Cheikh Sarr

Sofiane Khalfallah
email: sofiane.khalfallah@hds.utc.fr

Isabelle Guérin-Lassous
email: isabelle.guerin-lassous@ens-lyon.fr

Dynamic Bandwidth Management in IEEE 802.11-Based Multihop Wireless Networks

In this paper, we propose a new protocol named DRBT (Dynamic Regulation of Best Effort Traffic) which supports QoS throughput guarantees and provides a distributed regulation mechanism for Best Effort traffic in multihop wireless networks. By adapting dynamically the rate of Best Effort traffic at the link layer, DRBT increases the acceptance ratio of QoS flows and provides a good use of the remaining resources through the network. Our protocol also provides an accurate method to evaluate the available bandwidth in IEEE 802.11-based ad hoc networks which is able to differentiate QoS applications from Best Effort traffic. Through extensive simulations, we compare the performance of our proposal scheme with some others protocols like QPART for instance.

Introduction

The large availability of small wireless devices has enabled the deployment of mobile multihop wireless networks also called ad hoc networks. Ad hoc networks are autonomous and self-organized networks. They do not require any fixed infrastructure and nodes themselves solve topology changes due to mobility. This lack of centralized infrastructure makes the design of QoS protocols for these networks a challenging task. Moreover, due to their multihop and mobile features, QoS protocols must use distributed algorithms and not rely on global information. Quality of Service (QoS) focuses on several metrics like for instance delay, bandwidth, loss probability, etc. Our proposed scheme focuses on the bandwidth parameter, which is a basic metric often used to perform admission control, flow management or congestion control in ad hoc networks.

We assume that two types of applications are transmitted in the network:

-The first one requires guarantees on their throughput like video transmissions for instance. They are called QoS traffic henceforth. -The second one is more tolerant to changes on their throughput like file transfer for instance. They are called Best Effort traffic henceforth or BE for short.

Many works focus on the guarantee of bandwidth for QoS traffic without dealing with Best Effort traffic. Most of these works in this area supply guarantees for QoS flows, thanks to an evaluation of the available bandwidth. However, these evaluations do not provide any differentiation between QoS and Best Effort data packets, which may lead to situations where there is not enough available bandwidth for a new QoS traffic just because most of the bandwidth is occupied by Best Effort traffic. Such approaches limit the number of accepted QoS flows.

A first solution could be to reserve a fixed proportion of the residual bandwidth for Best Effort traffic. However, such an approach is not efficient when few QoS flows are transmitted, since the throughputs of BE traffic are limited to a fixed value whereas they could have used the whole remaining bandwidth.

The main idea of our protocol (called DRBT for Dynamic Regulation of Best Effort Traffic) is to provide a QoS mechanism, which can regulate the throughput of Best Effort traffic (when it is necessary) and provides throughput guarantees to QoS flows, according to an evaluation of the available bandwidth. This evaluation mainly relies on the possibility for nodes to decode local information in order to differentiate QoS and Best Effort traffic and uses some mechanisms already defined in ABE [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF].

The regulation scheme of DRBT proceeds in two phases:

decreasing the throughput of Best Effort traffic in order to increase the number of accepted flows, -increasing the throughput of Best Effort traffic to provide a maximal use of radio links when it is possible.

This regulation is coupled with a routing protocol in order to benefit of signaling packets to disseminate information required for our regulation mechanism. We have slightly modified AODV in order to transform it into DRBT. The remainder of this paper is organized as follows: Section 2 presents related work. Section 3 presents succinctly the different mechanisms of ABE used in DRBT. Section 4 describes our distributed protocol DRBT and finally simulations results are presented in Section 5.

Related work

To offer throughput guarantees to QoS flows, mobiles need first to evaluate the amount of bandwidth that is available in the network to ensure that the resource requirements of QoS admitted flows can be handled by the network. Different solutions have been proposed to evaluate the available bandwidth in IEEE 802.11-based multihop wireless networks, like for instance BRuIT [START_REF] Chaudet | BRuIT -Bandwidth Reservation under InTerferences influence[END_REF], CACP [START_REF] Yang | Contention Aware Admission Control for Ad Hoc Networks[END_REF] and ABE [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF].

BRuIT and CACP attempt to provide a good estimation of the carrier sensing area in order to derive an accurate per node available bandwidth estimation. Indeed, with CSMA protocols (like in IEEE 802.11), two nodes within carrier sensing range share the medium and thus the bandwidth, even if they cannot directly communicate. Therefore, each node needs to know the channel occupancy in its carrier sensing area in order to derive an accurate available bandwidth estimation per node. ABE, as we will see in Section 3, adds a mechanism to estimate the synchronization of idle periods between emitter and receiver and an estimation of link probability of collision in order to provide an accurate available bandwidth estimation per link.

These different estimations are then used in a routing protocol in order to compute QoS routes, i.e. routes that offer at least the requested bandwidth. These works mainly focus on the QoS traffic and do not optimize the cohabitation between QoS flows and Best Effort flows. For instance, BRuIT allocates a small fixed share of the bandwidth to Best Effort flows, which is not very efficient when there is no QoS flow in the network. The permanent channel sensing realized by CACP or ABE may limit the number of accepted QoS flows. Indeed, with sensing, Best Effort flows are considered in the computation of the used bandwidth, which can lead to situations where most of the bandwidth is used by Best Effort flows and new QoS flows consider that there is not enough bandwidth to be transmitted. To summarize none of these protocols presented above deal with the presence of both QoS and Best Effort traffic.

In [START_REF] Filali | Towards a fully distributed QoS-aware MAC protocol for multihop wireless networks[END_REF], a solution proposes to separate traffic by using two distinct channels of transmission. But this technique is difficult to implement and requires to change the current wireless cards.

SWAN [START_REF] Ahn | SWAN: Service Differentiation in Wireless Ad Hoc Networks[END_REF] is a distributed protocol that dynamically guarantees bandwidth for QoS traffic without the use of QoS routing protocol. To achieve this goal, SWAN performs three mechanisms. An admission control mechanism and two regulation mechanisms, one for QoS flows and the other one for Best Effort traffic. The first admission control mechanism uses a passive approach to estimate the available bandwidth. Before each transmission of a QoS flow, a probe packet is sent from the sender towards the receiver to estimate the available bandwidth along this path. In function of this value, the admission control performed at the sender decides or not to send the QoS flow. There is no routing process because the route is supposed known. When SWAN estimates that the network is congested, the regulation mechanisms are started. The first mechanism of regulation handles QoS traffic. According to the modeling of the IEEE 802.11 DCF performed by Bianchi [START_REF] Bianchi | Performance Analysis of the IEEE 802.11 Distributed Coordination Function[END_REF], available bandwidth can be evaluated. Consequently, as soon as a congestion is detected, the sender of the QoS flows simply reduces its throughput. The second mechanism of regulation concerns Best Effort traffic. The delay of ACK packets is measured. If this delay is higher than a predefined threshold, SWAN considers the network in a congested state and reduces the throughput of Best Effort flows.

The estimation of the residual bandwidth performed by probe packets sent from the sender to the receiver consumes an non-negligible amount of bandwidth. Moreover, for the first regulation mechanism, the assumptions of Bianchi's model are not valid in a multihop wireless environment. Therefore, the protocol SWAN provides an approximative residual bandwidth estimation, which makes more difficult the decision to regulate Best Effort traffic.

The protocol QPART [START_REF] Yang | Distributed QoS Guarantees for Realtime Traffic in Ad Hoc Networks[END_REF] does not use any admission control protocol, but regulates the different flows according to a congestion threshold. The available bandwidth estimation is based on RED theory [START_REF] Floyd | Random Early Detection Gateways for Congestion Avoidance[END_REF] which stipulates that to maintain the throughput constant over a multihop path, the queues size should be kept under a predefined threshold. If this threshold is exceeded, the regulation mechanisms are automatically activated. In case of congestion, the contention window size of Best Effort traffic in 802.11 is increased in order to make the transmission of QoS flows prior. QPART suffers from the same limitations as SWAN. Firstly, the available bandwidth based on RED theory is not adapted in the multihop wireless context where the presence of asymetric hidden terminals makes this theory inaccurate. Secondly, increasing the contention window size for Best Effort traffic just allows QoS data to have a probabilistic prior access on the radio channel.

To sum up, none of the described protocols in this section takes advantage of the differentiation between different kinds of traffic to provide an accurate regulation mechanism for the different traffic. In this work, we start with ABE that is, from our point of view, the most accurate protocol so far for evaluating the available bandwidth on a link, which is important for guaranteeing throughput. Then we add a differentiation mechanism to ABE in order to provide a more efficient bandwidth management. For instance, let's consider the scenario depicted on Figure 1. In this configuration, all the nodes are within communication range and the capacity (the maximum rate in the communication area) corresponds to 1600 Kb/s. Two flows are transmitted: a QoS flow of 500 Kb/s and a Best Effort of 1000 Kb/s. A third flow attempts to transmit data on the medium. With ABE, the perceived available bandwidth is almost null. Therefore this new flow can not be accepted. But, if we provide an estimation that differentiates QoS flows from Best Effort flows and that takes into account only transmissions of QoS flows in the evaluation, then we obtain a remaining bandwidth for the new flow almost equal to 1000 Kb/s. Therefore, the third traffic can be transmitted without degrading the existing QoS flow providing that a mechanism reducing the throughput of the Best Effort flow is used. Thus, we can accept more QoS flows, offer guarantees to the accepted QoS flows and provide the remaining bandwidth to the Best Effort flows.

Before describing the protocol DRBT, we present in the next section some mechanisms used in ABE to evaluate the available bandwidth of wireless links.

ABE

For ensuring Best Effort traffic regulation, our solution relies on an accurate available bandwidth estimation. Hereafter, we define the available bandwidth between two neighbor nodes as the maximum throughput that can be transmitted between these two peers without disrupting any ongoing flow in the network. This term should not be confused with the link capacity (also called base bandwidth) that designates the maximum throughput a flow can achieve between two neighbor nodes, even at the cost of other flows' level of service degradation.

For the available bandwidth estimation, we choose the protocol ABE (Available Bandwidth Estimation), first proposed in [START_REF] Sarr | A node-based available bandwidth evaluation in ieee 802.11 ad hoc networks[END_REF] and then refined in [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF]. In [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF], the authors show that ABE is more accurate than several protocols with the same goal while requiring a small overhead. By considering the overlapping of the silence periods of both emitter and receiver of a link, the collision probability of the link and the backoff window size correlated to this collision probability, ABE reaches an accuracy in the estimation that is often not achieved by the other protocols.

As our regulation mechanism depends on this available bandwidth estimation, this section is devoted to the description of ABE. Of course, due to space limitation, we can not include all the details of ABE that is not the novelty of our proposition. The interested reader can refer to [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF]. For providing an accurate evaluation, some phenomena need to be taken into account when the IEEE 802.11 MAC protocol operates:

-Carrier sense mechanism prevents two close emitters from transmitting simultaneously. Therefore, an emitter shares the channel bandwidth with all these close emitters. The channel utilization has to be monitored to evaluate the capacity of a node to emit a given traffic volume. As in many protocols, this channel utilization is computed by each node by monitoring the radio medium in its surroundings and measuring the total amount of time that is idle for emitting frames. Therefore, this method does not only take into account the bandwidth used in the transmission range of the nodes but also in the whole carrier sensing area. -For a transmission to take place, both emitter and receiver need that no jamming occurs during the whole transmission. Therefore, the value of the available bandwidth on a link depends on both peers' respective channel utilization ratios but also on the idle periods synchronization. In [START_REF] Sarr | A node-based available bandwidth evaluation in ieee 802.11 ad hoc networks[END_REF], we propose a probabilistic method to estimate this synchronization. This estimation, for the link (s, r), is denoted E(b (s,r) ) in the following. -Collision detection is difficult in a wireless environment. Therefore, whenever collisions happen, both colliding frames are completely emitted, maximizing the bandwidth loss. It is thus necessary to integrate this bandwidth loss in the available bandwidth estimation. In [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF], we provide an estimation of the collision probability on each link. This estimation combines two approaches: i) A on line approach that computes the impact of the medium occupancy distribution at the receiver side thanks to the collision probability on Hello packets. These Hello packets are used in many ad hoc routing protocols and are required for computing the previous estimation E(b (s,r) ) on each link; ii) A off line approach that takes into account the size of the packets sent by the source thanks to an interpolation. The goal of this last approach is to compute the collision probability that packets of known and fixed size will undergo on a link from the collision probability of Hello packets deduced from real measurements on the same link. This collision probability estimation is denoted p, in the following, and depends on the size of packets that will be sent. -Finally, when collisions happen on unicast frames, the IEEE 802.11 protocol automatically retries to emit the same frame, drawing the backoff counter in a doublesized contention window. The time lost in additional overhead may also have an impact on the available bandwidth. In [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF], we compute the mean backoff, backof f according to p the collision probability computed in the previous estimation. It is then possible to deduce the proportion of bandwidth consumed by the backoff mechanism. This proportion is denoted by K in the following and computed as

K = DIF S+backof f T
, where T is the time separating the emission of two consecutive frames (see [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF] for more details).

These different estimations are then combined to estimate the available bandwidth on a wireless link, i.e. between an emitter s and a receiver r:

E f inal " b (s,r) " = (1 -K) • (1 -p) • E " b (s,r) " (1)
Finally, the intraflow contention issue is also considered in ABE. The available bandwidth of link (i, j) will take into account the neighboring hops in the flow path and will be reduced accordingly. See [START_REF] Sarr | Bandwidth Estimation for IEEE 802.11-Based ad hoc network[END_REF] for all the details.

DRBT: a Dynamic Regulation protocol for Best Effort Traffic

This section describes how we introduce the differentiation in the available bandwidth estimation and how we use this estimation for a regulation of Best Effort traffic. The first step is the differentiated estimation of the available bandwidth. It will allow us to quantify the proportion of the available bandwidth which is occupied by Best Effort flows. This estimation relies on the protocol ABE (Section 3). However, in the current state, ABE is not able to differentiate between QoS and Best Effort data packets. Therefore we present hereafter how we perform this differentiation in order to enhance the measurement accuracy.

Differentiation between QoS and Best effort traffic

As explained previously, a differentiation between QoS and Best Effort flows allows a better use of the available bandwidth for new QoS transmissions. We assume that each packet is marked in its IP header in order to know to which kind of flow it belongs, i.e. a QoS flow or a BE flow. The differentiation in the remaining bandwidth estimation is simply done at the MAC layer and consists in measuring only medium occupancy of QoS data packets during the monitoring phase of ABE.

Note that this differentiation is only possible if the node is able to decode data sensed over the medium since packets IP header has to be examined. Packets sent in the carrier sensing area of this node will not be decoded because the signal perceived is below the transmission range threshold. Consequently, in DRBT, the estimation of the available bandwidth is differentiated if the packet sensed over the medium can be decoded. In other words, Best Effort traffic that can not be decoded are included in the used bandwidth during the monitoring.

To summary, each node computes its differentiated remaining bandwidth by removing the bandwidth consumed by QoS flows in its communication range and consumed by all the flows in its carrier sensing area. Then this differentiated remaining bandwidth per node is used to compute the differentiated remaining bandwidth per link with the ABE method (Eq. ( 1)).

Regulation of Best Effort traffic

The previous available bandwidth estimation is not enough to provide guarantees to QoS flows. The BE traffic needs also to be regulated. In DRBT, the regulation scheme concerns only the Best Effort traffic. This regulation is done in two steps:

-Decreasing the throughput of Best Effort flows when a new QoS flow wishes to be transmitted and does not find enough available bandwidth because this one is partially consumed by Best Effort transmissions. -Increasing the throughput of Best Effort flows when a QoS flow releases its bandwidth or moves to another transmission area.

This regulation is coupled with a routing protocol in order to benefit of signaling packets to disseminate information required for our regulation mechanism. We have slightly modified AODV in order to transform it into a QoS protocol, called DRBT henceforth. The choice of AODV is driven by the possibility to use broadcasted route request messages and route reply messages to inform and regulate BE flows and to find adequate (constrained) routes for QoS flows.

Reduction of Best Effort traffic

In this section we explain how we decrease the throughput of Best Effort flows. The regulation process of BE flows is triggered when a new QoS flow asks to be accepted in the network. Therefore, the search of an adequate route for a QoS flow is intimately linked to the possible regulation of some BE flows. To do this, DRBT does not introduce additional message overhead but uses classical RREQ (Route Request) and RREP (Route Reply) packets found in AODV. Every time a new QoS flow wants to transmit data, it ckecks the resources availability using these RREQ and RREP packets. The information stored on these packets with DRBT are:

-The throughput requested by the new QoS flow (T hroughputQoS).

-The number of Best Effort flows (nbBE) within the neighborhood of the path on which the QoS flow is transmitted. For this, each Best Effort flow has a single identifier propagated on Hello messages. Therefore, each node can be able to know the number of Best Effort flows in its vicinity by analyzing these identifiers. -The differentiated remaining bandwidth (Dif f Bandwidth) which only takes into account the QoS transmissions (when possible) as described in Section 4.1.

The RREQ packet contains, in addition to the fields described previously, the address of the sender, the destination address and a sequence number. The sequence number is used in order to avoid cycles in the routing process, therefore a RREQ is just examined during its first passage. Each intermediate mobile that receives a RREQ performs an admission control by simply comparing whether the bandwidth requirement carried in the RREQ packet is lower than the differentiated available bandwidth of the link (previous sender, this node). If it is the case, the node updates nbBE and Dif f Bandwidth (if necessary) and forwards the RREQ. Otherwise it discards it. nbBE is incremented at each node by the number of Best Effort flows that the node knows. If the differentiated remaining bandwidth of the node that receives the RREQ is lower than the differentiated remaining bandwidth given in the RREQ, then the node modifies this field with its value. It allows us to know the available bandwidth computed along a path when considering only QoS transmissions (when possible).

When the destination receives a RREQ, it also needs to do the checking procedure as described above. Finally the destination sends a unicast route reply (RREP) to the initiator of the request along the reverse path to ensure that mobiles along the reverse path are still reachable.

Every time a Best Effort sender intercepts a RREQ or a RREP, it checks whether there is enough available bandwidth to carry the QoS flow without degrading it, by comparing its own throughput (T hroughputBE) with the parameter AvailableBandwidth computed as:

AvailableBandwidth = Dif f Bandwidth -T hroughputQoS nbBE (2)
If it is not the case, it reduces its throughput by sending a packet called DRP for Dynamic Regulation Packet. This packet, sent from the the IP layer towards the LL layer, activates the throughput reduction mechanism for Best Effort traffic. A leacky bucket algorithm is then applied, at the LL level, on the Best Effort packets with a rate corresponding to the AvailableBandwidth value1 . Equation 2 computes the new available bandwidth allocated to the Best Effort flow if the new QoS flow is accepted. This equation is conservative, because it considers that all the Best Effort flows, neighbors of a QoS flow on the path, share the same radio medium, which is not necessarily the case, and that it uses the minimum differentiated bandwidth on the path.

Increase of Best Effort traffic

When a QoS flow stops transmitting or moves to another transmission area, all the Best Effort flows that have reduced their bandwidth should increase their throughput to its initial value in order to use the maximum of the available bandwidth when possible. To address this issue, we use the Hello messages. Indeed, each node carrying a QoS flow encapsulates, in its Hello messages, information about the identifier of this flow and the differentiated available bandwidth.

When a QoS flows stops transmitting or releases its bandwidth, it indicates this information in these Hello packets. The Best Effort emitter which is in the vicinity of this QoS flow will intercept these Hello messages indicating that a QoS transmission has stopped or that the available bandwidth has increased. Finally, the Best Effort flow can increase their throughput according to the updated available bandwidth. When a node moves, the nodes carrying a Best Effort flow do not received Hello messages anymore and can thus use their previous allocated bandwidth providing that they have stored it, according to this updated available bandwidth.

Simulations

In this section, we evaluate the performances of DRBT and compare it with other approaches. We use the network simulator 2 (NS-2.27)2 and the IEEE 802.11 implementation provided with the simulator. The parameters used for all scenarios are presented on Table 1, except when specified. We use different physical rates in different scenarios in order to test the effectiveness of our approach under different physical rates.

We compare the performance of our regulation mechanism through the DRBT protocol described above with AODV [START_REF] Perkins | Ad hoc On-Demand Distance Vector (AODV) Routing[END_REF], ABE [START_REF] Sarr | Improving Accuracy in Available Bandwidth Estimation for 802.11-based Ad Hoc Networks[END_REF] and QPART [START_REF] Yang | Distributed QoS Guarantees for Realtime Traffic in Ad Hoc Networks[END_REF]. We chose QPART because as far as we know, it is one of the most recent solutions that provide both differentiation between QoS and Best Effort in addition to a regulation scheme as DRBT. The comparison between DRBT and ABE will allow us to evaluate the added value of a differentiation between QoS and Best Effort data packets. We also compare DRBT with AODV as a baseline for comparison. When AODV is used, as shown on Figure 3(a), each connection tries to send its data packets when possible without any regulation mechanism and this situation leads to a shared medium between the Best Effort and the QoS flows. The throughput of these two flows are consequently degraded.

When ABE is performed (Figure 3(b)), the admission control step estimates that there is not enough available bandwidth to carry the QoS flow with its bandwidth requirement. Hence, only Best Effort traffic is transmitted.

QPART is not able to enough decrease the throughput of the Best Effort flow. Therefore, the throughput request is not guaranteed to the QoS flow, as shown in Figure 3(d).

With DRBT, the decrease is more accurate and consequently, the QoS flow is carried with its bandwidth requirement without any degradation, as shown in Figure 3(c).

A random topology

To compare the different protocols and illustrate the effectiveness of DRBT to provide a better regulation mechanism, we generate random topologies with random constant bit-rate flows (random source, random destination and random throughput with fixed 1000 bytes frames). For each of these protocols, similar scenarios (same number of nodes and same number of flows) lead to similar behaviors. The scenario consists of a static network involving 10 randomly positioned nodes. Two QoS and three Best Effort CBR connections and established. The physical rate is set to 2 Mb/s. Table 2 sums up the throughputs requested by all flows in the network.

As in the previous simulation, the same observations can be done. AODV (Figure 4(a)) involves a sharing of resources among the different flows without any priority for QoS flows. ABE estimates that there is not enough available bandwidth to carry QoS flows with their bandwidth requirements. Hence, only Best Effort traffic are transmitted (Figure 4(b)).

QPART is not able to enough decrease the throughputs of the Best effort flows. Therefore, QoS flows are degraded as shown in Figure 4(d). Table 2 Throughputs desired by all flows

Finally, DRBT reduces effectively the throughputs of the Best Effort flows so that the QoS flows are carried with their bandwidth requirements without any degradation.

Accuracy of the regulation mechanism of DRBT

Let us now investigate the general case. To reflect the accuracy of DRBT, we define a new metric accounting for the number of right admissions. A right admission happens when the admission control protocol allows the routing of a QoS flow and this flow's throughput is not degraded by more than 5 % when it gets transfered. This definition implies that the differentiated available bandwidth estimation and the admission control are reliable. The acceptance ratio is defined by the following expression: A falsely admitted flow either degrades the throughput of close flows or is not able to achieve its desired throughput. Hence, the value of β decreases. So β is able to characterize as well an under-estimation as an over-estimation.

We measured the value of β by simulation on networks composed of 10 to 40 nodes, using an 11 Mb/s medium capacity. Each simulation lasts 100 seconds and 15 randomly chosen pairs of nodes try to establish CBR connections (5 Best Effort connections and 10 QoS connections). Each flow starts according to the Poisson model with average arrival rate of one flow every 5 seconds. The throughput of each connection is uniformly drawn between 0 kb/s and 300 kb/s. Results presented are the average of 30 simulations for a defined number of nodes with a confidence interval of 95%. Influence of the used protocol: Figures 5(a When the network is not too dense (between 10 and 20 nodes), the acceptance ratio of DRBT is high (about 60%) while this value is lower for QPART (about 41%). Therefore, the differentation between QoS and Best Effort data packets and the regulation scheme provided by both QPART and DRBT allows these two protocols to have better performances than ABE and AODV. When the network becomes denser (between 20 and 30 nodes), β decreases, which is expected as the available bandwidth per link decreases and lower quality routes are established. However, DRBT can also transmit about 51% of QoS flows.

Finally, when the network becomes very dense (between 30 and 40 nodes), the residual bandwidth becomes low and even a reduction of throughput of Best Effort traffic can not release enough available bandwidth to allow QoS flows to be transmitted with their throughput requirements. Nevertheless, DRBT still correctly transmits about 29% of QoS flows while all the others protocols transmit in the best case almost than 11%. Influence of the carrier sensing range: For this simulation, we reduce the carrier sensing range until it is equal to the communication range. Results are presented in Figure 5(b). The acceptance rate of DRBT is higher than previously for a same number of nodes and is always higher than for QPART, ABE and AODV. In fact, reducing the carrier sensing range allows DRBT to decode more information stored on packets. This facilitates the differentiation between QoS and Best Effort packets in DRBT, and thus the regulation of more Best Effort flows. On the other hand, the disadvantage of reducing the carrier sensing is that more hidden terminals configurations are created which poses problems to AODV, since it does not take into account in their measurement configurations like hidden terminals contrary to DRBT.

Loss rate

Other parameters like loss rate or delay are of some importance for QoS applications. We have focused our study on the loss rate parameter because this parameter seems to be more critical than delay in multihop wireless networks of medium size (see [START_REF] Pucha | The performance impact of traffic patterns on routing protocols in mobile ad hoc networks[END_REF]). We study the loss rate on QoS flows obtained with AODV, QPART and DRBT. We have tested the same scenarios than the ones tested in Section 5.2 with a carrier sensing range two times larger than the communication range. In Figure 6, we see that loss rate is smaller with DRBT than with AODV and QPART. For instance, with 40 nodes, this packet loss with DRBT is two times smaller than QPART and three times smaller than AODV. This is mainly explained by the fact that with DRBT, more QoS flows are transmitted with less packet losses thanks to a good estimation of the available bandwidth and to a better regulation of close Best-effort flows.

Note that, however, the values obtained on loss rate are still high, even with DRBT, for some QoS applications. It seems that a smart bandwidth management decreases the loss rate but is not enough to achieve small target loss rates required by some applications.

Signaling overhead

We study the signaling overhead induced by the use of the broadcast technique of AODV in which we have added new fields in the control packets. We have tested the same scenarios than the ones tested previously. Figure 7 shows the signaling overhead with the control packets RREQ, RREP and RRER (when there is an error at the routing layer) in bytes of DRBT and AODV. Even though larger control packets are used with DRBT compared to AODV (e.g. 48 bytes for RREQ packets or 68 bytes for RREP packets with DRBT versus 32 bytes for RREQ and RREP packets with AODV), the signaling overhead in bytes is much smaller with DRBT than with AODV. For instance, with 40 nodes, this signaling overhead is three times smaller. This is mainly explained by the fact that with DRBT, less RREQ packets are forwarded due to the bandwidth requests of the QoS flows. It is also interesting to note that the number of RRER packets is much less with DRBT than with AODV. For instance, with 40 flows, there are around 200 RERR packets compared to the 4800 RERR packets with AODV. It tends to show that DRBT has an efficient bandwidth management which results in less collisions and thus in less route recoveries than with AODV. The same remarks as the ones provided in Section 5.1.1 can be done: AODV involves a sharing of resources among the different flows without any priority for QoS flows (Figure 8(a)) ; ABE prevents one QoS flow from being transmitted because it estimates that there is not enough available bandwidth (Figure 8(b)) ; QPART is less effective than with CBR BE flows (Figure 8(d)). It seems that the alternance of burst/idle periods prevents QPART from adapting and converging towards the right size of the congestion window of each kind of traffic ; on the other hand, DRBT reduces effectively the throughputs of the Best Effort flows so that the QoS flows are transmitted with their bandwidth requirements without any degradation. Compared to the results of Section 5.1.1, the throughputs of the QoS flows with DRBT are more variable. This is probably due to the variations of the BE traffic that imply variations in the remaining

Conclusions and future works

In this paper, we have presented DRBT (Dynamic Regulation of Best Effort Traffic), a protocol which guarantees bandwidth of QoS flows by adpating effectively and dynamically the throughputs of Best Effort transmissions when it is necessary. Our protocol relies on an estimation of the available bandwith differentiated according to the type of packets (QoS or Best Effort data packets). With these features, DRBT increases the acceptance ratio of QoS flows, while providing a better usage of the radio medium. Furthermore, the effectiveness of our protocol is shown through simulations, where DRBT effectively manages the throughputs of QoS transmissions by dynamically adapating rate of close Best Effort traffic, compared to other protocols like AODV, ABE and QPART. We show also that DRBT reduces the loss rate compared to the other tested protocols. However, the values achieved may be still too high for some QoS applications.

In future works, we intend to improve some features of DRBT and more particularly to deal with the problems induced by the carrier sensing mechanism. We also plan to design a smarter share of the medium between Best Effort flows. Indeed, the current share is very conservative since it considers that all the BE flows neighbors of a QoS flow share the same medium, which it may not be the case. Therefore, more capacity could be used for BE flows. Another important point is to deal with mobility which prevents from guaranteeing resources due to the disappearance of link radio.

Fig. 1

 1 Fig. 1 Differentiation between QoS and Best Effort flow

Fig. 2

 2 Fig. 2 Two pairs

Fig. 3

 3 Fig. 3 Throughputs obtained withAODV, ABE, DRBT et QPART Flow Type Desired throughput (Kb/s) Date of emission (s) CBR1 Best Effort 319 10 CBR2 Best Effort 164 20 CBR3 Best Effort 386 30 CBR4 QoS 129 40 CBR5 QoS 281 50

Fig. 4

 4 Fig. 4 Throughputs obtained with AODV, ABE, DRBT et QPART in a random topology

  ) represents the values of β for AODV, ABE, QPART and DRBT in function of the number of nodes in the network.

Fig. 5

 5 Fig. 5 Acceptance ratio of QoS flows with AODV, ABE, QPART and DRBT

Fig. 6

 6 Fig. 6 Loss rate on QoS flow packet with AODV, QPART and DRBT

Fig. 7

 7 Fig. 7 Signaling overhead in bytes with AODV and DRBT

Fig. 8

 8 Fig. 8 Throughputs obtained with AODV, ABE, DRBT et QPART in a random topology with VBR BE traffic

Table 1

 1 General parameters for simulationsFor this simulation, we consider a simple topology of two pairs of nodes within the same communication range as depicted on Figure2.At the date t=1s, a Best Effort connection of throughput 1000 Kb/s is established between A and B. Four seconds later, a new QoS flow of throughput 1000 Kb/s is also established between C and D. At date t=30s, the QoS flow stops its transmission. The

	Parameters	Values
	HELLO interval	1 s
	Packet size	1000 bytes
	Physical rate	2 or 11 Mb/s
	Communication range	250 m
	Carrier sensing range	550 m
	Grid size	1000×1000 m
	LL queue size	100 packets
	5.1 First topologies
	5.1.1 A simple scenario

Table 3

 3 sums up the throughputs requested by all flows in the network.

	Flow	Type	Desired throughput (Kb/s) Date of emission (s)
	VBR1 Best Effort	266	10
	VBR2 Best Effort	268	20
	VBR3 Best Effort	258	30
	CBR1	QoS	315	40
	CBR2	QoS	591	50

Table 3

 3 Throughputs desired by all flows

Note that the throughput regulation is applied at the LL level. Packets may be dropped because the application rate is higher than the rate applied at the LL level. Regulate the rate at the application layer would remove this packet loss but implies another cross-layer interaction that we decided to not consider in this study and to let to future work.

http://www.isi.edu/nsnam/ns/


Pierre Dolbeault

BOUNDARIES OF LEVI-FLAT HYPERSURFACES: SPECIAL HYPERBOLIC POINTS

Let S ⊂ C n , n ≥ 3 be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface whose boundary is S, possibly as a current. Our goal is to get examples of such S containing at least one special 1-hyperbolic point: sphere with two horns; elementary models and their gluing. The particular cases of graphs are also described.

Introduction

Let S ⊂ C n , be a compact connected 2-codimensional submanifold having the following property: there exists a Levi-flat hypersurface M ⊂ C n \ S such that dM = S (i.e. whose boundary is S, possibly as a current). The case n = 2 has been intensively studied since the beginning of the eighties, in particular by Bedford, Gaveau, Klingenberg; Shcherbina, Chirka, G. Tomassini, Slodkowski, Gromov, Eliashberg; it needs global conditions: S has to be contained in the boundary of a srictly pseudoconvex domain.

We consider the case n ≥ 3; results on this case has been obtained since 2005 by Dolbeault, Tomassini and Zaitsev, local necessary conditions recalled in section 2 have to be satisfied by S, the singular CR points on S are supposed to be elliptic and the solution M is obtained in the sense of currents [START_REF] Dolbeault | On boundaries of Levi-flat hypersurfaces in C n[END_REF][START_REF] Dolbeault | On Levi-flat hypersurfaces with prescribed boundary[END_REF]. More recently a regular solution M has been obtained when S satisfies a supplementary global condition as in the case n = 2 [START_REF] Dolbeault | Boundary problem for Levi flat graphs[END_REF], the singular CR points on S still supposed to be elliptic.

The problem we are interested in is to get examples of such S containing at least one special 1-hyperbolic point (section 2.4). The CR-orbits near a special 1-hyperbolic point are large and, assuming them compact, a careful examination has to be done (sections 2.6, 2.7). As a topological preliminary, we need a generalization of a theorem of Bishop on the difference of the numbers of special elliptic and 1-hyperbolic points (section 2.8); this result is a particular case of a theorem of Hon-Fei Lai [START_REF] Fei | Characteristic classes of real manifolds immersed in complex manifolds[END_REF].

The first considered example is the sphere with two horns which has one special 1-hyperbolic point and three special elliptic points (section 3.4). Then we consider elementary models and their gluing to obtain more complicated examples (section 3.5). Results have been announced in [START_REF] Dolbeault | On Levi-flat hypersurfaces with given boundary in C n[END_REF], and Date: January 7, 2013.

in more precise way in [START_REF] Dolbeault | Complex plateau problem: old and new results and prospects[END_REF]; the first aim of this paper is to give complete proofs. Finally, we recall in detail and extend the results of [START_REF] Dolbeault | Boundary problem for Levi flat graphs[END_REF] on regularity of the solution when S is a graph satisfying a supplementary global condition, as in the case n = 2, to the case of existence of special 1-hyperbolic points, and to gluing of elemetary smooth models (section 4).

2. Preliminaries: local and global properties of the boundary 2.1. Definitions. A smooth, connected, CR submanifold M ⊂ C n is called minimal at a point p if there does not exist a submanifold N of M of lower dimension through p such that HN = HM | N . By a theorem of Sussman, all possible submanifolds N such that HN = HM | N contain, as germs at p, one of the minimal possible dimension, defining a so called CR orbit of p in M whose germ at p is uniquely determined.

Let S be a smooth compact connected oriented submanifold of dimension 2n -2. S is said to be a locally flat boundary at a point p if it locally bounds a Levi-flat hypersurface near p. Assume that S is CR in a small enough neighborhood U of p ∈ S. If all CR orbits of S are 1-codimensional (which will appear as a necessary condition for our problem), the following two conditions are equivalent [START_REF] Dolbeault | On boundaries of Levi-flat hypersurfaces in C n[END_REF]:

(i) S is a locally flat boundary on U ;

(ii) S is nowhere minimal on U .

2.2. Complex points of S. (i.e. singular CR points on S) [START_REF] Dolbeault | On boundaries of Levi-flat hypersurfaces in C n[END_REF]. At such a point p ∈ S, T p S is a complex hyperplane in T p C n . In suitable local holomorphic coordinates (z, w) ∈ C n-1 ×C vanishing at p, with w = z n and z = (z 1 , . . . , z n-1 ), S is locally given by the equation

(1) w = Q(z) + O(|z| 3 ), Q(z) = 1≤i,j≤n-1 (a ij z i z j + b ij z i z j + c ij z i z j )
S is said flat at a complex point p ∈ S if b ij z i z j ∈ λR, λ ∈ C. We also say that p is flat.

Let S ⊂ C n be a locally flat boundary with a complex point p. Then p is flat.

By making the change of coordinates (z, w) → (z, λ -1 w), we get b ij z i z j ∈ R for all z. By a change of coordinates (z, w) → (z, w + a ′ ij z i z j ) we can choose the holomorphic term in (1) to be the conjugate of the antiholomorphic one and so make the whole form Q real-valued.

We say that S is in a flat normal form at p if the coordinates (z, w) as in (1) are chosen such that Q(z) ∈ R for all z ∈ C n-1 . 2.2.1. Properties of Q. Assume that S is in a flat normal form; then, the quadratic form Q is real valued. If Q is positive definite or negative definite, the point p ∈ S is said to be elliptic; if the point p ∈ S is not elliptic, and if Q is non degenerate, p is said to be hyperbolic. From section 2.4, we will only consider particular cases of the quadratic form Q.

Elliptic points.

2.3.1. Properties of Q.

Proposition 1. ( [START_REF] Dolbeault | On boundaries of Levi-flat hypersurfaces in C n[END_REF][START_REF] Dolbeault | On Levi-flat hypersurfaces with prescribed boundary[END_REF]). Assume that S ⊂ C n , (n ≥ 3) is nowhere minimal at all its CR points and has an elliptic flat complex point p. Then there exists a neighborhood V of p such that V \ {p} is foliated by compact real (2n -3)-dimensional CR orbits diffeomorphic to the sphere S 2n-3 and there exists a smooth function ν, having the CR orbits as the level surfaces.

Sketch of Proof. (see [START_REF] Dolbeault | On Levi-flat hypersurfaces with prescribed boundary[END_REF]). In the case of a quadric S 0 (w = Q(z)), the CR orbits are defined by w 0 = Q(z), where w 0 is constant. Using (1), we approximate the tangent space to S by the tangent space to S 0 at a point with the same coordinate z; the same is done for the tangent spaces to the CR orbits on S and S 0 ; then we construct the global CR orbit on S through any given point close enough to p.

2.4. Special flat complex points. From [START_REF] Bishop | Differentiable manifolds in complex Euclidean space[END_REF], for n = 2, in suitable local holomorphic coordinates centered at 0, Q(z) = (zz + λRe z 2 ), λ ≥ 0, under the notations of [START_REF] Bedford | On the envelope of holomorphy of a 2-sphere in C 2[END_REF]; for 0 ≤ λ < 1, p is said to be elliptic, and for 1 < λ, it is said to be hyperbolic. The parabolic case λ = 1, not generic, will be omitted [START_REF] Bedford | On the envelope of holomorphy of a 2-sphere in C 2[END_REF]. When n ≥ 3, the Bishop's reduction cannot be generalized.

We say that the flat complex point p ∈ S is special if in convenient holomorphic coordinates centered at 0, (2)

Q(z) = n-1 j=1
(z j z j + λ j Re z 2 j ), , λ j ≥ 0 Let z j = x j + iy j , x j , y j real, j = 1, . . . , n -1, then:

(3)

Q(z) = n-1 l=1 (1 + λ l )x 2 l + (1 -λ l )y 2 l + O(|z| 3
). A flat point p ∈ S is said to be special elliptic if 0 ≤ λ j < 1 for any j. A flat point p ∈ S is said to be special k-hyperbolic if 1 < λ j for j ∈ J ⊂ {1, . . . , n -1} and 0 ≤ λ j < 1 for j ∈ {1, . . . , n -1} \ J = ∅, where k denotes the number of elements of J.

Special elliptic (resp. special k-hyperbolic) points are elliptic (resp. hyperbolic).

Special flat complex points 2.5. Special hyperbolic points. S being given by (1), let S 0 be the quadric of equation w = Q(z).

Lemma 2. Suppose that S 0 is flat at 0 and that 0 is a special k-hyperbolic point. Then, in a neighborhood of 0, and with the above local coordinates, S 0 is CR and nowhere minimal outside 0, and the CR orbits of S 0 are the (2n -3)-dimensional submanifolds given by w = const. = 0.

Proof. The submanifolds w = const. = 0 have the same complex tangent space as S 0 and are of minimal dimension among submanifolds having this property, so they are CR orbits of codimension 1, and from the end of section 2.1, S 0 is nowhere minimal outside 0.

The section w = 0 of S 0 is a real quadratic cone Σ ′ 0 in R 2n whose vertex is 0 and, outside 0, it is a CR orbit Σ 0 in the neighborhood of 0. We will improperly call Σ ′ 0 a singular CR orbit.

2.6. Foliation by CR-orbits in the neighborhood of a special 1hyperbolic point. We first mimic and transpose the begining of the proof of Proposition 1, i.e. of 2.4.2. in ([DTZ05, DTZ09]).

2.6.1. Local 2-codimensional submanifolds. In order to use simple notations, we will assume n = 3. In C 3 , consider the 4-dimensional submanifold S locally defined by the equation

(1) w = ϕ(z) = Q(z) + O(|z| 3 )
and the 4-dimensional submanifold S 0 of equation

(4) w = Q(z) with Q = (λ 1 + 1)x 2 1 -(λ 1 -1)y 2 1 + (1 + λ 2 )x 2 2 + (1 -λ 2 )y 2 2
having a special 1-hyperbolic point at 0, (λ 1 > 1, 0 ≤ λ 2 < 1), and the cone Σ ′ 0 whose equation is: Q = 0. On S 0 , a CR orbit is the 3-dimensional submanifold K w 0 whose equation is

w 0 = Q(z). If w 0 > 0, K w 0 does not cut the line L = {x 1 = x 2 = y 2 = 0}; if w 0 < 0, K w 0 cuts L at two points. Lemma 3. Σ 0 = Σ ′ 0 \ 0 has two connected components in a neighborhood of 0. Proof. The equation of Σ ′ 0 ∩ {y 1 = 0} is (λ 1 + 1)x 2 1 + (1 + λ 2 )x 2 2 + (1 -λ 2 )
y 2 2 = 0 whose only zero , in the neighborhood of 0, is {0}: the connected components are obtained for y 1 > 0 and y 1 < 0 respectively.

Local 2-codimensional submanifolds 2.6.2. CR-orbits. By differentiating (1), we get for the tangent spaces the following asymptotics (5)

T (z,ϕ(z) )S = T (z,Q(z) )S 0 + O(|z| 2 ), z ∈ C 2
Here both T (z,ϕ(z) )S and T (z,Q(z) )S 0 depend continuously on z near the origin.

Consider (i) the hyperboloïd H

-= {Q = -1}, (then Q( z (-Q(z)) 1/2 ) = -1)
, and the projection:

π -: C 3 \ {z = 0} → H -, (z, w) → z (-Q(z)) 1/2 ,
(ii) for every z ∈ H -, a real orthonormal basis e 1 (z), . . . , e 6 (z) of

C 3 ∼ = R 6 such that e 1 (z), e 2 (z) ∈ H z H -, e 3 (z) ∈ T z H -, where HH -is the complex tangent bundle to H -.
Locally such a basis can be chosen continuously depending on z. For every (z, w) ∈ C 3 \ {z = 0}, consider the basis e 1 (π -(z, w)), . . . , e 6 (π -(z, w)). The unit vectors e 1 (π -(z, w 0 )), e 2 (π -(z, w 0 )), e 3 (π -(z, w 0 )) are tangent to the CR orbit K w 0 in (z, w 0 ) for w 0 < 0. Then, from (5), we have:

(6) H (z,ϕ(z) )S = H (z,Q(z) )S 0 + O(|z| 2 ), z = 0, z → 0.
As in [START_REF] Dolbeault | On Levi-flat hypersurfaces with prescribed boundary[END_REF], in the neighborhood of 0, denote by E(q), q ∈ S\{0}, w < 0 the tangent space to the local CR orbit K on S through q, and by E 0 (q 0 ), q 0 ∈ S 0 \ {0}, w < 0 the analogous object for S 0 . We have :

(7) E(z, ϕ(z)) = E 0 (z, Q(z)) + O(|z| 2 ), z = 0, z → 0
Given q ∈ S, by integration of E(q), q ∈ S, we get, locally, the CR orbit (the leaf), on S through q; given q 0 ∈ S 0 , by integration of E 0 (q 0 ), q 0 ∈ S 0 , we get, locally, the CR orbit (the leaf), on S 0 through q 0 (theorem of Sussman). On S 0 , a leaf is the 3-dimensional submanifold K q 0 = K w 0 = K 0 whose equation is w 0 = Q(z), with q = (z 0 , w 0 = Q(z 0 )). dπ -projects each E 0 (q), q ∈ S 0 , w < 0, bijectively onto T π(q) H -, then π -| K 0 is a diffeomorphism onto H -; this implies, from (7), that, in a suitable neighborhood of the origin, the restriction of π -to each local CR orbit of S is a local diffeomorphism.

We have:

ϕ(z) = Q(z) + Φ(z) with Φ(z) = O(|z| 3 ).
2.6.3. Behaviour of local CR orbits. Follow the construction of E(z, ϕ(z)); compare with E 0 (z, Q(z)). We know the integral manifold, the orbit of E 0 (z, Q(z)); deduce an evaluation of the integral manifold K of E(z, ϕ(z)).

Lemma 4. Under the above hypotheses, the local orbit Σ corresponding to Σ 0 has two connected components in the neighborhood of 0.

Proof. Using the real coordinates, as for Lemma 3, Σ ′ ∩ {y 1 = 0}. Locally, the connected components are obtained for y 1 > 0 and y 1 < 0 respectively, from formula (1).

We will improperly call Σ ′ = Σ a singular CR orbit and a singular leaf of the foliation.

We intend to prove: 1) K does not cross the singular leaf through 0;

2) the only separatrix is the singular leaf through 0.

From the orbit K 0 , construct the differential equation defining it, and using (7), construct the differential equation defining K.

In C 3 , we use the notations: x = x 1 , y = y 1 , u = x 2 , v = y 2 ; it suffices to consider the particular case: Q = 3x 2y 2 + u 2 + v 2 . On S 0 , the orbit K 0 issued from the point (c, 0, 0, 0) is defined by: 3x 2y 2u 2 + v 2 = 3c 2 , i.e., for x ≥ 0, x = 1 √ 3 (y 2u 2v 2 + 3c 2 ) 1 2 = A(y, u, v); the local coordinates on the orbit are (y, u, v). K 0 satisfies the differential equation: dx = dA. From (9), the orbit K, issued from (c, 0, 0, 0), satisfies dx = dA + Ψ with Ψ(y, u, v; c) = O(|z| 2 ); hence Ψ = dΦ, then x = A + Φ, with Φ = O(|z| 3 ). More explicitly, K is defined by:

x = x K,c = 1 √ 3 (y 2 -u 2 -v 2 + 3c 2 ) 1 2 + Φ(y, u, v; c), Φ(y, u, v; c) = O(|z| 3 )
The cone Σ ′ 0 whose equation is: Q = 0 is a separatrix for the orbits K 0 . The corresponding object Σ ′ = {ϕ(z) = 0} for S has the singular point 0 and for x > 0, y > 0, u > 0, v > 0 is defined by the differential equation dx = d(A + Φ), with c = 0, i.e. the local equation of Σ ′ is

x = x K,0 = 1 √ 3 (y 2 -u 2 -v 2 ) 1 2 + Φ(y, u, v; 0), Φ(y, u, v; 0) = O(|z| 3 ) For given (y, u, v), x K,c -x K,0 = x K 0 ,c -x K 0 ,0 + Φ(y, u, v; c) -Φ(y, u, v; 0). But x K 0 ,c -x K 0 ,0 = O(1) and Φ(y, u, v; c) -Φ(y, u, v; 0) = O(|z| 3 ).
As a consequence, for x > 0, y > 0, u > 0, v > 0, locally, Σ ′ is a separatrix for the orbits K, and the only one. Same result for x < 0. 2.6.4. What has been done from the hyperboloïd H -= {Q = -1} can be repeated from the hyperboloïd H + = {Q = 1}.

As at the beginning of the section 2.6.2, we consider (i) the hyperboloïd H + {Q = 1} and the projection:

π + : C 3 \ {z = 0} → H + , (z, w) → z (Q(z)) 1/2 ,
(ii) for every z ∈ H + , a real orthonormal basis e 1 (z), . . . , e 6 (z) of

C 3 ∼ = R 6 such that e 1 (z), e 2 (z) ∈ H z H + , e 3 (z) ∈ T z H + ,
where HH + is the complex tangent bundle to H + .

2.6.5.

Lemma 5. Given ϕ, there exists R > 0 such that, in B(0, R) ∩ {x > 0, y > 0, u > 0, v > 0} ⊂ C 2 , the CR orbits K have Σ ′ as unique separatrix.

Proof. When c tends to zero, ,

x K,c -x K,0 = x K 0 ,c -x K 0 ,0 = O(|z|), Φ(y, u, v; c) -Φ(y, u, v; 0) = O(|z| 3 ). For ϕ(z) = Q(z) + Φ(z) with Φ(z) = O(|z| 3 ) given, in (9), E(z, ϕ(z)) -E 0 (z, Q(z)) = O(|z| 2
) and Φ(y, u, v; c) -Φ(y, u, v; 0) = O(|z| 3 ) are also given. Then there exists R such that, for |z| < R, x K,cx K,0 > 0.

2.7. CR-orbits near a subvariety containing a special 1-hyperbolic point.

2.7.1. In the section 2.7, we will impose conditions on S and give a local property in the neighborhood of a compact (2n -3)-subvariety of S.

Assume that S ⊂ C n (n ≥ 3), is a locally closed (2n -2)-submanifold, nowhere minimal at all its CR points, which has a unique 1-hyperbolic flat complex point p, and such that:

(i) Σ being the orbit whose closure Σ ′ contains p, then Σ ′ is compact. Let q ∈ S, q = p; then, in a neighborhood U of q disjoint from p, S is CR, CR-dim S = n -2, S is non minimal and Σ is 1-codimensional. To show that the CR orbits contitute a foliation on S whose separatrix is Σ ′ : this is true in U since Σ ∩ U is a leaf. Moreover, let U 0 the ball B(0, R) centered in p = 0 in Lemma 5, if U ∩ U 0 = ∅, the leaves in U glue with the leaves in U 0 on U ∩ U 0 . Since Σ ′ is compact, there exists a finite number of points q j ∈ Σ ′ , j = 0, 1, . . . , J, and open neighborhoods U j , as above, such that (U j ) J

j=0 is an open covering of Σ ′ . Moreover the leaves on U j glue respectively with the leaves on

U k if U j ∩ U h = ∅.

2.7.2.

Proposition 6. Assume that S ⊂ C n (n ≥ 3), is a locally closed (2n -2)-submanifold, nowhere minimal at all its CR points, which has a unique special 1-hyperbolic flat complex point p, and such that:

(i) Σ being the orbit whose closure Σ ′ contains p, then Σ ′ is compact;

(ii) Σ has two connected components σ 1 , σ 2 , whose closures are homeomorphic to spheres of dimension 2n -3.

Then, there exists a neighborhood V of Σ ′ such that V \ Σ ′ is foliated by compact real (2n -3)-dimensional CR orbits whose equation, in a neighborhood of p is (3), and, the w(= x n )-axis being assumed to be vertical, each orbit is diffeomorphic to the sphere S 2n-3 above Σ ′ , the union of two spheres S 2n-3 under Σ ′ , and there exists a smooth function ν, having the CR orbits as the level surfaces.

Proof. From subsection 2.7.1 and the following remark: When x n tends to 0, the orbits tends to Σ ′ , and because of the geometry of the orbits near p, they are diffeomorphic to a sphere above Σ ′ , and to the union of two spheres under Σ ′ . The existence of ν is proved as in Proposition 1, namely, consider a smooth curve γ : [0, ε) → S such that γ(0) = q, where q is a point of Σ close to p, and γ is a diffeomorphism onto its image Γ = γ([0, ε)). Let ν = γ -1 on the image of γ, then, close enough to q, every CR orbit cuts Γ at a unique point q(t), t ∈ [0, ε). Hence there is a unique extension of ν from γ([0, ε)) to V \ p where V is a neighborhood of Σ ′ having CR orbits as its level surfaces. ν being smooth away from p, it is smooth on the orbit Σ and, if we set ν(p) = ν(q) = 0, ν is smooth on a neighborhood of Σ ∪ {p} = Σ ′ . 2.8. Geometry of the complex points of S. The results of section 2.8 are particular cases of theorems of H-F Lai [START_REF] Fei | Characteristic classes of real manifolds immersed in complex manifolds[END_REF], that I learnt from F. Forstneric in July 2011.

In [BK91] E. Bedford & W. Klingenberg cite the following theorem of E. Bishop [START_REF] Bishop | Differentiable manifolds in complex Euclidean space[END_REF][section 4, p.15]: On a 2-sphere embedded in C 2 , the difference between the numbers of elliptic points and of hyperbolic points is the Euler-Poincaré characteristic, i.e. 2. For the proof, Bishop uses a theorem of ([CS 51], section 4).

We extend the result for n ≥ 3 and give proofs which are essentially the same than in the general case of [Lai72, Lai74] but simpler.

2.8.1. Let S be a smooth compact connected oriented submanifold of dimension 2n -2. Let G be the manifold of the oriented real linear (2n -2)subspaces of C n . The submanifold S of C n has a given orientation which defines an orientation o(p) of the tangent space to S at any point p ∈ S. By mapping each point of S into its oriented tangent space, we get a smooth Gauss map t : S → G Denote -t(p) the tangent space to S at p with opposite orientation -o(p).

2.8.2. Properties of G. (a) dim G = 2(2n -2).
Proof. G is a two-fold covering of the Grassmannian M m,k , of the linear k-subspaces of R m [Ste99][Part, section 7.9], for m = 2n and k = 2n -2; they have the same dimension. We have:

M m,k ∼ = O m /O k × O m-k But dim O k = 1 2 k(k -1), hence dim M m,k = 1 2 m(m -1) -k(k -1) - (m -k)(m -k -1) = k(m -k).
(b) G has the complex structure of a smooth quadric of complex dimension (2n -2) of CP 2n-1 L74, [START_REF] Polo | Grassmanniennes orientées réelles[END_REF].

(c) There exists a canonical isomorphism h : ]): Let S 1 , S 2 be generators of H 2n-2 (G, Z); we assume that S 1 and S 2 are fundamental cycles of complex projective subspaces of complex dimension (n -1) of the complex quadric G. We also denote S 1 , S 2 the ordered two factors CP n-1 , so that h : G → S 1 × S 2 .

G → CP n-1 × CP n-1 . (d) Homology of G (cf [ Pol08

2.8.3.

Proposition 7. For n ≥ 2, in general, S has isolated complex points.

Proof. Let π ∈ G be a complex hyperplane of C n whose orientation is induced by its complex structure; the set of such π is

H = G C n-1,n = CP n-1 * ⊂ G, as real submanifold. If p is a complex point of S, then t(p) ∈ H or -t(p) ∈ H. The set of complex points of S is the inverse image by t of the intersections t(S) ∩ H and -t(S) ∩ H in G. Since dim t(S) = 2n -2, dim H = 2(n -1), dim G = 2(2n -2), the intersection is 0-dimensional, in general.
2.8.4. Denoting also S, the fundamental cycle of the submanifold S and t * the homomorphism defined by t, we have:

t * (S) ∼ u 1 S 1 + u 2 S 2
where ∼ means homologous to.

2.8.5.

Lemma 8 (proved for n = 2 in [START_REF] Shiing-Shen Chern | A theorem on orientable surfaces in fourdimensional space[END_REF]). With the above notations, we have:

u 1 = u 2 ; u 1 + u 2 = χ(S), Euler-Poincaré characteristic of S.
The proof for n = 2 works for any n ≥ 3, namely: Let G ′ be the manifold of the oriented real linear 2-subspaces of C n . Let α : G → G ′ map each oriented 2(n -1)-subspace R onto its normal 2subspace R ′ oriented so that R, R ′ determine the orientation of C n . α is a canonical isomorphism. Let n : S → G ′ the map defined by taking oriented normal planes; then: n = αt and t = α -1 n, hence the mapping hαh -1 : S 1 ×S 2 → S 1 ×S 2 . Let (x, y) be a point of S 1 ×S 2 , then ( †) hαh -1 (x, y) = (x, -y).

Over G, there is a bundle V of spheres obtained by considering as fiber over a real oriented linear (2n -2)-subspace of C n through 0 the unit sphere S 2n-3 of this subspace. Let Ω be the characteristic class of V , and let Ω t , Ω n denote the characteristic classes of the tangent and normal bundles of S. Then t * Ω = Ω t , n * Ω = Ω n .

V is the Stiefel manifold of ordered pairs of orthogonal unit vectors through in

R 2n ∼ = C n . Let f : V → G the projection.
From the Gysin sequence, we see that the kernel of f * :

H 2n-2 (G) → H 2n-2 (V ) is generated by Ω. To find the kernel of f * , we determine the morphism f * : H 2n-2 (V ) → H 2n-2 (G). A generating 2n -2)-cycle of in V is S 2 × e
where S 2 ∼ = CP n-1 and e is a point. Let z be any point of S 2 , then from ( †), we have hf (z, e) = (z, -z) Therefore, we see that

f * (S 2 × e) = S 1 -S 2 . Then, the kernel of f * is Z-generated by S * 1 + S * 2 .
With convenient orientation for the fibre of the bundle V , we get: Ω = S * 1 + S * 2 . For convenient orientation of S, we get Ω t .S = χ S = Euler characteristic of S. We have

Ω t = t * (S * 1 + S * 2 ) = t * S * 1 + t * S * 2 Ω n = n * (S * 1 + S * 2 ) = t * α * (S * 1 + S * 2 ) = t * (S * 1 -S * 2 ) = t * S * 1 -t * S * 2
Since Ω n = 0, we get:

(t * S * 1 ).S = (t * S * 2 ).S = 1 2 χ S
2.8.6. Local intersection numbers of H and t(S) when all complex points are flat and special. H is a complex linear (n -1)-subspace of G, then is homologous to one of the S j , j = 1, 2, say S 2 when G has its structure of complex quadric. The intersection number of H and S 1 is 1 and the intersection number of H and S 2 is 0. So, the intersection number of H and

u 1 S 1 + u 2 S 2 is u 1 .
In the neighborhood of a complex point 0, S is defined by equation (1), with w = z n and

(1 ′ ) Q(z) = n-1 j=1 µ j (z j z j + λ j Re z 2 j ), µ j > 0, λ j ≥ 0 Let z j = x 2j-1 + ix 2j , j = 1, .
. . , n, with real x l . Let e l the unit vector of the x l axis, l = 1, . . . , 2n.

For simplicity assume n = 3:

Q(z) = µ 1 (z 1 z 1 + λ 1 Re z 2 1 ) + µ 2 (z 2 z 2 + λ 2 Re z 2
2 ), with µ 1 = µ 2 = 1. Then, up to higher order terms, S is defined by:

z 1 = x 1 + ix 2 ; z 2 = x 3 + ix 4 ; z 3 = (1 + λ 1 )x 2 1 + (1 -λ 1 )x 2 2 + (1 + λ 2 )x 2 3 + (1 -λ 2 )x 2
4 . In the neighborhood of 0, the tangent space to S is defined by the four independent vectors

ν 1 = e 1 +2(1+λ 1 )x 1 e 5 ; ν 2 = e 2 +2(1-λ 1 )x 2 e 5 ; ν 3 = e 3 +2(1+λ 2 )x 3 e 5 ; ν 4 = e 4 + 2(1 -λ 2 )x 4 e 5
Then, if 0 is special elliptic or special k-hyperbolic with k even, the tangent plane at 0 has the same orientation; if 0 is special elliptic or special k-hyperbolic with k odd the tangent space has opposite orientation.

2.8.7.

Proposition 9 (known for n = 2 [START_REF] Bishop | Differentiable manifolds in complex Euclidean space[END_REF], here for n ≥ 3). Let S be a smooth, oriented, compact, 2-codimensional, real submanifold of C n whose all complex points are flat and special elliptic or special 1-hyperbolic. Then, on S, ♯ (special elliptic points) -♯ (special 1-hyperbolic points = χ(S). If S is a sphere, this number is 2. Proof. Let p ∈ S be a complex point and π be the tangent hyperplane to S at π. Assume that (**) the orientation of S induces, on π, the orientation given by its complex structure, then π ∈ H.

If p is elliptic, the intersection number of H and t(S) is 1; if p is 1hyperbolic, the intersection number of H and t(S) is -1 at p.

From the beginning of section 2.8.6, the sum of the intersection numbers of H and t(S) at complex points p satisfying (**) is u 1 . Reversing the condition (**), and using Lemma 8, we get the Proposition.

3. Particular cases: horned sphere; elementary models and their gluing 3.1. We recall the following Harvey-Lawson theorem with real parameter to be used later.

3.1.1. Let E ∼ = R × C n-1
, and k : R × C n-1 → R be the projection. Let N ⊂ E be a compact, (oriented) CR subvariety of C n+1 of real dimension 2n -2 and CR dimension n -2, (n ≥ 3), of class C ∞ , with negligible singularities (i.e. there exists a closed subset τ ⊂ N of (2n -2)-dimensional Hausdorff measure 0 such that N \ τ is a CR submanifold). Let τ ′ be the set of all points z ∈ N such that either z ∈ τ or z ∈ N \ τ and N is not transversal to the complex hyperplane k -1 (k(z)) at z. Assume that N , as a current of integration, is d-closed and satisfies: (H) there exists a closed subset L ⊂ R x 1 with H 1 (L) = 0 such that for every x ∈ k(N ) \ L, the fiber k -1 (x) ∩ N is connected and does not intersect τ ′ .

3.1.2.

Theorem 10 ([DTZ10] (see also [START_REF] Dolbeault | On boundaries of Levi-flat hypersurfaces in C n[END_REF])). Let N satisfy (H) with L chosen accordingly. Then, there exists, in E ′ = E \ k -1 (L), a unique C ∞ Levi-flat (2n -1)-subvariety M with negligible singularities in E ′ \ N , foliated by complex (n -1)-subvarieties, with the properties that M simply (or trivially) extends to E ′ as a (2n -1)-current (still denoted M ) such that dM = N in E ′ .1 The leaves are the sections by the hyperplanes E x 0 1 , x 0 1 ∈ k(N ) \ L, and are the solutions of the "Harvey-Lawson problem" for finding a holomorphic subvariety in

E x 0 1 ∼ = C n with prescribed boundary N ∩ E x 0 1 . 3.1.3.
Remark 11. Theorem 10 is valid in the space E ∩ {α 1 < x 1 < α 2 }, with the corresponding condition (H). Moreover, since N is compact, for convenient coordinate x 1 , we can assume x 1 ∈ [0, 1].

3.2.

To solve the boundary problem by Levi-flat hypersurfaces, S has to satisfy necessary and sufficient local conditions. A way to prove that these conditions can occur is to construct an example for which the solution is obvious.

3.3. Sphere with one special 1-hyperbolic point (sphere with two horns): Example.

3.3.1. In C 3 , let (z j ), j = 1, 2, 3, be the complex coordinates and z j = x j + iy j . In R 6 ∼ = C 3 , consider the 4-dimensional subvariety (with negligible singularities) S defined by:

y 3 = 0 0 ≤ x 3 ≤ 1; x 3 (x 2 1 + y 2 1 + x 2 2 + y 2 2 + x 2 3 -1) + (1 -x 3 )(x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 -2y 2 1 + x 2 2 + y 2 2 ) = 0 -1 ≤ x 3 ≤ 0; x 3 = x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 -2y 2 1 + x 2 2 + y 2 2
The singular set of S is the 3-dimensional section x 3 = 0 along which the tangent space is not everywhere (uniquely) defined. S being in the real hyperplane {y 3 = 0}, the complex tangent spaces to S are {x 3 = x 0 } for convenient x 0 .

The tangent space to the hypersurface

f (x 1 , y 1 , x 2 , y 2 , x 3 ) = 0 in R 5 is X 1 f ′ x 1 + Y 1 f ′ y 1 + X 2 f ′ x 2 + Y 2 f ′ y 2 + X 3 f ′
x 3 = 0, Then, the tangent space to S in the hyperplane {y 3 = 0} is:

for 0 ≤ x 3 , 2x 1 [x 3 + 2(1 -x 3 )(x 2 1 + 2)]X 1 + 2y 1 [x 3 + 2(1 -x 3 )(y 2 1 -1)]Y 1 + 2x 2 [x 3 + (1 -x 3 )(2x 2 2 + 1)]X 2 + 2y 2 [x 3 + (1 -x 3 )(2y 2 2 + 1)]Y 2 + [(x 2 1 + y 2 1 + x 2 2 + y 2 2 + 3x 2 3 -1) -(x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 -2y 2 1 + x 2 2 + y 2 2 )]X 3 = 0; for x 3 ≤ 0, 4(x 2 1 + 2)x 1 X 1 + 4(y 2 1 -1)y 1 Y 1 + 2(2x 2 2 + 1)x 2 X 2 + 2(2y 2 2 + 1)y 2 Y 2 -X 3 = 0.
3.3.3. The complex points of S are defined by the vanishing of the coefficients of X j , j=1,2,3,4 in the equation of the tangent spaces for 0

≤ x 3 ≤ 1, x 1 [x 3 + 2(1 -x 3 )(x 2 1 + 2)] = 0, y 1 [x 3 + 2(1 -x 3 )(y 2 1 -1)] = 0, x 2 [x 3 + (1 -x 3 )(2x 2 2 + 1)] = 0, y 2 [x 3 + (1 -x 3 )(2y 2
2 + 1)] = 0. We have the solutions h: x j = 0, y j = 0, (j = 1, 2), x 3 = 0; e 3 : x j = 0, y j = 0, (j = 1, 2), x 3 = 1. for x 3 ≤ 0, (x 2 1 + 2)x 1 = 0, (y 2 1 -1)y 1 = 0, (2x 2 2 + 1)x 2 = 0, (2y 2 2 + 1)y 2 = 0. We have the solutions h: x j = 0, y j = 0, (j = 1, 2), x 3 = 0; e 1 , e 2 : x 1 = 0, y 1 = ±1, x 2 = 0, y 2 = 0, x 3 = -1.

Remark that the tangent space to S at h is well defined. Moreover, the set S will be smoothed along its section by the hyperplane {x 3 = 0} by a small deformation leaving h unchanged. In the following S will denote this smooth submanifold.

3.3.4.

Lemma 12. The points e 1 , e 2 , e 3 are special elliptic; the point h is special {1}-hyperbolic.

Proof. Point e 3 : Let x ′ 3 = 1-x 3 , then the equation of S in the neighborhood of e 3 is:

(1

-x ′ 3 )(x 2 1 + y 2 1 + x 2 2 + y 2 2 + x ′ 2 3 -2x ′ 3 ) -x ′ 3 (x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 - 2y 2 1 + x 2 2 + y 2 2 ) = 0, i.e. 2x ′ 3 = x 2 1 + y 2 1 + x 2 2 + y 2 2 ) + O(|z| 3 ), or w = zz + O(|z| 3
) then e 3 is special elliptic. Points e 1 , e 2 : Let y ′ 1 = y 1 ± 1, x ′ 3 = x 3 + 1, then the equation of S in the neighborhood of e 1 , e 2 is:

x ′ 3 -1 = x 4 1 + (y ′ 1 ∓ 1) 4 + x 4 2 + y 4 2 + 4x 2 1 -2(y ′ 1 ∓ 1) 2 + x 2 2 + y 2 2 = x 4 1 + y ′ 4 1 ∓ 4y ′ 3 1 + 6y ′ 2 1 ∓ 4y ′ 1 + 1 + x 4 2 + y 4 2 + 4x 2 1 -2(y ′ 1 ∓ 1) 2 + x 2 2 + y 2 2 , then x ′ 3 = x 4 1 + y ′ 4 1 ∓ 4y ′ 3 1 + 4y ′ 2 1 + x 4 2 + y 4 2 + 4x 2 1 + x 2 2 + y 2 2 , i.e. x ′ 3 = 4x 2 1 + 4y ′ 2 1 + x 2 2 + y 2 2 + O(|z| 3
), or w = 4z 1 z 1 + z 2 z 2 , then e 1 , e 2 are special elliptic. Point h: The equation of S in the neighborhood of h is: for x 3 ≥ 0,

x 3 (x 2 1 + y 2 1 + x 2 2 + y 2 2 + x 2 3 -1) + (1 -x 3 )(x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 -2y 2 1 + x 2 2 + y 2 2 ) = 0 for x 3 ≤ 0, x 3 = x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 -2y 2 1 + x 2 2 + y 2 2 , i.e. x 3 = 4x 2 1 -2y 2 1 + x 2 2 + y 2 2 + O(|z| 3
), in both cases, up to the third order terms, i.e.: w = z 1 z 1 + z 2 z 2 + 3Re z 2 1 , then h is special {1}-hyperbolic.

3.3.5. Section Σ ′ = S ∩ {x 3 = 0}. Up to a small smooth deformation, its equation is:

x 4 1 + y 4 1 + x 4 2 + y 4 2 + 4x 2 1 -2y 2 1 + x 2 2 + y 2 2 = 0, in {x 3 = 0}. The tangent cone to Σ ′ at 0 is: 4x 2 1 -2y 2 1 + x 2 2 + y 2 2 = 0. Locally, the section of S by the coordinate 3-space

x 1 , y 1 , x 3 is:

x 3 = 4x 2 1 -2y 2 1 + O(|z| 3 ) x 2 , y 2 , x 3 is: x 3 = x 2 2 + y 2 2 + O(|z| 3 ) 3.3.1'. Shape of Σ ′ = S ∩ {x 3 = 0} in the neighborhood of the origin 0 of C 3 .
Lemma 13. Under the above hypotheses and notations, (i) Σ = Σ ′ \ 0 has two connected components σ 1 , σ 2 .

(ii) The closures of the three connected components of S \ Σ ′ are submanifolds with boundaries and corners.

Proof. (i) The only singular point of Σ ′ is 0. We work in the ball B(0, A) of C 2 (x 1 , y 1 , x 2 , y 2 ) for small A and in the 3-space π λ = {y 2 = λx 2 }, λ ∈ R. For λ fixed, π λ ∼ = R 3 (x 1 , y 1 , x 2 ), and Σ ′ ∩ π λ is the cone of equation 4x 2 1 -2y 2 1 + (1 + λ 2 )x 2 2 + O(|z| 3 ) = 0 with vertex 0 and basis in the plane x 2 = x 0 2 the hyperboloid H λ of equation 4x 2 1 -2y 2 1 +(1+λ 2 )x 02 2 +O(|z| 3 ) = 0; the curves H λ have no common point outside 0. So, when λ varies, the surfaces Σ ′ ∩ π λ are disjoint outside 0. The set Σ ′ is clearly connected; Σ ′ ∩ {y 1 = 0} = {0}, the origin of C 3 ; from above:

σ 1 = Σ ∩ {y 1 > 0}; σ 2 = Σ ∩ {y 1 < 0}.
(ii) The three connected components of S \ Σ ′ are the components which contain, respectively e 1 , e 2 , e 3 and whose boundaries are σ 1 , σ 2 , σ 1 ∪ σ 2 ; these boundaries have corners as shown in the first part of the proof.

The connected component of C 2 ×R\S containing the point (0, 0, 0, 0, 1/2) is the Levi-flat solution, the complex leaves being the sections by the hyperplanes x 3 = x 0 3 , -1 < x 0 3 < 1. The sections by the hyperplanes x 3 = x 0 3 are diffeomorphic to a 3-sphere for 0 < x 0 3 < 1 and to the union of two disjoint 3-spheres for -1 < x 0 3 < 0, as can be shown intersecting S by lines through the origin in the hyperplane x 3 = x 0 3 ; Σ ′ is homeomorphic to the union of two 3-spheres with a common point.

The connected component of C 2 ×R\S containing the point (0, 0, 0, 0, 1/2) is the Levi-flat solution, the complex leaves being the sections by the hyperplanes x 3 = x 0 3 , -1 < x 0 3 < 1. The sections by the hyperplanes x 3 = x 0 3 are diffeomorphic to a 3-sphere for 0 < x 0 3 < 1 and to the union of two disjoint 3-spheres for -1 < x 0 3 < 0, as can be shown intersecting S by lines through the origin in the hyperplane x 3 = x 0 3 ; Σ ′ is homeomorphic to the union of two 3-spheres with a common point.

3.4. Sphere with one special 1-hyperbolic point (sphere with two horns). The example of section 3.3 shows that the necessary conditions of section 2 can be realised. Moreover, from Proposition 2.8.7, the hypothesis on the number of complex points is meaningful.

3.4.1.

Proposition 14. [cf [START_REF] Dolbeault | On Levi-flat hypersurfaces with given boundary in C n[END_REF][Proposition 2.6.1]] Let S ⊂ C n be a compact connected real 2-codimensional manifold such that the following holds:

(i) S is a topological sphere; S is nonminimal at every CR point;

(ii) every complex point of S is flat; there exist three special elliptic points e j , j = 1, 2, 3 and one special 1-hyperbolic point h;

(iii) S does not contain complex manifolds of dimension (n -2);

(iv) the singular CR orbit Σ ′ through h on S is compact and Σ ′ \ {h} has two connected components σ 1 and σ 2 whose closures are homeomorphic to spheres of dimension 2n -3;

(v) the closures S 1 , S 2 , S 3 of the three connected components S ′ 1 , S ′ 2 , S ′ 3 of S \ Σ ′ are submanifolds with (singular) boundary.

Then each S j \ {e j ∪ Σ ′ }, j = 1, 2, 3 carries a foliation F j of class C ∞ with 1-codimensional CR orbits as compact leaves.

Proof. From conditions (i) and (ii), S satisfying the hypotheses of Proposition 1, near any elliptic flat point e j , and of Proposition 6 near Σ ′ , all CR orbits being diffeomorphic to the sphere S 2n-3 . The assumption (iii) guarantees that all CR orbits in S must be of real dimension 2n -3. Hence, by removing small connected open saturated neighborhoods of all special elliptic points, and of Σ ′ , we obtain, from S \ Σ ′ , three compact manifolds S j ", j = 1, 2, 3, with boundary and with the foliation F j of codimension 1 given by its CR orbits whose first cohomology group with values in R is 0, near e j . It is easy to show that this foliation is transversely oriented.

Recall the Thurston's Stability Theorem ([ CaC], Theorem 6.2.1).

Proposition 15. Let (M, F) be a compact, connected, transversely-orientable, foliated manifold with boundary or corners, of codimension 1, of class C 1 . If there is a compact leaf L with H 1 (L, R) = 0, then every leaf is homeomorphic to L and M is homeomorphic to L × [0, 1], foliated as a product, Then, from the above theorem, S j " is homeomorphic to S 2n-3 × [0, 1] with CR orbits being of the form S 2n-3 × {x} for x ∈ [0, 1]. Then the full manifold S j is homeomorphic to a half-sphere supported by S 2n-2 and F j extends to S j ; S 3 having its boundary pinched at the point h.

3.4.3.

Theorem 16. Let S ⊂ C n , n ≥ 3, be a compact connected smooth real 2codimensional submanifold satisfying the conditions (i) to (v) of Proposition 15. Then there exists a Levi-flat (2n -1)-subvariety M ⊂ C × C n with boundary S (in the sense of currents) such that the natural projection π : C × C n → C n restricts to a bijection which is a CR diffeomorphism between S and S outside the complex points of S.

Proof. By Proposition 1, for every e j , a continuous function ν ′ j , C ∞ outside e j , can be constructed in a neighborhood U j of e j , j = 1, 2, 3, and by Proposition 6, we have an analogous result in a neighborhood of Σ ′ . Furthermore, from Proposition 15, a smooth function ν" j whose level sets are the leaves of F j can be obtained globally on S ′ j \ {e j ∪ Σ ′ }. With the functions ν ′ j and ν" j , and analogous functions near Σ ′ , then using a partition of unity, we obtain a global smooth function ν j : S j → R without critical points away from the complex points e j and from Σ ′ .

Let σ 1 , resp. σ 2 be the two connected, relatively compact components of Σ \ {h}, according to condition (iv); σ 1 , resp. σ 2 are the boundary of S 1 , resp. S 2 , and σ 1 ∪ σ 2 the boundary of S 3 . We can assume that the three functions ν j are finite valued and get the same values on σ 1 and σ 2 . Hence a function ν : S → R.

The submanifold S being, locally, a boundary of a Levi-flat hypersurface, is orientable. We now set S = N = gr ν = {(ν(z), z) : z ∈ S}. Let S s = {e 1 , e 2 , e 3 , σ 1 ∪ σ 2 }.

λ : S → S z → ν((z), z) is bicontinuous; λ| S\Ss is a diffeomorphism; moreover λ is a CR map. Choose an orientation on S. Then N is an (oriented) CR subvariety with the negligible set of singularities τ = λ(S s ).

At every point of S \ S s , d x 1 ν = 0, then condition (H) (section 3.1.1) is satisfied at every point of N \ τ .

Then all the assumptions of Theorem 10 being satisfied by N = S, in a particular case, we conclude that N is the boundary of a Levi-flat (2n -2)variety (with negligible singularities) M in R × C n .

Taking π : C × C n → C n to be the standard projection, we obtain the conclusion.

3.5. Generalizations: elementary models and their gluing.

3.5.1.

The examples and the proofs of the theorems when S is homeomorphic to a sphere (sections 3.4) suggest the following definitions.

3.5.2. Definitions. Let T ′ be a smooth, locally closed (i.e. closed in an open set), connected submanifold of C n , n ≥ 3. We assume that T ′ has the following properties:

(i) T ′ is relatively compact, non necessarily compact, and of codimension 2.

(ii) T ′ is nonminimal at every CR point.

(iii) T ′ does not contain complex manifold of dimension (n -2).

(iv) T ′ has exactly 2 complex points which are flat and either special elliptic or special 1-hyperbolic.

(v) If p ∈ T ′ is special 1-hyperbolic, the singular orbit Σ ′ through p is compact, Σ ′ \ p has two connected components σ 1 , σ 2 , whose closures are homeomorphic to spheres of dimension 2n -3.

(vi) If p ∈ T ′ is special 1-hyperbolic, in the neighborhood of p, with convenient coordinates, the equation of T ′ , up to third order terms is

z n = n-1 j=1 (z j z j + λ j Re z 2 j ); λ 1 > 1; 0 ≤ λ j < 1 for j = 1
or in real coordinates x j , y j with z j = x j + iy j ,

x n = (λ 1 + 1)x 2 1 -(λ 1 -1)y 2 1 + n-1 j=2 (1 + λ j )x 2 j + (1 -λ j )y 2 j + O(|z| 3 ) (vii) the closures, in T ′ , T 1 , T 2 , T 3 of the three connected components T ′ 1 , T ′ 2 , T ′ 3 of T ′ \ Σ ′ are
submanifolds with (singular) boundary. Let T " j , j = 1, 2, 3 be neighborhoods of the T ′ j in T ′ . up-and down-1-hyperbolic points. Let τ be the (2n -2)-submanifold with (singular) boundary contained into T ′ such that either σ 1 (resp. σ 2 ) is the boundary of τ near p, or Σ ′ is the boundary of τ near p. In the first case, we say that p is 1-up, (resp. 2-up), in the second that p is down. If T ′ is contained in a small enough neighborhood of Σ ′ in C n , such a T ′ will be called a local elementary model, more precisely it defines a germ of elementary model around Σ.

The union T of T 1 , T 2 , T 3 and of the germ of elementary model around the singular orbit at every special 1-hyperbolic point is called an elementary model. T behaves as a locally closed submanifold still denoted T . 3.5.4. Properties of elementary models. For instance, T is 1-up and has one special elliptic point, we solve the boundary problem as in S 1 in the proof of Theorem 16.

Proposition 17. Let T be a local elementary model. Then, T carries a foliation F of class C ∞ with 1-codimensional CR orbits as compact leaves.

Proof. From the definition at the end of section 3.5.2 and Proposition 6. Proof. By removing small connected open saturated neighborhoods of every special elliptic point, and of Σ ′ , the singular orbit through every special 1hyperbolic point p, we obtain, from S \ Σ ′ , three compact manifolds S j ", j = 1, 2, 3, with boundary, (a) S 1 and S 2 containing one special elliptic point e or one special 1hyperbolic point with the foliations F 1 , F 2 , from Propositions 1 and 17, (b) S 3 " with the foliation F 3 of codimension 1 given by its CR orbits whose first cohomology group with values in R is 0, near e, or p. It is easy to show that this later foliation is transversely oriented.

From the Thurston's Stability Theorem (see section 3.4.2), S 3 " is homeomorphic to S 2n-3 × [0, 1], foliated as a product, with CR orbits being of the form S 2n-3 × {x} for x ∈ [0, 1]; hence smooth functions ν 1 , ν 2 , ν 3 , whose level sets are the leaves of the foliations F 1 , F 2 , F 3 respectively, and using a partition of unity the desired function ν on T .

3.6.

Theorem 19. Let T be an elementary model. Then there exists a Levi-flat (2n -1)-subvariety M ⊂ C × C n with boundary T (in the sense of currents) such that the natural projection π : C × C n → C n restricts to a bijection which is a CR diffeomorphism between T and T outside the complex points of T .

Proof. The submanifold T being, locally, a boundary of a Levi-flat hypersurface, is orientable. We now set T = N = gr ν = {(ν(z), z) : z ∈ S} ⊂ E ∼ = R × C n-1 . Let T s be the union of the flat complex points of T .

λ : T → T z → ν((z), z) is bicontinuous; λ| T \Ts is a diffeomorphism; moreover λ is a CR map. Choose an orientation on T . Then N is an (oriented) CR subvariety with the negligible set of singularities τ = λ(T s ).

Using Remark 11, at every point of T \T s , d x 1 ν = 0, we see that condition (H) (section 3.1.1) is satisfied at every point of N \ τ .

Then all the assumptions of Theorem 10 being satisfied by N = T , in a particular case, we conclude that N is the boundary of a Levi-flat (2n -2)variety (with negligible singularities) M in R × C n .

Taking π : C × C n → C n to be the standard projection, we obtain the conclusion.

3.7. Gluing of elementary models. 3.7.1. The gluing happens between two compatible elementary models along boundaries, for instance down and 1-up. Remark that the gluing can only be made at special 1-hyperbolic points. More precisely, it can be defined as follows.

The assumed properties of the submanifold S in section 2 in C n have a meaning in any complex analytic manifold X of complex dimension n ≥ 3, and are kept under any holomorphic isomorphism.

We will define a submanifold S ′ of X obtained by gluying of elementary models by induction on the number m of models. An elementary model T in X is the image of an elementary model T 0 in C n by an analytic isomorphism of a neighborhood of T 0 in C n into X.

3.7.2. Let S ′ be a closed smooth real submanifold of X of dimension 2n -2 which is non minimal at every CR point. Assume that S ′ is obtained by gluing of m elementary models. a) S ′ has a finite number of flat complex points, some special elliptic and the others special 1-hyperbolic; b) for every special 1-hyperbolic p ′ , there exists a CR-isomorphism h induced by a holomorphic isomorphism of the ambient space C n from a neighborhood of p in T ′ onto a neighborhood of p ′ in S ′ . c) for every CR-orbit Σ p ′ whose closure contains a special 1-hyperbolic point p ′ , there exists a CR-isomorphism h induced by a holomorphic isomorphism of the ambient space C n from a neighborhood of Σ p = Σ ′ p \ p in T ′ onto a neighborhood V of Σ p ′ in S ′ .

Every special 1-hyperbolic point of S ′ which belongs to only one elementary model in S ′ will be called free.

We will define the gluing of one more elementary model to S ′ .

3.7.3. Gluing an elementary model T of type (d 1 ) to a free down-1-hyperbolic point of S ′ . Let h 1 be a CR-isomorphism from a neighborhood V 1 of σ ′ 1 induced by a holomorphic isomorphism of the ambient space C n onto a neighborhood of σ 1 in S ′ . Let k 1 be a CR-isomorphism from a neighborhood T "

1 of T ′ 1 into X such that k 1 |V 1 = h 1 . 3.7.4.
Theorem 20. The compact manifold or the manifold with singular boundary S ′ , obtained by the gluing of a finite number of elementary models, is the boundary of a Levi-flat hypersurface of X in the sense of currents.

Proof. From Theorem 19 and the definition of gluing.

Case of graphs

(see [START_REF] Dolbeault | Boundary problem for Levi flat graphs[END_REF] for the case of elliptic points only, and dropping the property of the function solution to be Lipschitz).

4.1. We want to add the following hypothesis: S is embedded into the boundary of a strictly pseudoconvex domain of C n , n ≥ 3 , and more precisely, let (z, w) be the coordinates in C n-1 ×C, with z = (z 1 , . . . , z n-1 ), w = u + iv = z n , let Ω be a strictly pseudoconvex domain of C n-1 × R u (i.e. the second fundamental form of the boundary bΩ of Ω is everywhere positive definite); let S be the graph gr(g) of a smooth function g : bΩ → R v . notice that bΩ × R v contains S and is strictly pseudoconvex.

Assume that S is a horned sphere (section 3.4), satisfying the hypotheses of Theorem 16. Denote by p j , j = i, . . . , 4 the complex points of S. Our aim is to prove 4.2. Theorem 21. Let S be the graph of a smooth function g : bΩ → R v . Let Q = (q 1 , . . . , q 4 ) ∈ bΩ be the projections of the complex points P = (p 1 , . . . , p 4 ) of S, respectively. Then, there exists a continuous function f : Ω → R v which is smooth on Ω \ Q and such that f |bΩ = g, and M 0 = graph(f ) \ S is a smooth Levi flat hypersurface of C n . Moreover, each complex leaf of M 0 is the graph of a holomorphic function φ : Ω ′ → C where Ω ′ ⊂ C n-1 is a domain with smooth boundary (that depends on the leaf ) and φ is smooth on Ω

′ .

The natural candidate to be the graph M of f is π M where M and π are as in Theorem 16. We prove that this is the case proceeding in several steps.

Behaviour near S.

4.3.1. Assume that D is a strictly pseudoconvex domain and that S ⊂ bD.

Recall ([HL75][Theorem 10.4]: Let D be a strictly pseudoconvex domain of C n , n ≥ 3 with boundary bD, Σ ⊂ bD be a compact connected maximally complex smooth (2d -1)-submanifold with d ≥ 2. Then, Σ is the boundary of a uniquely determined relatively compact subset V ⊂ D such that V \ Σ is a complex analytic subset of D with finitely many singularities of pure dimension ≤ d -1, and near Σ, V is a d-dimensional complex manifold with boundary.

V is said to be the solution of the boundary problem for Σ.

Lemma 22 ([DTZ09]

). Let Σ 1 , Σ 2 be compact connected maximally complex (2d-1)-submanifolds of bD. Let V 1 , V 2 be the corresponding solutions of the boundary problem.

If d ≥ 2, 2d ≥ n + 1 and Σ 1 ∩ Σ 2 = ∅, then V 1 ∩ V 2 = ∅.
Let Σ be a CR orbit of the foliation of S \ P . Then Σ is a compact maximally complex (2n -3)-dimensional real submanifold of C n contained in bD. Let V = V Σ be the solution of the boundary problem corresponding to Σ. From Theorem 16, V = π( Ṽ ), where Ṽ = ( M \ S) ∩ (C n × {x}) for suitable x ∈ (0, 1), the projection on the x-axis being finite, we can always assume that it lies into (0, 1). Moreover π | Ṽ is a biholomorphism Ṽ ∼ = V and M \ S ⊂ D.

Let Σ 1 , Σ 2 be two distinct orbits of the foliation of S \ P , and V 1 , V 2 the corresponding leaves, then, from Lemma 22, V 1 ∩ V 2 = ∅. (a i ) the set Γ(ϕ) \ Γ(ϕ) is the union of a disjoint family of complex discs {D α };

(a ii ) for each α, there is a simply connected domain Ω α ⊂ C z and a holomorphic function w = f α , defined on Ω α , such that D α is the graph of f α .

(a iii ) For each f α , there exists an extension

f * α ∈ C(Ω α ) and bD α = {(z, w) ∈ bΩ α × C w : w = f * α (z)}. (b) Lemma 25. Let {G n } ∞ n=0 , G n ⊂ C z × R u , be a sequence of bounded strictly convex domains such that G n → G 0 . Let {ϕ n } ∞ n=0 , ϕ n : ∂G n → R v be a sequence of continuous functions such that Γ(ϕ n ) → Γ(ϕ 0 ) in the Hausdorff metric. Then, if Φ n is the continuous function : G n → R v such that Γ(ϕ) = Γ(Φ), we have Γ(Φ n ) → Γ(Φ 0 ) in the Hausdorff metric. (c)
Lemma 26. Let U be a smooth connected surface which is properly embedded into some convex domain G ⊂ C z × R u . Suppose that near each point of this surface, it can be defined locally by the equation u = u(z). Then the surface U can be represented globally as a graph of some function u = U (z), defined on some domain Ω ⊂ C z .

4.4.2.

Proposition 27. M is the graph of a continuous function f : Ω → R v .

Proof. We will intersect the graph S with a convenient affine subspace of real dimension 4 to go back to the situation of Shcherbina. By Proposition 27, M is the graph of a continuous function over Ω, and, by Lemma 28, each leaf V t is a complex smooth hypersurface and π| Vt is a submersion.

→

Since Ω is strictly convex, as in Shcherbina (see 4.4.1, c)), π |Vt is 1-1, then, by Corollary 24, π sends V t onto a domain Ω t ⊂ C n-1 z with smooth boundary. Let × R u ′ . As in section 3.7.1, we will define a submanifold S ′ of X obtained by gluing of elementary smooth models by induction on the number m of models. An elementary smooth model T in X is the image of an elementary smooth model T 0 of C n by an analytic isomorphism of a neighborhood of T 0 in C n into X.

π u : (C n-1 z × R u ) × iR v → R u π v : (C n-1 z × R u ) × iR v → R v then π u|Lt = a t .
Gluing an elementary smooth model T of type (d 1 ) to a free down-1-hyperbolic point of S ′ .

Every elementary smooth model is contained in a cylinder bΩ × R v determined by Ω and an (n -1, 1)-frame. Two sets Ω are compatible if either they coincide or one is part of the other.

The announced gluing is defined in the following way: there exists a CRisomorphism h 1 from a neighborhood V 1 of σ ′ 1 induced by a holomorphic isomorphism of the ambient space C n onto a neighborhood of σ 1 in S ′ . Let k 1 be a CR-isomorphism from a neighborhood T " 1 of T ′ 1 into X such that k 1 |V 1 = h 1 , and there exists a common (n -1, 1)-frame on which the corresponding sets Ω are compatible. The existence of such a situation is possible as the example of the horned (almost everywhere) smooth sphere shows (Theorem 21.).

Remark that the gluing implies that the obtained submanifold S ′ is C 0 and smooth except at the complex points.

Other gluing are obtained in a similar way. Hence:

Theorem 30. The manifold S ′ obtained by gluing of elementary smooth models is of class C 0 , and smooth except at the complex points.

Corollary 31. The manifold S ′ is the boundary of manifold M of class C ∞ whose interior is a Levi-flat smooth hypersurface.

3.5. 3 .

 3 Examples of elementary models. We will say that T is a elementary model of type: (a) if it has: two elliptic points; (b) if it has: one special elliptic point and one down-{1}-hyperbolic point; (c 1 ) if it has: one special elliptic point and one 1-up-{1}-hyperbolic point; (c 2 ) if it has: one special elliptic point and one 2-up-{1}-hyperbolic point; (d 1 ) if it has: two special 1-up-{1}-hyperbolic points; (d 2 ) if it has: two special 2-up-{1}-hyperbolic points; (e) if it has: two special down-{1}-hyperbolic points; Other configurations are easily imagined. The prescribed boundary of a Levi-flat hypersurface of C n in [DTZ05] and [DTZ10], whose complex points are flat and elliptic, is an elementary model of type (a).

  Let T be the elementary model there exists an open neighborhood T " in T ′ carrying a smooth function ν : T " → R whose level sets are the leaves of a smooth foliation.

3. 8 .

 8 Examples of gluing. Denoting the gluing of the two models of type (d 1 ) and (d 2 ) to a free down-1-hyperbolic point of S ′ by: → (d 1 ) -(d 2 ), and the converse by: (d 1 ) -(d 2 ) →, and, also, analogous configurations in the same way, we get: torus: (b) → (d 1 ) -(d 2 ) → (b); the Euler-Poincaré characteristic of a torus is χ(T k ) = 0: 2 special elliptic and 2 special 1-hyperbolic points. bitorus: (b) → (d 1 ) -(d 2 ) → (e) → (d 1 ) -(d 2 ) → (b).

4.3. 3 .

 3 Assume that S satisfies the full hypotheses of Theorem 21. Set m 1 = min S g, m 2 = max S g and r ≫ 0 such that D = Ω × [m 1 , m 2 ] ⊂⊂ B(r) ∩ Ω × iR v where B(r) is the ball {|(z, w)| < r}. 4.3.4. Lemma 23. Let p ∈ S be a CR point. Then, near p, M is the graph of a function φ on a domain U ⊂ C n-1 z × R u which is smooth up to the boundary of U . Proof. Near p, each CR orbit Σ is smooth and can be represented as the graph of a CR function over a strictly pseudoconvex hypersurface and V Σ as the graph of the local holomorphic extension of this function. From Hopf lemma, V is transversal to the strictly pseudoconvex hypersurface dΩ × iR v near p. Hence the family of the V Σ , near p, forms a smooth real hypersurface with boundary on S that is the graph of a smooth function φ from a relative open neighborhood U of p on Ω into R v . Finally, Lemma 22 garantees that this family does not intersect any other leaf V from M . 4.3.5. Corollary 24. If p ∈ S is a CR point, each complex leaf V of M , near p, is the graph of a holomorphic function on a domain Ω V ⊂ C n-1 z , which is smooth up to the boundary of Ω V . 4.4. Solution as a graph of a continuous function. 4.4.1. Recall results of Shcherbina [Shc93] from: (a) the Main Theorem: Let G be a bounded strictly convex domain in C z × R u (z ∈ C) and ϕ : bG → R v be a continuous function. Then the following properties hold, where Γ = gr, and Γ(ϕ) means polynomial hull of Γ(ϕ):

  Fix a ∈ (C n-1 z \ 0) and, for a given point (ζ, ξ) ∈ Ω, with ζ ∈ C n-1 z and ξ ∈ R u , let H (ζ,ξ) ⊂ C n-1 z× {ξ} be the complex line through (ζ, ξ) in the direction (a, 0). Set:L (ζ,ξ) = H (ζ,ξ) +R u (0, 1), Ω (ζ,ξ) = L (ζ,ξ) ∩Ω, S (ζ,ξ) = (H (ζ,ξ) +C w (0, 1))∩S Then S (ζ,ξ) is contained in the strictly convex cylinder (H (ζ,ξ) + C w (0, 1)) ∩ (bΩ × iR v )and is the graph of g |bΩ (ζ,ξ) .From (a ii ), the polynomial hull of S (ζ,ξ) is a continuous graph over Ω (ζ,ξ) . Consider M = π( M ) and set M ζ,ξ) = (H (ζ,ξ) + C w (0, 1)) ∩ M.

4. 7 .

 7 Elementary smooth models. 4.7.1. Definition. An elementary smooth model in C n is an elementary model in the sense of section 3.5.2 and satisfying the further condition which makes sense from Theorem 21:(G) Let (z, w) be the coordinates in C n-1 ×C, with z = (z 1 , . . . , z n-1 ), w = u + iv = z n , let Ω be a strictly pseudoconvex domain of C n-1 × R u ; assume that T ′ is the graph of a smooth function g : bΩ → R v . 4.7.2. Theorem 29. Let T be an elementary smooth model. Then, there exists a continuous function f : Ω → R v which is smooth on Ω \ Q and such that f |bΩ = g, and M 0 = graph(f ) \ S is a smooth Levi flat hypersurface of C n ; in particular, S is the boundary of the hypersurface M = graph(f ) Proof. similar to the proof of Theorem 21. 4.7.3. Gluing of elementary smooth models. In an open set of C n , a coordinate system (z, w) of C n-1 z × R u defines an (n -1, 1)-frame. To define the gluing of elementary models (section 3.7) we considered a CR-isomorphism from an open set of C n induced by a holomorphic isomorphism of the ambient space C n onto a an open set of C n . To define the gluing of elementary smooth models, we have to consider a holomorphic isomorphism of the ambient space C n onto an open set of C n sending an (n -1, 1)-frame of C n-1 z × R u onto an (n -1, 1)-frame of C n-1 z ′

  π |Lt and π v|Lt = b t .π |Lt where a t , b t are smooth functions on bΩ t . Moreover bΩ t , a t , b t depend smoothly on t. If (z t , w t ) ∈ M , then w t varies on V t , so w t is the holomorphic extension of a t + ib t to Ω t . In particular u t and v t are smooth in (z, t), from the Bochner-Martinelli formula. ∂u t ∂t is harmonic on Ω t for each t and has a smooth extension on bΩ t . From Lemma 23 and Corollary 24, ∂u t ∂t does not vanish on bΩ t . Since the CR orbits L t are connected from Proposition 14, bΩ t is also connected, hence ∂u t ∂t has constant sign on bΩ t . Then, by the maximum principle, also ∂u t ∂t on Ω t and, in particular does not vanish. This implies that M \ S is the graph of a smooth function over Ω which smoothly extends to Ω \ Q. From Proposition 27, M is the graph of a continuous function over Ω.

It follows that M ζ,ξ) is contained in the polynomial hull Ŝ(ζ,ξ) . From (a iii ), Ŝ(ζ,ξ) is a graph over Ω (ζ,ξ) foliated by analytic discs, so M ζ,ξ) is a graph over a subset U of Ω (ζ,ξ) .

Every analytic disc ∆ of Ŝ(ζ,ξ) had its boundary on S (ζ,ξ) . Since all the the complex points of S are isolated, b∆ contains a CR point p of S; from Lemma 23, near

It follows that M is the graph of a function f : Ω → R v . One proves, using (b), that f is continuous on Ω, whence on Ω \ Q, by Lemma 23. Then continuity at every q j is proved using the Kontinuitätsatz on the domain of holomorphy Ω × iR v . 4.5. Regularity. The property: M \P = (p 1 , . . . , p 4 ) is a smooth manifold with boundary results from:

Let π : C n z,w → C n-1 z be the projection. The local structure theorem for analytic sets gives:

for some neighborhood U of 0 in C n-1 z , there exists an analytic hypersurface ∆ ⊂ U such that:

It is easy to show that the covering π :

Then we may define d holomorphic functioons τ 1 , . . . , τ d : U \ ∆ → C such that A ∆ is the union of the graphs of the τ j . By the Riemann extension theorem, the functions τ j extend as holomorphic functions τ j ∈ O(U ). Suppose that τ j = τ k , for j = k, then for some disc D ⊂ U centered at 0, we have

4.6.

Proof of the Theorem 21. Consider the foliation of S \ P given by the level sets of the smooth function ν : S → [0, 1] (sections 2.3 and 2.7) and set L t = {ν = t} for t ∈ (0.1). Let V t ⊂ Ω × iR v ⊂ C n be the complex leaf of M bounded by L t .