaeilzadeh | Machine Learning Approach for Flight Departure Delay Prediction and Analysis[END_REF], a support vector machine (SVM) model is used. Based on 20 days, this latter study examines some causes of air traffic delays at the three major airports in New York City. In the study [START_REF] Cai | A Deep Learning Approach for Flight Delay Prediction Through Time-Evolving Graphs[END_REF], a deep learning approach for flight delay prediction considering a multi-airport scenario is proposed. Regarding regression-based approaches, the authors [START_REF] Rebollo | Characterization and prediction of air traffic delays[END_REF] have proposed approaches based on classification and regression with random forests for US airports.

To our knowledge, there is only one study that attempts to predict takeoff delay, but it only attempts to predict one hour before the estimated takeoff time, and it is intended for the Maastricht [START_REF] Dalmau-Codina | Improving the predictability of take-off times with machine learning a case study for the maastricht upper area control centre area of responsibility[END_REF].

Paris-CDG AIRPORT

In this section, we provide factual information about Paris-CDG airport. Paris-CDG airport is the most important airport in France. It was opened in 1974 to cope with the saturation of Paris Orly airport (the main Parisian airport before the opening of Paris-CDG). It is located north of Paris and is the hub of Air France. This company represents 50% of the traffic at Paris-CDG. Three main terminals numbered 1 to 3 makeup Paris-CDG.

At Paris-CDG, there is more than one flight departure per minute. There are about 720, 000 flights per year, or about 2, 000 flights per day. On average, there are 145 passengers per flight. At Paris-CDG, resources are currently planned using solutions powered by constraint solvers. Among the critical resources are the parking lots (or "stands") assigned to the flights. When a parking lot is released late, this can lead to complex scheduling changes and cascading delays. It is, therefore, essential to anticipate and predict these delays as accurately as possible, explain them, and propose actions to limit them. Before presenting the problem, we will first introduce some terms. We call rotation the set composed of an arrival flight and a departure flight. This set generally consists of two flights, but there may be only one, in which case the flight begins a new rotation. The rotation period is the time between the arrival of an aircraft (landing) and its departure (takeoff). A flight has a scheduled departure time, called SOBT (Scheduled Off-Block Time), at which it is supposed to leave its parking area. The moment when the flight leaves its parking position is called AOBT. The delay is then the period between AOBT and SOBT.

Milestones Before Off-Block

We present here the main milestones preceding the pushback of an aircraft. These milestone stands are presented in Figure.

Off-Block Delays at CDG

Punctuality is a sensitive issue in large airports and hubs for passenger experience and controlling costs at the airport level. Paris-CDG is ranked 1 in 2018 in 10th place in terms of punctuality. Around half of the flights arrive on time, but only 20% take off on time.

A study of delays at Paris-CDG has highlighted different causes (eg. extreme weather conditions, congestion, breakdowns, incidents at the airport, passenger processes, etc.) of these delays at different phases (parking/pushback, taxi-ing, etc.). Figure . 2 gives an overview of off-block delays over the year considered in our study (March 2021-March 2022). It should be noted that during this period, some terminals were closed and are still closed, while some other terminals have reopened. Over this period, the proportion of flights with off-block delays is 77%. Figure. 2b shows the number of delayed flights per terminal. We can observe that the two terminals corresponding to Air France (2E and 2F) have the most delayed flights. Figure . 2c shows the number of flights and the mean of delay for each terminal. Terminal 2E has an average delay 37% above the airport average. Figure . 2e shows the cumulated delays (in minutes) each day of the considering period. We can observe an increase during summer 2022, corresponding to a resumption of traffic and some terminals' reopening.

1 According to OAG Flightview Figure 2f depicts the sum and the mean of the delays for each time band of the day, ranging from p1 (6am -8am) to p6 (8pm -11pm). We can observe that the morning periods, particularly p2 (9am -11am) and p3 (12pm -2pm), accumulate most of the delay. Since the majority of delays occur in the morning and there is a cascading delay effect, it is crucial to predict and manage these delays in these time slots accurately.

PROBLEM STATEMENT AND OBJECTIVES

This section presents the problem and the objectives of our off-block delay prediction approach at Paris-CDG. Recall that we call off-block delay the time separating the moment that the aircraft leaves the boarding gate (this operation is called pushback) from the scheduled off-block time (between Actual Off-Block Time and Scheduled Off Block Time).

Let Y be the target variable to predict for an input sample describing the flight under study. We distinguish two regression tasks:

Real-Time Off-Block Delay Prediction

The problem considered here is the one of predicting at any time t, the off-block delay Y (expressed in minutes) that this flight will indeed have. These predictions are updated every 5 minutes until the flight leaves its stand. Indeed, for every slot (each slot lasts 5 minutes), we can acquire new relevant data from the Paris-CDG operational information system. These data can be used to update the predictions (this is valid for dynamic variables such as weather conditions, the progress of passenger processes, etc.). Real-time offblock delay predictions are intended mainly to:

• Draw managers' attention in real-time to flights likely to have significant delays and which may have cascading consequences.

• Explain and identify actionable causes if necessary to resolve the situation.

Off-Block Delay Forecast

We call forecasts the prediction of delays before the opening of the flights. This may be a few hours or several days before the flight. In our case, the forecast cannot rely on dynamic information on the progress of passenger processes, weather conditions, Etc. A such forecast may serve to :

• Establish several plans and mitigation measures according to the expected delays. Then managers can activate the plan provided for each situation when it occurs.

• Identify the causes and anticipate the chaining effect and the consequences.

• Use forecasts to make plausible simulations on congestion and queues according to delay forecasts, then consider solutions and management plans.

MODEL DEFINITION

In this section, we present and motivate the essential information currently available in the Paris-CDG operational information system and which is likely to be relevant for predicting off-block delays. The proposed features come from the analysis of a recent report on takeoff delays at Paris-CDG and the analysis of an entire year of real data.

Basic Flight Features (BFF)

These are the basic characteristics of a flight, and they do not change over time (these features are listed in Table 1). For example, the name of the airline operating the flight, the type of the aircraft, the IATA code of the destination of the flight, the terminal, the type of customs (national, Schengen, or international), and the IATA season (Summer or Winter).

Off-Block Milestone Features (OMF)

These are milestones corresponding to each off-block and its management by the various stakeholders, such as air traffic control, the airline, and the airport.

For this study, we focused on milestones around the SOBT (-2h30 to +2h30, see Figure . 1). This period is split into 60 slots of 5 minutes. Each new milestone (EIBT, TOBT, CTOT, TSAT) has a timestamp. The OMF features are listed in Table . 2.

Previous and Current Flights Delay Features (PCFDF)

Relevant information on the probability of a delay for a given flight is the proportion of flights scheduled just before the flight under study and which are late. This could, for example, be due to congestion, a breakdown at the airport, or extreme weather conditions. We compute the proportion of late flights and the average duration of these delays for each flight slot. At a time t, these features are calculated during a time window w, ranging from a few minutes to a few hours. As we will show empirically, the optimal window duration is a few minutes. For example, if w =10 minutes and the current slot of the flight is 25, we compute the different values from the flights whose AOBT is between slot 23 and slot 25.

Weather Condition Features (WCF)

Certain weather conditions, such as low visibility and strong winds, are known to be factors that can cause takeoff delays and therefore delay the departure of the flight from its stand. Table 4 shows examples of weather-related faults.

Passenger Flow Features (PFF)

These features provide information at any time on the progress of specific passenger processes, which may cause an off-block delay. In particular, the relevant information is the percentage at slot t of passengers who have already passed boarding or passed security checkpoints. These features are used only to predict and update the off-block delay prediction in real-time (each flight slot).

DATA AND FEATURE EXTRACTION, PREPROCESSING AND SELECTION

In this section, we present our main findings concerning the selection of variables and the selection of data (in particular, the choice of the best parameters for the time window duration, the training data amount, Etc.).

Data Extraction and Preprocessing

Paris-CDG's operational information system (called AOP for Airport Operation Plan) collects much information about each flight and its progress. For our prediction tasks, a new flight entry is created with the associated time stamp and updated data at each time slot. It is, therefore, possible to trace the status of a flight back to its departure. Therefore, for static characteristics, we extract them only once. For dynamic characteristics, such as delays of other flights, these are calculated variables that we perform with queries on past flights. For example, to compute the proportion of flights that have been delayed in the last w minutes, we need to review all flights involved in the w time window. The delay characteristics of previous and current flights (PCFDF) are computed after extraction with different w windows. In our study, we considered one year of data (August 2021-September 2022) and constructed a dataset with 10, 633, 920 rows and 31 columns (each flight is repeated 60 times with dynamic values for each slot).

Feature Selection

Once our data set was extracted and preprocessed we proceeded to the selection of variables in order to confirm our intuitions and to eliminate attributes that would prove irrelevant to our prediction tasks. We first performed a simple correlation analysis between each characteristic and the target variable (delay to parking departure). The Figures. 3 show the results of the Pearson correlation coefficient.

It can be seen in Figure . 3 that the most relevant variables at the slot 0 are :

• the difference between the SOBT and the TSAT with a score of 0.4, • the difference between the TOBT between TOBT and SOBT with a Pearson score of 0.33. In contrast, the variables representing the rotation time (Rotation) and the arrival delay of a flight (Arrival delay) seem to have little relevance. These variables have a negative Pearson score of -0.66 and -0.29, respectively. For the slot 30, the order of the essential variables is confirmed. The durations between SOBT and TSAT or TOBT are the essential variables with a Pearson score of 0.74 and 0.68.

The importance of the TSAT variable is explained by the fact that it is one of the last pieces of information obtained for a flight before it leaves its parking lot and that departure is most often at the time indicated by the TSAT. Nevertheless, air traffic control may send this information very early before the departure of the aircraft. The variable Percent Flight Airport, representing the proportion of delayed flights airport (for the calculation of the values, we used the 10-minute time window), becomes more important with a progression of its score from 0.16 to 0.23. Finally, we can note that the dynamic variables (in particular, the variable concerning the progression of the boarding) have a score in progression. This progression shows the importance of using dynamic flight data for real-time predictions.

In order to validate the findings on the correlation scores obtained, we performed another empirical analysis by varying the set of variables used for delay prediction. We also varied the window w used to calculate the dynamic variables from the nearby history and the amount of history to use (twelve months, nine months, six months, or three months). We used an ensemble model of boosted regression trees regression trees called LightGbm. In the following, we denote by D m w the dataset with m the number of months used (m ∈ {3, 6, 9, 12}) and w the duration of the window used for the computation of the dynamic variables (w ∈ {10, 30, 60}). V denote the set of variables used for the dataset (V ∈ {{BFF}, {BFF,WCF, OMF, PFF}, {BFF, PCFDF,WCF, OMF, PFF}}).

For our study, we tested our configuration on 40 days (from August 12 to September 21, 2022). For each day, we trained the model until the day before the test day and evaluated it on the test day. Table 5 shows the optimal configurations. The errors and R2 score presented in this table are for the 40 days tested. The hyperparameters are noted as #Tree/#Leaves/LearningRate.

Table 5 presents the results for the different configurations. The use of dynamic variables dynamic variables bring a real gain, reducing the error by half and significantly improving the R2 score. Thus the best configuration uses all of the history (12 months), with a time window of 60 minutes.

REAL TIME PREDICTIONS

Table 6 presents the mean absolute error (MAE) for a subset of the days (for space reasons) tested with the best model selected in the previous section. We notice that the proposed model is more accurate and stable in terms of error than the reference model (e.g., a model predicting the average delay at CDG all the time). However, we can also notice that September 16 significantly increased errors. 2 . The delays could go up to 4h30 (the average delay over the day was 1h20). Addition, as the period covered for each flight is between -2h30 and +2h30, we only collect some of the data on these delays. Figure 4 gives an overview of the five most important features for our model 3 . The features are sorted by the sum of the SHAP values over all samples at slot 0 and slot 30. The color represents the value of the feature (red corresponds to high, blue to low). This reveals, for example, that when the slot is 30, a high 2 On Thursday, September 16, the delays are due to an air traffic controllers' strike 3 Full figures are available in appendices.

value for TSAT increases the predicted delay. Finally, the SHAP values largely confirm the coefficients of Pearson from the 6.2 section and the importance of the impact of the TSAT and TOBT variables.

CONCLUSIONS

Punctuality is a sensitive issue in major airports and hubs for the passenger experience. In this article, we have addressed the problem of predicting delays at the departure of parking lots at Paris-CDG airport, one of the largest airports in the world and the hub of the airline Air France. Our study started with analyzing the problem (its magnitude, form, causes, Etc.) and the needs (real-time forecasting and prediction) at Paris-CDG. Based on this analysis, we proposed two types of feature categories that could be useful for delay prediction: static features used for the forecasting task and dynamic features (which can be updated in real-time) used for real-time delay prediction. The next step was to build a pipeline to extract the raw data we needed from the operational information sys- tem of Paris-CDG. This allowed us to build a dataset representing one year of authentic activity. We conducted an empirical study to select the characteristics and the data and then to select the models. One of the specificities of our work is that we worked with very fluctuating data due to the COVID-19 pandemic pandemic and its consequences in terms of air travel restrictions on several occasions, as well as other air traffic hazards. The results show that some delays can be predicted much better than the baseline model. This result can be significantly improved by systematically exploring other models and their best hyperparameters. In addition to the accuracy improvement, one of the crucial elements for our application will be the explicability of the predictions, in particular, the identification of explanations that can help in delay management.

Figure 1 :

 1 Figure 1: Milestones of a flight at Paris-CDG.

  (a) number of on-time flights vs. number of late flights. (b) proportion of off-block delayed flights per terminal. (c) number of delayed flights and mean delay per terminal. Number of flights by wake category and mean of the delay by wake category.(e) Cummulative delay for each day of the considering period.(f) Cummulative and mean delay for each period of a day.

Figure 2 :

 2 Figure 2: Overview of off-block delays at Paris-CDG from August 2021 to September 2022.

Figure 3 :

 3 Figure 3: Results of statistical correlation measure at slots 0 (left side figure) and at slot 30 (right side figure).

Figure 4 :

 4 Figure 4: SHAP values for 16th September 2022 at slot 0 (top figure) and slot 30 (bottom figure).

Table 1 :

 1 Basic flight features (BFF).

	Feature	Description	Type	Example
	Airline	airline unique company code	Categorical	AF
	AircraftType	aircraft type code	Categorical	77W
	Destination	IATA code of the destination	Categorical	JFK
		airport
	Terminal	CDG Terminal	Categorical	C2E
	Customs	Customs Criteria	Categorical	Schengen
	Season	The IATA Season	Categorical	W
	Week	A week index calculated	Numeric	1000
		from a reference date.
	Day	Day number in the week (1-	Numeric	1
		7)
	Bus	True if the flight has bus ac-	Boolean	True
		cess.
	Parking	True if the arrival flight and	Boolean	False
		the departure flight have the
		same parking
	SOBT	The Scheduled off-block	Timestamp	360
		time in minutes since mid-
		night
	Rotation	duration between landing	Numeric	300
		and takeoff in minutes
	Pax Count	Number of passengers on the	Numeric	140
		flight (estimated)
	Total Pif Passenger	Number of passengers that	Numeric	75
		must pass through the secu-
		rity point (estimated)
	Service Type	Transport category	Categorical	J

Table 2 :

 2 Off-block milestone features (OMF).

	Feature	Description	Type
	Arrival Delay	Time in minutes between the last	Numeric
		EIBT and the SIBT.
	T OBT di f f	time in minutes between the last	Numeric
		TOBT and the SOBT.
	T OBT count	The number of TOBTs	Numeric
	CT OT di f f	time in minutes between the last	Numeric
		CTOT and the SOBT.
	T SAT di f f	time in minutes between the last	Numeric
		TSAT and the SOBT.

Table 3 :

 3 Previous and current flights delay features (PCFDF).

Table 4 :

 4 Weather condition features (WCF).

	Feature	Description	Type
	Low Visibility Procedures	These procedures are applied at	Boolean
		an airport to ensure safe opera-
		tions when there is low visibility.
	Humidity rate (in percent)	Humidity rate	Numeric
	Wind speed (in meter/sec)	Wind speed	Numeric
	Air pressure (in hectoPascal)	Air pressure	Numeric
	Temperature (in degrees Celsius)	Temperature	Numeric

Table 5 :

 5 Results of feature selection and historical data.

	Dataset Features (V )	Hyperparameters	MAE RMSE	R2
	D 12 60 D 12 10 D 12 60 D 12 60 D 3 60 D 12 10 D 9 60	BFF, PCFDF,WCF, OMF, PFF 75/256/0.05 BFF,WCF, OMF, PFF 75/256/0.05 BFF, PCFDF,WCF, OMF, PFF 32/64/0.05 BFF, PCFDF,WCF, OMF, PFF 32/128/0.05 BFF, PCFDF,WCF, OMF, PFF 32/128/0.05 BFF,WCF, OMF, PFF 32/64/0.05 BFF 75/256/0.05	9.645 13.858 9.709 14.009 10.376 14.500 10.473 14.584 11.637 15.414 13.922 20.812 17.443 26.576 0.0108 0.731 0.725 0.705 0.702 0.667 0.393

Table 6 :

 6 MAE for off-block delay forecast.

	Date	MAE Baseline model	2022-09-10	7.429709 16.339208
	2022-08-13 8.196	18.814	2022-09-11	8.311289 19.677116
	2022-08-14 8.490	18.176	2022-09-12	7.088036 16.854864
	2022-08-15 8.026	16.877	2022-09-13	7.252672 15.125720
	2022-08-16 9.911	20.869	2022-09-14	8.613775 17.317014
	2022-08-17 8.288	18.728	2022-09-15	7.553985 17.969956
	2022-09-05 9.340	19.727	2022-09-16 31.250517 65.622123


Harold Rosenberg

Rabah Souam

Eric Toubiana

GENERAL CURVATURE ESTIMATES FOR STABLE H-SURFACES IN 3-MANIFOLDS AND APPLICATIONS

Keywords: Mathematics Subject Classification. 53A10, 53C42. Key words and phrases. constant mean curvature, stability, curvature estimates, gradient estimates, bounded sectional curvature

We obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds with bounded sectional curvature. Our estimate depends on the distance to the boundary of the surface and on the bound on the sectional curvature but not on the manifold itself. We give some applications, in particular we obtain an interior gradient estimate for H-sections in Killing submersions.

Introduction

Let (M, g) be a smooth Riemannian 3-manifold. Consider an immersed surface Σ M with trivial normal bundle. Call A its second fundamental form and N a global unit normal on it. We denote by H the length of the mean curvature vector of Σ. When Σ has constant mean curvature H, we say that Σ is an H-surface.

Recall that an H-surface Σ is said strongly stable if for any u ∈ C ∞ 0 (Σ) we have

Σ |∇u| 2 dΣ ≥ Σ (|A| 2 + Ric(N ))u 2 dΣ,
where dΣ and ∇ stand respectively for the area element and the gradient on Σ, and Ric denotes the Ricci curvature of M . Throughout this paper, we will use the term stable to mean strongly stable. Curvature estimates for stable H-surfaces in 3-manifolds are an important tool in the study of H-surfaces, see for instance Meeks-Perez-Ros [START_REF] Meeks | Stable constant mean curvature surfaces[END_REF]. Graphs are stable and more generally, surfaces transverse to an ambient Killing field are stable. Curvature estimates for graphs were obtained in 1952 by Heinz [START_REF] Heinz | ber die Lsungen der Minimalflchengleichung[END_REF]. In 1983, Schoen [START_REF] Schoen | Estimates for stable minimal surfaces in three-dimensional manifolds[END_REF] obtained an estimate for the norm of the second fundamental form of stable minimal surfaces in a 3-manifold M depending on the distance to the boundary on the surface and an upper bound on the curvature tensor of M and its covariant derivative. In particular, in R 3 , he showed the existence of a constant C > 0 such that for any stable and orientable minimal surface Σ one has In 1999, Bérard and Hauswirth [START_REF] Bérard | General curvature estimates for stable H-surfaces immersed into a space form[END_REF] extended Schoen's work to stable H-surfaces with trivial normal bundle in space forms. In 2005, using different methods based on ideas of Colding and Minicozzi [START_REF] Colding | Estimates for parametric elliptic integrands[END_REF], Zhang [START_REF] Zhang | Curvature estimates for CMC surfaces in three dimensional manifolds[END_REF] extended Schoen's result to stable H-surfaces (with trivial normal bundle) in a general 3-manifold, again the estimate depends on H, an upper bound on the sectional curvature and on the covariant derivative of the curvature tensor of M .

In this paper we obtain an estimate for the norm of the second fundamental form of stable H-surfaces in Riemannian 3-manifolds assuming only a bound on the sectional curvature. Our estimate depends on the distance to the boundary of the surface and only on the bound on the sectional curvature of the ambient manifold. More precisely, our main result is the following: Main Theorem. Let (M, g) be a complete smooth Riemannian 3-manifold of bounded sectional curvature |K| ≤ Λ < +∞.

Then there exists a universal constant C which depends neither on M nor on Λ, satisfying the following:

For any immersed stable H-surface Σ M with trivial normal bundle, and for any p ∈ Σ we have

|A(p)| ≤ C min d(p, ∂Σ), π 2 √ Λ .
We also obtain a local version of our result, see Theorem 2.5. Our method of proof is completely different from the above mentioned works and is based on a blow-up argument. For the readers convenience we sketch the idea of the proof.

Assuming by contradiction the result is not true, we have for each n a stable H n -surface Σ n in some 3-manifold M n with bounded sectional curvature, |K| ≤ Λ, admitting a point p * n ∈ Σ n satisfying |A n (p * n )| min d(p, ∂Σ), π 2 √ Λ > n. In addition, there exists a real number r n > 0 such that the geodesic disk D * n on Σ n centered at p * n with radius r n lies in the domain of some chart of M n which allows us to treat this disk as a surface in a Euclidean ball of fixed radius in R 3 endowed with the pull-back metric, with p * n at the origin. Furthermore in our construction we can make r n |A n (p * n )| → +∞. We then blow-up the latter metric multiplying by the factor |A n (p * n )|. This sequence of new metrics converges to the Euclidean metric on compact sets of R 3 . The sequence (D * n ) gives rise to a new sequence of surfaces still denoted by (D * n ) with controlled geometry and second fundamental form at the origin of norm 1. We can then construct a complete surface S, passing through the origin, in the accumulation set of the sequence (D * n ), whose universal cover S is a stable H-surface in the Euclidean space R 3 . It is well-known that S is then a plane which contradicts the fact that the second fundamental form of S has norm 1 at the origin.

However, this blow-up argument requires some care because of technical difficulties we explain in the proof.

It is worth noticing that an estimate of the type (1) cannot be expected in general, even for minimal surfaces, as such an estimate would imply that a complete stable H-surface is totally geodesic. For example, in H 3 there are complete non-totally geodesic stable H-surfaces, see Silveira [START_REF] Da Silveira | Stability of complete noncompact surfaces with constant mean curvature[END_REF]. Also in H 2 × R, for any 0 ≤ H ≤ 1/2 there are non-constant entire vertical H-graphs, and therefore stable, see Nelli and Rosenberg, [START_REF] Nelli | Minimal surfaces in H 2 × R[END_REF] and [START_REF] Nelli | Global properties of constant mean curvature surfaces in H 2 × R[END_REF].

As a consequence of the Main Theorem we have the following:

Corollary 1.1. Let (M, g) be a complete smooth Riemannian 3-manifold of bounded sectional curvature |K| ≤ Λ < +∞.
Then there exists a constant C, which depends neither on M nor on Λ, such that for any immersed complete stable H-surface Σ M , with trivial normal bundle, we have: This corollary was enunciated by Ros [START_REF] Ros | One-sided complete stable minimal surfaces[END_REF] for minimal surfaces (even with non trivial normal bundle) with the additional hypotheses that the derivatives of the curvature tensor are bounded and the injectivity radius is bounded from below.

• if Σ is non compact, then |A(p)| ≤ C √ Λ for any p ∈ Σ, • if Σ is compact, then |A(p)| min diam Σ, π √ Λ ≤ C
We are grateful to G. Besson, E. Hebey, J. Jost and M. Vaugon for useful information about harmonic coordinates.

Proof of the Main Theorem

In the blow-up argument we sketched before, the first idea which comes to mind is to use a chart given by the exponential map, that is, to use geodesic normal coordinates. However in those coordinates bounded geometry guarantees only a C 0 -control of the metric. For example consider the metrics g n , n ∈ N, on R 2 given in polar coordinates by

g n = dr 2 + G 2 n (r, θ)dθ 2 , with G n (r, θ) = r(1 + ar 2 + e -1/r 2 cos nθ),
where a > 0 is big enough. In Cartesian coordinates x = r cos θ, y = r sin θ, we have

g n = x 2 x 2 + y 2 + y 2 G 2 n (x 2 + y 2 ) 2 dx 2 + 2xy 1 x 2 + y 2 - G 2 n (x 2 + y 2 ) 2 dxdy + y 2 x 2 + y 2 + x 2 G 2 n (x 2 + y 2 ) 2 dy 2 = g n 11 dx 2 + 2g n 12 dxdy + g n 22 dy 2 .
It is easily seen that g n is a complete and smooth metric on R 2 and that g n ij (0) = δ ij for any n ∈ N, so that, (x, y) are geodesic normal coordinates at the origin for the metrics g n . The Gauss curvature of g n is given by

K n = -1 Gn ∂ 2 Gn ∂r 2
. Thus, a computation shows that

K n = - 6a + ( 4 r 6 -2 r 4 )e -1/r 2 cos nθ 1 + ar 2 + e -1/r 2 cos nθ .
Therefore, for a > 0 big enough, the Gaussian curvatures K n are negative and uniformly bounded on R 2 with respect to n. Consequently, the injectivity radius of (R 2 , g n ) is infinite for any n ∈ N. We can show directly (or as a consequence of Rauch comparaison theorem) that for any r > 0 there exists Q 0 > 0 depending on r, a and not on n such that on the geodesic balls of (R 2 , g n ) centered at the origin with radius r we have

Q -1 0 δ ij ≤ g n ij ≤ Q 0 δ ij , as quadratic forms,
this gives a uniform C 0 -control of the metrics g n . Nevertheless we do not have uniform C 1 -control. Indeed, consider for example the coefficient g n 11 , we have

∂g n 11 ∂x = ∂ ∂x x 2 x 2 + y 2 + ∂ ∂x y 2 (x 2 + y 2 ) 2 G 2 n + 2G n y 2 (x 2 + y 2 ) 2 ∂G n ∂x .
Observe that at any fixed point (x, y) = (0, 0), there is only one term which is not uniformly bounded with respect to n: the one involving the partial derivative of G n with respect to x. In coordinates (x, y) we have

G n (x, y) = x 2 + y 2 + a(x 2 + y 2 ) 3/2 + x 2 + y 2 e -1/(x 2 +y 2 ) Re(x + iy) n (x 2 + y 2 ) n/2 .
Also, at any fixed point (x, y) = (0, 0), in the partial derivative ∂G n /∂x, the only term which is not uniformly bounded with respect to n is the one involving the derivative of Re(x + iy) n /(x 2 + y 2 ) n/2 . We have, using the relations x = r cos θ, y = r sin θ, ∂ ∂x

Re(x + iy) n (x 2 + y 2 ) n/2 = n Re(x + iy) n-1 (x 2 + y 2 ) n/2 -nx Re(x + iy) n (x 2 + y 2 ) 1+n/2 = n r cos(n -1)θ - n r cos θ cos(n -1)θ = n r sin θ sin nθ.
This shows that at any fixed point (x, y) = (0, 0) whose argument θ is not rational with π the partial derivative (∂g n 11 /∂x)(x, y) is not uniformly bounded with respect to n. Consequently uniform estimates of the Gaussian curvature do not imply a local C 1 -control of the metric in geodesic normal coordinates.

For this reason we will instead use harmonic coordinates. More precisely, we will need the following result which can be deduced from [START_REF] Hebey | Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds[END_REF]Theorem 6].

Theorem 2.1. Let α ∈ (0, 1) and δ > 0. Let (M, g) be a smooth Riemannian 3manifold, without boundary, of bounded sectional curvature |K| ≤ Λ < +∞ and Ω an open subset of M . Set

Ω(δ) = x ∈ M | d g (x, Ω) < δ
where d g is the distance asociated to g. Suppose there exists i > 0 such that for all x ∈ Ω(δ) we have inj (M,g) (x) ≥ i, where inj (M,g) (x) is the injectivity radius at x.

Then there exist a constant Q 0 > 1 and a real number r 0 > 0, depending only on i, δ, Λ and α, and not on M , such that for any x ∈ Ω, there exists a harmonic coordinate chart (U, ϕ, B(x, r 0 )), U being an open subset of R 3 containing the origin and B(x, r 0 ) the geodesic ball in M centered at x of radius r 0 , with ϕ(0) = x, and such that the metric tensor g is C 1,α -controlled. Namely the components g ij , i, j = 1, 2, 3 of g satisfy:

Q -1 0 δ ij ≤ g ij ≤ Q 0 δ ij , as quadratic forms, (2) 3 k=1 sup y∈U |∂ k g ij (y)| + 3 k=1 sup y =z |∂ k g ij (y) -∂ k g ij (z)| d g (y, z) α ≤ Q 0 . (3)
We will also need the following result. Lemma 2.2. Let (M, g) be a smooth complete Riemannian manifold of bounded sectional curvature, |K| ≤ Λ < +∞ -thus for each x ∈ M the exponential map

exp x : B 0 ( π √ Λ ) ⊂ T x M → B(x, π √ Λ ) is a local diffeomorphism. Then in the closed ball B 0 ( π 4 √ Λ ), the injectivity radius in B 0 ( π √ Λ ) endowed with the pull-back metric exp * x g is at least π 4 √ Λ .
Proof. For any p ∈ B 0 ( π 4 √ Λ ) we denote by i(p) the injectivity radius of p and by C(p) its cut-locus in (B 0 ( π √ Λ ), exp * x g). We denote by i 0 the infimum of i(p), p ∈ B 0 ( π

4 √ Λ ). Assume by contradiction that i 0 < π 4 √ Λ . Let (p n ) ∈ B 0 ( π 4 √ Λ ) be a sequence such that i(p n ) → i 0 and i(p n ) < π 4 √ Λ . For each n we take a point q n ∈ B 0 ( π 2 √ Λ ) such that i(p n ) = d(p n , q n )(= d(p n , C(p n )), where d is the distance in (B 0 ( π √ Λ ), exp *
x g), we can assume that q n ∈ B 0 ( 3π 8 √ Λ + i 0 2 ). Up to choosing subsequences, we can assume that (p n )

converges to p ∞ ∈ B 0 ( π 4 √ Λ ) and that (q n ) converges to q ∞ ∈ B 0 ( 3π 8 √ Λ + i 0 2 ) ⊂ B 0 ( π 2 √ Λ ) and so d(p ∞ , q ∞ ) = i 0 .
Note that i 0 > 0. Indeed, there is a neighborhood W ⊂ B 0 ( π √ Λ ) of p ∞ and there exists δ > 0 such that for any points x, y in W there is a unique minimizing geodesic of length < δ joining x to y, see do Carmo [START_REF] Carmo | Francis Flaherty. Mathematics: Theory & Applications[END_REF]Remark 3.8,p.72]. It follows easily that there exist a neighborhood W ′ ⊂ W of p ∞ and δ ′ > 0 such that for any x ∈ W ′ we have i(x) ≥ δ ′ , and so i 0 ≥ δ ′ .

Since d(p n , q n ) < d(p n , ∂B 0 ( π √ Λ )), there exists for each n a minimizing geodesic α n in B 0 ( π √ Λ ) joining p n to q n . Actually, as q n ∈ C(p n ) and is not conjugate to p n (since d(p n , q n ) < π 4 √ Λ ) there is another minimizing geodesic β n joining p n to q n . As d(p n , q n ) = i(p n ), by a usual argument, cf. for instance Petersen [START_REF] Petersen | Riemannian geometry[END_REF]Lemma 16,p.142], α n and β n fit smoothly together at q n to form a geodesic loop γ n (with possibly a corner at p n ). As the length of γ n is bounded from below by 2i 0 , up to taking a subsequence,

γ n converges to a geodesic loop γ ∞ at p ∞ of length 2i 0 = 2d(p ∞ , q ∞ ) with midpoint q ∞ . Therefore q ∞ ∈ C(p ∞ ) and consequently i(p ∞ ) = i 0 .
We now discuss two cases.

Case 1. Assume that q ∞ ∈ B 0 ( π 4 √ Λ )
. Consider the geodesic sphere S centered at 0 of smallest radius enclosing the loop γ ∞ . Then γ ∞ is tangent to S at some smooth point z. We now see this is contra-

dictory. For x ∈ B 0 ( π 2 √ Λ ), set r(x) = d(x, 0). Consider the function f (t) = r(γ ∞ (t)), where t is an arc length parameter of γ ∞ with γ ∞ (0) = z. We have f ′ (0) = 0 and f ′′ (t) = ∇ 2 r(γ ′ ∞ , γ ′ ∞ ) + ∇r, D dt γ ′ ∞ = ∇ 2 r(γ ′ ∞ , γ ′ ∞ ). By comparison theorems we know that ∇ 2 r is positive definite since r(z) ≤ π 4 √ Λ + i 0 ≤ π 2 √
Λ , Petersen [16, Chapter 6, Section 5, Theorem 27]. Therefore f has a local strict minimum at 0 contradicting the fact that S encloses the loop γ ∞ .

Case 2. Assume that q ∞ ∈ B 0 ( π 4 √ Λ ).
By the symmetry of the cut-locus, p ∞ ∈ C(q ∞ ). Observe that we have by con-

struction i 0 ≤ i(q ∞ ) ≤ d(p ∞ , q ∞ ) = i 0 . Therefore i(q ∞ ) = d(p ∞ , q ∞ )
and so p ∞ is a closest point to q ∞ in C(q ∞ ). As before we deduce that γ ∞ is also smooth at p ∞ .

Taking the smallest geodesic sphere centered at 0 enclosing the closed geodesic γ ∞ , we arrive to a contradiciton as in case 1.

We now start the proof of our Main Theorem.

We first show that the conclusion of the theorem is true with a constant C(Λ) depending on Λ. Later we will show that this constant may be chosen independant of Λ.

Assume by contradiction that for any n ∈ N * there are: a complete smooth Riemannian 3-manifold (M n , g n ) with bounded sectional curvature, |K| ≤ Λ, a stable

H n -surface Σ n M n and a point p n ∈ Σ n satisfying (4) |A n (p n )| min(d Σn (p n , ∂Σ n ), π 2 √ Λ ) ≥ n,
A n being the second fundamental form of Σ n . We denote by

D n ⊂ Σ n the open geodesic disk of radius min(d Σn (p n , ∂Σ n ), π 2 √ Λ ) centered at p n .
For any n ∈ N * let us denote by 0 n the origin in

T pn M n . Call D n the connected component, containing 0 n , of exp -1 pn (D n ) ∩ B 0n ( π 8 √ Λ ) ⊂ T pn M n .
We endow the ball B 0n ( π √ Λ ) with the pull-back metric exp * pn g n . By abuse of notation, we still denote by g n this metric and note that its sectional curvature satisfies |K| ≤ Λ.

Observe that D n is a stable H n -surface with trivial normal bundle in B 0n ( π √ Λ ). Indeed, by a result of Fischer-Colbrie and Schoen [START_REF] Fischer-Colbrie | The structure of complete stable minimal surfaces in 3manifolds of nonnegative scalar curvature[END_REF], there exists a positive Jacobi function u on D n , therefore u • exp pn is a positive Jacobi function on D n , this implies stability of D n (cf. [START_REF] Fischer-Colbrie | The structure of complete stable minimal surfaces in 3manifolds of nonnegative scalar curvature[END_REF]). By abuse of notation we still denote by A n the second

fundamental form of D n in (B 0n ( π √ Λ ), g n ). Note that (5) |A n ( 0 n )| min d e Dn ( 0 n , ∂ D n ), π 2 √ Λ ≥ n. Indeed, we have |A n ( 0 n )| = |A n (p n )| and one can check that for any x ∈ ∂ D n we have d e Dn ( 0 n , x) ≥ d Dn (p n , ∂D n ) (consider the two cases x ∈ ∂B 0n ( π 4 √ Λ ) and x ∈ ∂B 0n ( π 4 √ Λ )
). Then we conclude by [START_REF] Carmo | Francis Flaherty. Mathematics: Theory & Applications[END_REF]. Let α ∈ (0, 1) be a fixed number. Consider the constants Q 0 > 1 and r 0 given in Theorem 2.1 applied to (M, g)

= (B 0n ( π √ Λ ), g n ), Ω = B 0n ( π 8 √ Λ ), δ = π 8 √ Λ and i = π 4 √ Λ (cf. Lemma 2.
2). We can assume that r 0 ≤ π 4 √ Λ . Let p * n be a point in the closure of D n where the function f :

D n → R defined by f (x) = |A n (x)| min d e Dn (x, ∂ D n ), r 0 reaches its maximum. Note that p * n is an interior point of D n since f ≡ 0 on ∂ D n . One can check that f ( 0 n ) ≥ r 0 √ Λ π n. As f ( 0 n ) ≤ f (p * n ) ≤ |A n (p * n )|r 0 we get (6) |A n (p * n )| ≥ √ Λ π n. Put ρ n = min d e Dn (p * n , ∂ D n ), r 0 . Consider the geodesic disk D * n ⊂ D n of radius ρ n /2 centered at p * n . Consider for any n ∈ N * the harmonic chart of M n , (U n , ϕ n , B(p * n , r 0 )),
given by Theorem 2.1, where U n ⊂ R 3 is an open neighborhood of the origin and ϕ n (0) = p * n . By the property (2) the Euclidean ball of radius r 0 /Q 0 centered at the origin, B e (0, r 0 /Q 0 ), is contained in U n for any n.

Call

λ n = |A n (p * n )|, by (6), λ n → +∞. Since λ n = |A n (p * n )| ≥ H n ,
the sequence H n /λ n is bounded and so, up to taking a subsequence, we can assume that [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] H

n λ n → H * < ∞.
We can also assume that (λ n ) is nondecreasing. Let F n be the homothety of 3 and, up to taking a subsequence, we can assume that the sequence (V n ) is monotone for the inclusion.

R 3 of ratio 1/λ n . Put V n = F -1 n (U n ) and observe that B e (0, λ n r 0 /Q 0 ) ⊂ V n ⊂ B e (0, Q 0 λ n r 0 ) for any n. Therefore ∪ n V n = R
We call (x 1 , x 2 , x 3 ) the cartesian coordinates of x ∈ U n and (y 1 , y 2 , y 3 ) those of y = F -1 n (x) ∈ V n . Now we use a blow-up argument, we endow V n with the metric

h n = λ 2 n F * n (ϕ * n g n ).
In the coordinates (x 1 , x 2 , x 3 ) the metric ϕ * n g n reads as

(ϕ * n g n )(x) = i,j g n ij (x)dx i dx j .
For any C 1,α -function w on an open set Ω ⊂ R 3 we set:

w C 1,α (Ω) = w ∞ + 3 k=1 sup y∈Ω ∂ k w(y) + 3 k=1 sup y =z ∂ k w(y) -∂ k w(z) |y -z| α .
Observe that, because of property (2) of Theorem 2.1, up to passing to a subsequence, we can assume that the sequence of inner products (g n ij (0)) converges to some inner product on R 3 . Up to a linear change of coordinates we can assume that this limit is the Euclidean inner product (δ ij ).

From properties (2) and (3) of Theorem 2.1, there exists a constant Q > 1 such that :

(8) g n ij C 1,α (Un) ≤ Q.
For each n and each y ∈ V n we have

h n (y) = i,j g n ij ( y λ n )dy i dy j .
Thus the components h n ij of h n in the coordinates y = (y 1 , y 2 , y 3 ) are given by h n ij (y) = g n ij ( y λn ) for any y ∈ V n . Observe that

∂h n ij ∂y k (y) = 1 λn ∂g n ij ∂x k ( y λn ), so we get on V n : (9) ∂h n ij ∂y k ≤ Q λ n , k = 1, 2, 3.
Therefore we have for any y ∈ V n (10)

|h n ij (y) -δ ij | ≤ √ 3 Q λ n |y| + |h n ij (0) -δ ij |.
Thus on any Euclidean ball B R ⊂ V n of radius R > 0 centered at the origin, we get [START_REF] Lpez | Ros Complete minimal surfaces with index one and stable constant mean curvature surfaces[END_REF] h

n ij -δ ij C 1,α (B R ) ≤ √ 3 Q λ n R + 3 Q λ n + Q 1+α λ 1+α n + |h n ij (0) -δ ij |.
It follows that the sequence (V n , h n ) converges uniformly on compact subsets of R 3 for the C 1,α -Euclidean topology to (R 3 , g euc ), where g euc stands for the Euclidean metric.

Composing with the diffeomorphism (

ϕ n • F n ) -1 we view D * n as immersed in V n . Note that the image (ϕ n • F n ) -1 (p * n ) = 0 ∈ R 3
for any n and that D * n is a geodesic disk of radius λ n ρ n /2 centered at p * n for the metric induced by h n . Note that

λ n ρ n = f (p * n ) ≥ f ( 0 n ) ≥ r 0 √ Λ π n, thus (12)
λ n ρ n → +∞. For x ∈ D * n we have d e Dn (p * n , ∂ D n ) ≤ 2d e Dn (x, ∂ D n ). Since f (x) ≤ f (p * n ) we deduce that the second fundamental form A * n of D * n in (V n , h n ) satisfies (13) |A * n (x)| = |A n (x)| λ n ≤ min d e Dn (p * n , ∂ D n ), r 0 min d e Dn (x, ∂ D n ), r 0 ≤ 2.
It is important to observe that ( 14)

|A * n (p * n )| = 1 for any n ∈ N * .
Let us call II n the second fundamental form of D * n in (R 3 , g euc ). Given m ∈ N * denote by ∆ n,m the connected component of D * n ∩ B m passing through 0 (that is, containing p * n ). As the sequence of metrics h n converges on compact sets of R 3 to g euc for the C 1,α -Euclidean topology, we deduce from Proposition 4.1 in the appendix that there exists n m ∈ N * such that for any n ≥ n m and any x ∈ ∆ n,m , we have

|II n (x)| -|A * n (x)| ≤ 1 m . Furthermore we have d ∆n,m (p * n , ∂∆ n,m ) ≥ m, if n m is big enough, since λ n ρ n → +∞. Observe that if m ′ > m then ∆ n,m ⊂ ∆ n,m ′ .
Moreover we can assume that the sequence (n m ) is increasing. We set ∆ m = ∆ nm,m . Thus, we have constructed in this way a sequence of connected surfaces ∆ n R 3 passing through 0 (that is, containing p * n ) with the following property: for any k ∈ N * there exists n k such that for any n ≥ n k and any x ∈ ∆ n ∩ B k we have

(15) |II n (x)| -|A * n (x)| ≤ 1 k and d ∆n (p * n , ∂∆ n ) ≥ k.
In particular we get for n ≥ n k

(16) |II n (p * n )| -1 ≤ 1 k .
Furthermore, for any k, any n ≥ n k and any x ∈ ∆ n ∩ B k we get from ( 13) and ( 15):

(17) |II n (x)| ≤ 5.
We will use the following well known result which can be deduced from Colding-Minicozzi [START_REF] Colding | Minimal Surfaces[END_REF] or Perez-Ros [START_REF] Pérez | Properly embedded minimal surfaces with finite total curvature, The global theory of minimal surfaces in flat spaces[END_REF]. Proposition 2.3. Let Σ R 3 be an immersed surface whose second fundamental form A satisfies |A| < 1 4δ for some constant δ > 0. Then for any x ∈ Σ with d Σ (x, ∂Σ) > 4δ there is a neighborhood of x in Σ which is a graph of a function u over the Euclidean disk of radius √ 2δ centered at x in the tangent plane of Σ at x. Moreover u satisfies [START_REF] Ros | One-sided complete stable minimal surfaces[END_REF] |u| < 2δ, |∇u| < 1 and |∇ 2 u| < 1 δ .

We deduce from the estimate ( 17) and Proposition 2.3 that for each n big enough, a part of the surface ∆ n is the graph of a function u n over a Euclidean disk of radius √ 2δ, with δ = 1/20, centered at the origin in the tangent plane of ∆ n at the origin. Furthermore the functions u n satisfy the uniform estimates [START_REF] Ros | One-sided complete stable minimal surfaces[END_REF]. Up to passing to a subsequence, still denoted ∆ n , and up to a rotation in R 3 , we can assume that the tangent planes T 0 ∆ n converge to the horizontal plane P through the origin. Consequently, for n big enough, a part of ∆ n is the graph of a function still denoted u n , over the Euclidean disk D δ of radius δ centered at the origin in P . By continuity, note that the new functions u n satisfy the following uniform estimates for n big enough:

(19) |u n | < 3δ, |∇u n | < 2 and |∇ 2 u n | < 1 δ (with δ = 1 20
).

Thus we have obtained uniform C 2 -estimates for the functions u n on D δ . To go further we need C 2,α -estimates, 0 < α < 1. This is the content of the next lemma.

Lemma 2.4. For any δ ′ ∈ (0, δ) there exits a constant C which does not depend on n such that for n big enough we have

(20) u n C 2,α (D δ ′ ) < C,
where D δ ′ denotes the disk in P of radius δ ′ centered at the origin.

Proof. Since the mean curvature of ∆ n for the metric h n is H n /λ n , we infer, see for instance Colding-Minicozzi [2, p. 99-100], that the function u n is solution of an elliptic PDE of the form: ( 21)

L(u) := a ij (u, ∇u, h n )u ij +b i (u, ∇u, h n , ∂ m h n )u i = 2 H n λ n W (∇u, h n )+c(u, ∇u, h n , ∂ m h n ),
where by h n we mean the components (h n ) αβ of the metric h n and by ∂ m h n we mean the partial derivatives

∂(hn) αβ ∂ym , 1 ≤ α, β, m ≤ 3.
Moreover the functions a ij , b i , W and c depend in a smooth way on their arguments.

Recall that the sequence (u n ) has uniform C 2 -estimates on D δ and so has uniform C 1,α -estimates. Moreover the sequence of metrics (h n ) converges to the Euclidean metric g euc for the C 1,α -topology on compact sets.

Recall also that H n /λ n is bounded, see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. Thus the functions

a ij (u, ∇u, h n ), b i (u, ∇u, h n , ∂ m h n ) and 2(H n /λ n )W (∇u, h n )+ c(u, ∇u, h n , ∂ m h n ) have uniform C 0,α - estimates on D δ , that is, C 0,α -estimates independant on n.
By Schauder estimates, see Chapter 6] or Petersen [16, Chapter 10], for any δ ′ ∈ (0, δ), there exists a constant C 0 > 0 which does not depend on n such that:

(22) u n C 2,α (D δ ′ ) < C 0 L(u n ) C α (D δ ) + u n C α (D δ ) .
Consequently, there exists a constant C > 0 which does not depend on n such that:

(23) u n C 2,α (D δ ′ ) < C,
which concludes the proof of the lemma.

Fix some δ ′ ∈ (0, δ). As the sequence (u n ) is C 2,α -bounded on the disk D δ ′ , by Arzela-Ascoli's theorem there is some subsequence, still denoted (u n ), which converges to a C 2 -function u in the C 2 -topology. Since the mean curvature, H n /λ n , of the graph of u n for the metric h n tends to H * , see [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], thanks to the Proposition 4.1 we know that the mean curvature of the graph of u n for the Euclidean metric also tends to H * . We deduce that the graph of u, denoted by S, is an H * -surface for the Euclidean metric.

Note that this graph contains the origin and that its second fundamental form A verifies: |A(0)| = 1 and |A| < 5 on D δ ′ (by ( 13), ( 14) and Proposition 4.1). Let x 0 ∈ D δ ′ at Euclidean distance δ ′ /2 from 0 and fix once for all some δ ′′ ∈ (δ ′ , δ). Note that for n big enough, a part of the surface ∆ n is the graph of a function v n over the Euclidean disk D δ ′′ (x 0 ) centered at (x 0 , u(x 0 )) ∈ S in the tangent plane T (x 0 ,u(x 0 )) S. Using the same arguments as before, we can extract a subsequence of (v n ) whose graphs converge to a H * -graph over the disk D δ ′ (x 0 ). Observe that this new graph is not contained in S because |∇u| ≤ 1. We have in this way obtained a new H * -surface extending S. We will still denote this extended H * -surface by S.

As d ∆n (0, ∂∆ n ) → +∞, using a standard diagonal process, we obtain a complete H * -surface S immersed in R 3 , passing through the origin and whose second fundamental form A satisfies: Consider the universal covering S of S, observe that S is naturally immersed in R 3 and its second fundamental form is bounded. Consequently, there exists an ε > 0 such that the map Π : W := S × (-ε, +ε) → R 3 given by Π(p, t) = p + t N (p) is an immersion, where N is the Gauss map of S. Therefore we can endow W with a flat metric which makes Π a local isometry. Note that S is a complete H * -surface in W .

Claim: S is stable in W .

For any geodesic disk in S of radius δ ′ , for n big enough, a piece of ∆ n is a graph over this geodesic disk of S. By construction of S, using this fact, for any compact and simply connected domain U in S, a piece of ∆ n will be a graph over U for n big enough. Using a continuation argument, for any compact and simply connected domain U in S, we can lift a part G n of ∆ n , n big enough, to get a surface G n in W which will be a graph over U . Moreover the graphs G n converge to U in the C2 -topology.

Observe that each graph G n is a stable H-surface (with H = H n /λ n ) in (a part of) W endowed with the metric Π * (h n ). Indeed, since G n is two-sided and stable for the metric h n , by a result of Fischer-Colbrie and Schoen [START_REF] Fischer-Colbrie | The structure of complete stable minimal surfaces in 3manifolds of nonnegative scalar curvature[END_REF], there exists a positive Jacobi function v n on G n . The function v n = v n • Π is thus a positive Jacobi function on G n for the metric Π * (h n ). Again by [START_REF] Fischer-Colbrie | The structure of complete stable minimal surfaces in 3manifolds of nonnegative scalar curvature[END_REF] this implies that G n is stable.

For n big enough, G n is the graph over U of a function which tends to 0 in the C 2 -norm as n goes to +∞. In this way we may see the stability operator of G n :

J n = ∆ n + | B * n | 2 + Ric n (ν)
as an operator on U. Here B * n stands for the second fundamental form of G n , ∆ n denotes the Laplacian on G n and Ric n (ν) denotes the Ricci curvature in the direction of the unit normal field ν to G n , all of them with respect to the metric Π * (h n ).

As the metrics Π * (h n ) converge to the flat metric Π * (g euc ) in the C 1,α -topology and |Ric n (ν)| ≤ 2Λ/λ 2 n , it follows that the domain U is stable for the flat metric. This implies that S is stable in W for the flat metric as claimed.

As the immersion Π is a local isometry we infer that the complete immersion S → S ⊂ R 3 is stable. Thanks to results of do Carmo-Peng [START_REF] Carmo | Stable complete minimal surfaces in R 3 are planes[END_REF], Fischer-Colbrie and Schoen [START_REF] Fischer-Colbrie | The structure of complete stable minimal surfaces in 3manifolds of nonnegative scalar curvature[END_REF], Pogorelov [START_REF] Pogorelov | On the stability of minimal surfaces[END_REF], Lopez-Ros [START_REF] Lpez | Ros Complete minimal surfaces with index one and stable constant mean curvature surfaces[END_REF] and Silveira [START_REF] Da Silveira | Stability of complete noncompact surfaces with constant mean curvature[END_REF], we know that S is a plane. This contradicts the fact that |A(0)| = 1, see (24).

It remains to check that C(Λ) can be chosen independant of Λ. First we can assume that C(Λ) is the infimum among the constants satisfying the conclusion of the theorem. Let Σ be any stable H-surface in (M, g). Observe that Σ is a stable H τ -surface in (M, τ 2 g) for any τ > 0 and that the quantity

|A(p)|min d(p, ∂Σ), π is scale invariant. It follows that C(Λ) = C(Λ/τ 2 ) for
any τ > 0 and so C(Λ) does not depend on Λ, which concludes the proof of the Main Theorem.

The same proof gives the following local result.

Theorem 2.5. Let (M, g) be a smooth Riemannian 3-manifold (not necessarily complete), with bounded sectional curvature |K| ≤ Λ < +∞. Let Ω be an open subset of M such that there exists δ > 0 for which Ω(δ) is relatively compact in M (cf. Theorem 2.1 for the notation).

Then there exists a constant C = C(δ 2 Λ) > 0 depending only on the product δ 2 Λ and neither on M nor on Ω, satisfying the following:

For any immersed stable H-surface Σ Ω, with trivial normal bundle, and for any p ∈ Σ we have

|A(p)| < C min d(p, ∂Σ), π 2 √ Λ , δ .

Applications

In this section we consider Riemannian 3-manifolds (M 3 , g) which fiber over a Riemannian surface (M 2 , h). We assume that the fibration Π : (M 3 , g) → (M 2 , h) is a Riemannian submersion with the following properties.

(1) Each fiber is a complete geodesic of infinite length.

(2) The fibers of the fibration are the integral curves of a unit Killing vector field ξ on M 3 .

A fibration satisfying ( 1) and ( 2) will be called a Killing submersion. It can be shown that such a fibration is (topologically) trivial. Indeed, there always exists a global section s : M 2 → M 3 (see Steenrod [START_REF] Steenrod | The Topology of Fibre Bundles[END_REF]Theorem 12.2]). Considering the flow ϕ t of ξ, a trivialization of the fibration is given by the diffeomorphism:

(p, t) ∈ M 2 × R → ϕ t (s(p)) ∈ M 3 .
Notice that there are many such 3-manifolds, including R 3 , PSL(2, R) (which fibers over the hyperbolic plane H 2 ), the Heisenberg space Nil 3 (which fibers over the Euclidean plane R 2 ) and the metric product spaces M 2 × R for any Riemannian surface (M 2 , h). Definition 3.1. Let Π : (M 3 , g) → (M 2 , h) be a Killing submersion.

(1) Let Ω ⊂ M 2 be a domain. An H-section over Ω is an H-surface in M 3 which is the image of a section s : Ω → M 3 , with s of class C 2 on Ω and C 0 on Ω. (2) Let γ ⊂ M 2 be a smooth curve with geodesic curvature 2H. Observe that the surface Π -1 (γ) ⊂ M 3 has mean curvature H. We call such a surface a vertical H-cylinder.

We may also consider H-sections without boundary.

Remark 3.2. Let s : Ω → Σ ⊂ Ω be a H-section where Ω ⊂ M 2 is a relatively compact domain. We make the following observations.

(1) For any interior point p ∈ Ω the H-surface Σ is transversal at s(p) to the fiber. Indeed, assume by contradiction that Σ is tangent to the fiber at the interior point s(p). Let S ⊂ M 3 be the vertical H-cylinder tangent to Σ at s(p) with the same mean curvature vector. Then, in a neighborhood of s(p), the intersection Σ∩S is composed of n ≥ 2 smooth curves passing through s(p).

But the union of those curves cannot be a graph, contradicting the assumption that Σ is a graph over Ω. (2) Let s 1 : Ω → Σ 1 ⊂ Ω be another H-section over Ω such that the mean curvature vector field H 1 points in the same direction as the mean curvature vector field H of Σ, that is, the scalar products g( H, ξ) and g( H 1 , ξ) have the same sign. Assume that there exists a constant C > 0 such that |s(p)-s 1 (p)| < C at any p ∈ ∂Ω where |s(p)s 1 (p)| denotes the distance on the fiber over p between the points s(p) and s 1 (p). As ξ is a Killing field, the translated copies of Σ along the fibers are also H-surfaces whose mean curvature vector has the same orientation as that of Σ. Then applying the maximum principle to Σ 1 and a translation of Σ by ξ, we deduce that we have also |s(p)s 1 (p)| < C at any p ∈ Ω.

This gives for any such H-section Σ 1 height estimates, relative to Σ, depending on the vertical distance between the boundaries of Σ and Σ 1 .

Our first application is as follows.

Theorem 3.3. Let Π : (M 3 , g) → (M 2 , h) be a Killing submersion and let s : Ω → Σ ⊂ M 3 be an H-section over a domain Ω ⊂ M 2 . Let U 0 be a neighborhood of an arc γ ⊂ ∂Ω and s 0 : U 0 → M 3 a section.

Assume that for any sequence

(p n ) of Ω which converges to a point p ∈ γ, the height of s(p n ) goes to +∞, that is s(p n ) -s 0 (p n ) → +∞.
Then, γ is a smooth curve with geodesic curvature 2H. If H > 0 then γ is convex with respect to Ω if, and only if, the mean curvature vector H of Σ points up, that is, if g( H, ξ) > 0 along Σ. Moreover, Σ converges to the vertical H-cylinder Π -1 (γ) with respect to the C k -topology for any k ∈ N; this convergence will be made precise in the proof.

Proof. Let p ∈ γ and (p n ) a sequence in Ω converging to p. Let B(p, ρ) be a compact geodesic disc of M 2 centered at p, with the radius ρ small so that s(Ω ∩ B(p, ρ)) is far from ∂Σ. Let Σ n be the H-section obtained by translating Σ by the integral curves of ξ so that s(p n ) goes to s 0 (p n ) = p n . After passing to a subsequence, we can assume the tangent planes T e pn Σ n converge to a 2-plane P e p of the tangent space T e p M 3 ; here p = s 0 (p).

We deduce from the Main Theorem, Proposition 4.3 (H-sections are stable) and a continuity argument that there exist two real numbers δ, δ 0 > 0 such that for n big enough, a part of Σ n is the Euclidean graph (in the sense of Remark 4.2) of some function u n over the disk of P e p centered at p with Euclidean radius δ. Furthermore this part of Σ n contains the geodesic disk, denoted by D n , centered at p n with radius δ 0 .

Since the mean curvature of each D n is constant and equal to H, the functions u n satisfy an elliptic PDE. Therefore, using the Schauder theory, we can find a subsequence of the previous subsequence converging for the C k -topology, for any k ∈ N, to some H-surface D e p passing through p with tangent plane P e p , T e p D e p = P e p . Since D e p is the limit of vertical graph and p = Π( p) is a boundary point of Ω, the tangent plane at p must be vertical. Furthermore D e p contains the geodesic disk centered at p with radius δ 0 /2. Denoting by N the unit normal field along D e p given by the limit of the unit normal fields along the disks D n , observe that the function g(N, ξ) is a Jacobi function on D e p . As each D n is a vertical graph, this function has a sign, but it vanishes at an interior point of D e p : p. Therefore, the maximum principle, see Spivak [21, Chapter 10, Addendum 2, Corollary 19] shows that g(N, ξ) is the null function. This means that the normal field N is horizontal. Clearly, this implies that the limit surface D e p is a part of the vertical cylinder over some curve α ⊂ M 2 . Since D e p has mean curvature H and the fibers are geodesic lines of M 3 , the curve α must be smooth and must have geodesic curvature 2H in M 2 .

Since each D n is, up to a translation along the fibers, part of the graph Σ over Ω, we deduce that:

(1) α ⊂ Ω ⊂ M 2 .
(2) Each converging subsequence of D n must converge to the same H-surface D e p .

(3) The limit surface D e p doest not depend on the sequence (p n ) of Ω converging to p ∈ γ. Now we show that we have α ⊂ γ ⊂ ∂Ω, that is, the arc γ is smooth and has geodesic curvature 2H.

Assume by contradiction that there is an interior point q ∈ α ∩ Ω. Take any point q ∈ D e p in the fiber over q, Π( q) = q. By construction, q is the limit of some sequence ( q n ) with q n ∈ D n . Since the sequence ( D n ) converges to (a part of) the vertical cylinder Π -1 ( α) with the C k -topology for any k ∈ N, the surface Σ must be vertical at s(q), contradicting the fact that Σ is transversal to the fibers, see Remark 3.2- [START_REF] Bérard | General curvature estimates for stable H-surfaces immersed into a space form[END_REF].

The assertion about the convexity of the arc γ is obvious.

Remark 3.4. In the case where M 3 is the metric product space H 2 × R or S 2 × R, the result above was shown in an analytic way by Hauswirth, Rosenberg and Spruck [START_REF] Hauswirth | Infinite boundary value problems for constant mean curvature graphs in H 2 × R and S 2 × R, to appear in[END_REF]. Therefore, bounding |∇ h u| from above is equivalent to bounding |g(N, ξ)| from below away from 0.

Remark 3.5. In a Riemannian product (M 3 , g) = (M 2 , h) × R, consider a domain Ω ⊂ M
As a second application we obtain interior gradient estimates, see Remark 3.5, for H-sections. Theorem 3.6. Let Π : (M 3 , g) → (M 2 , h) be a Killing submersion. Let Ω ⊂ M 2 be a relatively compact domain and s 0 : Ω → Σ 0 a C 0 -section over Ω.

Then, for any C 1 , C 2 > 0, there exists a constant α = α(C

(p n )) ≤ 1 n ,
where N n is the unit normal field along Σ n oriented by H n .

Since Ω is compact, there exists a subsequence of (p n ) converging to a point p ∈ Ω, with d(p, ∂Ω) ≥ C 2 . For each n ∈ N * we denote by Σ n the vertically translated copy of Σ n passing through p n := s 0 (p n ). Observe that the sequence ( p n ) converges to p := s 0 (p).

Since each H n -surface Σ n is a vertical graph, Σ n is stable. Thus we can apply our Main Theorem and Proposition 4.3. Therefore, there exist positive constants δ and δ 0 such that for each n ∈ N * a part S n of Σ n is a Euclidean graph over the disk of T e pn Σ n centered at p n with Euclidean radius δ 0 , furthermore S n contains the geodesic disk D n of Σ n centered at p n with radius δ.

As usual, up to choosing a subsequence, we can assume that the sequence of tangent planes (T e pn Σ n ) converges to a 2-plane P e p ⊂ T e p M 3 at p. Taking into account (26), we deduce that P e p is vertical. Thus, there exists a positive number δ ′ 0 < δ 0 such that for n big enough, a part of S n is a Euclidean graph over the disk of P e p centered at p with Euclidean radius δ ′ 0 and this part contains the geodesic disk of Σ n centered at p n with radius δ/2.

Observe that the sequence (H n ) is bounded. Indeed, consider the geodesic sphere S( p) in M 3 centered at p with radius C 2 /2 and H 0 > 0 an upper bound of the absolute mean curvature of S( p).

Using the comparison principle applied to Σ n and a suitable translated copy of S( p) along the fibers, we get |H n | ≤ H 0 . Up to taking a subsequence, we can assume that H n → H.

As in the proof of the Main Theorem, using the Schauder theory, we can prove that a subsequence of ( S n ) converges in the C k -topology, for any k ∈ N, to a H-surface S shows that g(N, ξ) is the null function on S, where N is the limit unit normal field on S. This means that S is part of a vertical H-cylinder over some curve γ ⊂ Ω, that is, S ⊂ Π -1 ( γ). Since S is a H-surface, γ is a smooth curve with geodesic curvature 2H. Finally, S contains the geodesic disk of Π -1 ( γ) centered at p with radius δ/2.

Let us call q ∈ Π -1 (p) ⊂ M 3 the point in the same fiber than p with vertical distance δ/4, lying over p. Observe that, by construction, q is the limit of some sequence ( q n ), with q n ∈ D n ⊂ Σ n . Observe that Π( q) ∈ Ω and d(Π( q), ∂Ω) ≥ C 2 since Π( q) = p. Therefore, the same arguments used above show that the geodesic disk of the vertical H-cylinder Π -1 ( γ) centered at q with radius δ/2 is the limit of a sequence (Σ ′ n ), Σ ′ n ⊂ Σ n , extending in this way the part S of Π -1 ( γ). Repeating this argument, we can show that a connected part of the vertical Hcylinder Π -1 ( γ), which is as high as we want, is contained in the limit set of the sequence ( Σ n ). This gives a contradiction since the vertical distance between Σ 0 and Σ n is uniformly bounded by C 1 by hypothesis.

Appendix

Proposition 4.1. Let U ⊂ R 3 be an open set and let S ⊂ U be an immersed C 2surface without boundary. Let g be a metric on U , denote by g euc the Euclidean metric. Let respectively A and A be the second fundamental forms of S for the metrics g euc and g. Assume that |A| < C for some constant C > 0.

Then, for any n ∈ N * , there exists a constant C n > 0 which does not depend on S such that if g ijg euc,ij C 1 (U ) < C n , 1 ≤ i, j ≤ 3, then for any p ∈ S and any nonzero tangent vector v ∈ T p S we have:

(27) |λ p (v) -λ p (v)| < 1 n and |H -H|(p) < 1 n ,
where λ p (v), resp. λ p (v), denotes the normal curvature of S at p in the tangent direction v with respect to the metric g euc , resp. g, and H, resp. H, denotes the mean curvature of S at p for the metric g euc , resp. g, (the curvatures of both surfaces being computed with respect to normals inducing the same transversal orientation).

Consequently, for any n ∈ N * , there exists a constant C ′ n > 0 which does not depend on S such that if g ijg euc,ij C 1 (U ) < C ′ n , 1 ≤ i, j ≤ 3, then we have:

(28) |A| -|A| < 1 n .
Proof. Let us denote the normal curvatures by λ(v) and λ(v).

Observe that if we have g ijg euc,ij C 1 (U ) < C for some constant C > 0 then, for any Euclidean change of coordinates we certainly have g ijg euc,ij C 1 (U ) < 9C in the new coordinates. We can choose Euclidean coordinates (x 1 , x 2 , x 3 ) of R 3 such that the origin coincides with p, the tangent plane T p S coincides with the plane {x 3 = 0} and v is tangent to the x 1 -axis. Thus, a neighborhood S p of p in S is the graph of a function x 3 = u(x 1 , x 2 ) defined in a neighborhood V of the origin in the plane {x 3 = 0}. Therefore a parametrization of S p is given by (

x 1 , x 2 ) → F (x 1 , x 2 ) = (x 1 , x 2 , u(x 1 , x 2 )), (x 1 , x 2 ) ∈ V ⊂ {x 3 = 0}. Let us set F i = ∂F ∂x i , u i = ∂u ∂x i , i = 1, 2. We have (29) u i (0) = 0, i = 1, 2.
We denote by N the Gauss map of S for the metric g and by N the Gauss map of S for the metric g euc , both oriented so that at 0 we have N (0) = (0, 0, 1) and g euc (N, N )(0) > 0.

We have by definition:

λ(v) = g(N , ∇ F 1 F 1 ) g(F 1 , F 1 ) (0) and λ(v) = u 11 1 + u 2 1 (0),
where ∇ stands for the Levi-Civita connection with respect to the metric g.

Using (29) we get F 1 (0) = ∂ x 1 . Furthermore a straightforward computation shows that:

N (0) = 1 g 33 g 13 ∂ x 1 +g 23 ∂ x 2 +g 33 ∂ x 3 (0) and ∇ F 1 F 1 (0) = ∇ ∂x 1 ∂ x 1 (0)+u 11 (0)∂ x 3 ,
Therefore we deduce that

λ(v) = 1 g 11 g 33 g(∇ ∂x 1 ∂ x 1 , g 13 ∂ x 1 + g 23 ∂ x 2 + g 33 ∂ x 3 )(0) + u 11 (0) , (30) λ(v) = u 11 (0). ( 31
)
Observe that if the metrics g and g euc are close enough on U in the C 1 -topology then the expression g(∇ ∂x 1 ∂ x 1 , g 13 ∂ x 1 + g 23 ∂ x 2 + g 33 ∂ x 3 )(0) is as close to 0 as we want and the expression 1 g 11

√

g 33 is as close to 1 as we want. Observe that these expressions are independant of u, that is, independant of the surface S. By hypothesis, we have |λ(v)| < C. Therefore there exists a constant D n > 0 which does not depend on

S such that g ij -g euc,ij C 1 (U ) < D n , 1 ≤ i, j ≤ 3, implies that |u 11 (0)| < 2C. Consequently we have: |λ(v) -λ(v)| < 2C 1 - 1 g 11 g 33 + 1 g 11 g 33 g(∇ ∂x 1 ∂ x 1 , g 13 ∂ x 1 + g 23 ∂ x 2 + g 33 ∂ x 3 )(0) .
For the same reasons as before, there exists a constant C n > 0, with C n < D n , which does not depend on S such that g

ij -g euc,ij C 1 (U ) < C n , 1 ≤ i, j ≤ 3, implies that |λ(v) -λ(v)| < 1 n .
Let us denote by λ max , λ min , λ max and λ min the principal curvatures of S at P for the metrics g and g euc respectively. It follows that:

|λ max -λ max | < 1 n and |λ min -λ min | < 1 n .
Therefore:

|H -H| < 1 n .
Let k > 0 be some integer and let C k be the constant given in the first part of the proposition. Thus, if g ijg euc,ij C 1 (U ) < C k , 1 ≤ i, j ≤ 3, then we have: Remark 4.2. Let (M, g) be a smooth Riemannian 3-manifold and let S M be an immersed surface. Let p ∈ S be an interior point of S. Let (U, ϕ, B(p, r 0 )) be a harmonic coordinate chart at p, where B(p, r 0 ) is the geodesic ball in M centered at p with radius r 0 such that B(p, r 0 ) ∩ ∂S = ∅. Let us denote the harmonic coordinates on U by (x 1 , x 2 , x 3 ) and let us consider the Euclidean metric dx 2 1 + dx 2 2 + dx 2 3 on U . Viewing the connected component of B(p, r 0 ) ∩ S through p in U observe that a neighborhood of p in S is a Euclidean graph over a Euclidean disk in the tangent plane T p S centered at p. The arguments of the proof of the Proposition 4.1 can be used to show the following result which is very useful for the applications. Proposition 4.3. Let (M, g) be a smooth Riemannian 3-manifold (not necessarily complete), with bounded sectional curvature |K| ≤ Λ < +∞ and let r > 0. Let Ω ⊂ M be an open subset of M such that the injectivity radius in M at any x ∈ Ω is ≥ r.

Then for any C 1 > 0 and C 2 > 0 there exist constants δ, δ 0 > 0 depending only on C 1 , C 2 , Λ and r and neither on M nor on Ω satisfying the following:

For any immersed surface S Ω whose second fundamental form A satisfies |A| < C 1 and for any p ∈ S such that d S (p, ∂S) > C 2 then a part S 0 of S is contained in the image of a harmonic chart and is a Euclidean graph (see Remark 4.2) over the disk of T p S centered at p with Euclidean radius δ. Furthermore, the subset S 0 contains the geodesic disk of S centered at p with radius δ 0 .

Proof. Let us fix some constants α ∈ (0, 1) and Q 0 > 1. The Theorem 6 in Hebey-Herzlich [START_REF] Hebey | Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds[END_REF] shows that there exists some r 0 = r 0 (Q 0 , α, r, Λ) > 0 such that for any x ∈ Ω there exists a harmonic coordinate chart (U, ϕ, B(x, r 0 )) satisfying the assumption of the Theorem 2.1 where B(x, r 0 ) is the geodesic ball in M centered at x with radius r 0 .

Let C 1 , C 2 , S and p ∈ S be as stated. Consider a harmonic coordinate chart (U, ϕ, B(p, r 0 )) at p, and call x = (x 1 , x 2 , x 3 ) the coordinates on U ⊂ R 3 , we have

3 k=1 sup x∈U |∂ k g ij (x)| ≤ Q 0 and Q -1 0 δ ij ≤ g ij ≤ Q 0 δ ij as quadratic forms.
We want to show that there exists a constant C where ∂ k stands for the partial derivative with respect to y k . Let us denote by λ p (v), resp. λ p (v), the normal curvature of S at p in the tangent direction v with respect to the Euclidean metric g euc , resp. g. We get from the formulae (30) and (31) established in the proof of the Proposition 4.1 that λ p (v) = (g 11 g 33 )(0)λ(v)g(∇ ∂y 1 ∂ y 1 , g 13 ∂ y 1 + g 23 ∂ y 2 + g 33 ∂ y 3 )(0).

Taking into account the formulae (32) we get that the expressions (g 11 g 33 )(0) and |g(∇ ∂y 1 ∂ y 1 , g 13

( 1 )

 1 |A(p)| ≤ C d Σ (p, ∂Σ) ,for any p ∈ Σ, where d Σ denotes the intrinsic distance on Σ.

, π 2 √

 2 for any p ∈ Σ, where diam Σ denotes the intrinsic diameter of Σ. Proof. For non compact surfaces, this follows immediately from the Main Theorem. Suppose now that Σ is compact. Set d = diam Σ and let p ∈ Σ, then the geodesic disk D in Σ of center p and radius d/2 satisfies d(p, ∂D) = d/2. Applying the Main Theorem to D we get |A(p)| min d 2 Λ ≤ C. The result follows.

  | = 1 and |A| ≤ 5.

2 1 +

 21 and a smooth surface Σ ⊂ M 3 which is the vertical graph of a function u on Ω. Let N be a unit normal field on Σ and let ξ = ∂ ∂t be the unit vertical field. Then we have |g(N, ξ)| = 1 |∇ h u| 2 .

passing through p

  with vertical tangent plane P e p = T e p S. As in the proof of Theorem 3.3, the maximum principle, see Spivak [21, Chapter 10, Addendum 2, Corollary 19],

2 ≤ |A| 2 -

 22 |λ maxλ max | < 1 k and |λ minλ min | < 1 k on S. Therefore we get |A| 2 -|A| 2 = λ 2 max + λ 2 min -To conclude the proof just observe that |A| -|A| |A| 2 .

3 =+ dx 2 2 + dx 2 3 .

 323 C 3 (C 1 ) which does not depend on p ∈ S or on S such that |A(p)| < C 3 where A denotes the second fundamental form of S ∩ B(p, r 0 ) for the Euclidean metric dx2 1 Let v ∈ T p S be a nonzero tangent vector. Choose Euclidean coordinates y = (y 1 , y 2 , y 3 ) in U such that the tangent plane T p S coincides with the plane {y 3 = 0} and v is tangent to the y 1 -axis. In those new coordinates we have (32)3 k=1 sup y∈U |∂ k g ij (y)| ≤ 9Q 0 and Q -1 0 δ ij ≤ g ij ≤ Q 0 δ ij as quadratic forms.

∂ y 1 +

 1 g 23 ∂ y 2 + g 33 ∂ y 3 )(0)| are bounded by a constant Q 1 which only depends on Q 0 . Therefore the expression |λ p (v)| is bounded by a constant C 4 which depends only on C 1 and Q 0 and which does not depend on p ∈ S or on S. Consequently we have |A| < 2C 4 on S. Now the proof follows from the Proposition 2.3 setting δ = 1/8C 4 . The second part of the estimates (32) shows that we can choose δ 0 = δ/Q 0 .

  1 , C 2 , Ω) such that for any p ∈ Ω with d(p, ∂Ω) > C 2 and for any H-section s : Ω → Σ ⊂ M 3 over Ω satisfying |ss 0 | < C 1 on Ω, we have Proof. Assume by contradiction that there exist positive constants C 1 and C 2 such that for any n ∈ N * there exists a H n -section s n : Ω → Σ n ⊂ M 3 over Ω with |s ns 0 | < C 1 , and there exists a point p n ∈ Ω such that d(p n , ∂Ω) > C 2 and verifying

	(25)	g N, ξ (s(p)) > α,
	where N is a unit normal field along Σ.
	(26)	g N n , ξ (s n

√Λ