Mohamed Ben Ellefi
email: benellefi@lirmm.fr

Zohra Bellahsene

Konstantin Todorov
email: todorov@lirmm.fr

Stefan Dietze
email: dietze@l3s.de

Beyond Established Knowledge Graphs-Recommending Web Datasets for Data Linking

With the explosive growth of the Web of Data in terms of size and complexity, identifying suitable datasets to be linked, has become a challenging problem for data publishers. To understand the nature of the content of specific datasets, we adopt the notion of dataset profiles, where datasets are characterized through a set of topic annotations. In this paper, we adopt a collaborative filtering-like recommendation approach, which exploits both existing dataset profiles, as well as traditional dataset connectivity measures, in order to link arbitrary, non-profiled datasets into a global dataset-topic-graph. Our experiments, applied to all available Linked Datasets in the Linked Open Data (LOD) cloud, show an average recall of up to 81%, which translates to an average reduction of the size of the original candidate dataset search space to up to 86%. An additional contribution of this work is the provision of benchmarks for dataset interlinking recommendation systems.

Introduction

The web of data, in particular Linked Open Data (LOD) [START_REF] Bizer | Linked data -the story so far[END_REF], is growing constantly both in terms of size and impact. This growth introduces a wide variety and heterogeneity of Datasets with respect to represented resource types, currentness, coverage of topics and domains, size, used languages, coherence, accessibility [START_REF] Buil-Aranda | Sparql webquerying infrastructure: Ready for action?[END_REF] or general quality aspects [START_REF] Guéret | Assessing linked data mappings using network measures[END_REF]. The wide variety and heterogeneity of these dataset characteristics pose significant challenges for data consumers when attempting to find useful datasets without prior knowledge of available datasets. Dataset registries such as Datahub 1 or DataCite 2 aim at addressing this issue, for instance, by enabling users and data providers to annotate their datasets with some basic metadata, for instance, descriptive tags and access details. However, due to the reliance on human annotators, such metadata are often sparse and outdated [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF]. This has contributed to the fact that, the majority of data consumption, linking and reuse focuses on established datasets and knowledge graphs such as DBpedia [START_REF] Auer | Dbpedia: A nucleus for a web of open data[END_REF] or YAGO [START_REF] Suchanek | Yago: a core of semantic knowledge[END_REF], while a long tail of datasets has hardly been reused and adopted.

For these reasons, dataset recommendation is becoming an increasingly important task to support challenges such as entity interlinking [START_REF] Nunes | Combining a co-occurrence-based and a semantic measure for entity linking[END_REF], entity retrieval or semantic search [START_REF] Blanco | Effective and efficient entity search in rdf data[END_REF]. In line with [START_REF] Lopes | Two approaches to the dataset interlinking recommendation problem[END_REF], dataset recommendation is the problem of computing a rank score for each of a set of datasets D so that the rank score indicates the relatedness of a dataset from D to a given dataset, D 0 . In turn, this allows to determine the likelihood of datasets in D to contain linking candidates for D 0 .

While our approach is agnostic to the underlying data sharing principles, entity and data interlinking are of particular concern when considering Linked Open Data [START_REF] Bizer | Linked data -the story so far[END_REF], not least because its essential principles and reliance on IRIs for identifying any term or entity facilitates Web-scale linking of data. Here, the current topology of the LOD cloud underlines the need for practical and efficient means to recommend suitable datasets, as only very few, well established knowledge graphs show a high amount of inlinks with DBpedia being the most obvious target, while a large amount of datasets is largely ignored, often due to a lack of understanding of their content and characteristics and consequently, the challenge to identify suitable linking candidates.

For the dataset recommendation problem, one has to consider both schemalevel features, to take into account the overlap and complementarity of the actual schemas, as well as instance-level features, to consider the overlap and complementarity of described entities. Given the scale of available datasets, exhaustive comparisons of schemas and instances or some of their features are not feasible as an online process. Descriptive and reliable metadata, i.e. an index is required, which allows the efficient computation of suitable recommendations. Some approaches exist, which obtain such an index through topic modeling approaches. For instance, [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF] generates a weighted bipartite graph, where datasets and topics represent the nodes, related through weighted edges, indicating the relevance of a topic for a specific dataset. However, while computation of such topic profiles is costly, it is usually applied to a subset of existing datasets only, where any new or so far unannotated datasets require the pre-computation of a dedicated topic profile.

In our work, we provide a recommendation method which not only takes into account the direct relatedness of datasets as emerging from the topic-datasetgraph produced through the profiling in [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF], but instead, we adopt established collaborative filtering practices by considering the topic relationships emerging from the global topic-dataset-graph to derive specific dataset recommendations. We exploit dataset connectivity measures to relate non-profiled datasets to datasets in the dataset-topic-graph, enabling us to consider arbitrary datasets as part of our recommendations. This approach on the one hand significantly increases the recall of our recommendations, but at the same time improves recommendations through considering dataset connectivity as another relatedness indicator. The intuition is that this leads to more robust and less error-prone recommendations, since the consideration of global topic connectivity provides reliable connectivity indicators even in cases where the underlying topic profiles might be noisy. Our assumption is that even poor or incorrect topic annotations will serve as reliable relatedness indicator when shared among datasets.

While we adopt the topic profile graph in [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF] for our experiments, we would like to emphasize that our approach is agnostic to the underlying topic index. Topic profiles which are obtained by annotating samples of instances as in the chosen method, are shown to reflect both, instance-level as well as schema-level characteristics of a specific dataset. Even though topics are derived from instances, resources of particular types show characteristic topic distributions, which significantly differ across different types [START_REF] Taibi | Exploring type-specific topic profiles of datasets: a demo for educational linked data[END_REF].

In our experiments, we apply our approach to the LOD cloud as one scenario and use case, where dataset recommendation is of particular relevance. Our experiments show superior performance compared to three simple baselines, namely based on shared key-words, shared topics, and shared common links. In a series of experiments, we demonstrate the performance of our technique compared to the current version of the LOD as an evaluation data, achieving a reduction of the original (LOD) search space of up to 86% on average.

We proceed to present the theoretical grounds of our technique in Section 2, which contains two of the contributions of the paper -an efficient approach of propagating dataset profiles over the LOD cloud by starting off with a small set of profiled datasets and a dataset recommendation technique based on topicprofiles. Section 3 defines the evaluation framework that has been established and reports on our experimental results, providing a comparison to a set of baseline recommendation approaches, made available, as a third contribution of this work, to the community as a benchmark. Related approaches are presented and discussed in Section 4 before we conclude in Section 5.

Dataset Recommendation Framework

The current section introduces a novel approach to dataset recommendation based on dataset profiles with an aimed application in the entity interlinking process. In the current setting, the datasets profiles are generated by a topic modeling paradigm, which is briefly introduced in the following subsection together with some notation and basic definitions. A computationally efficient approach of propagating existing profiles towards new arbitrary datasets is presented in subsection 2.2. Our recommendation technique is given in detail in subsection 2.3. Fig. 2 provides an overview of the main steps of the approach that will be discussed in the sequel.

Preliminaries

We start by introducing notation and definitions. Let T 1 , ..., T N be a number of topics from a set of topics T and let D = {D 1 , ..., D M } be a set of datasets.

Dataset Topic Profile Topic modeling algorithms such as Latent Dirichlet allocation [START_REF] Blei | Latent dirichlet allocation[END_REF] are used to discover a set of topics from a large collection of documents, where a topic is a distribution over terms that is biased around those associated under a single theme. Topic modeling approaches have been applied to tasks such as corpus exploration, document classification, and information retrieval. Here, we will look into a novel application of this group of approaches, exploiting the topic structure in order to define and construct dataset profiles for dataset recommendation.

As a result of the topic modeling process, a bipartite-profile-graph is built, providing a relation between a document and a topic. Documents in our setting are the datasets to be considered, therefore the profile graph is induced by the relation between a dataset, D i , and a topic, T k , expressed by a weight, w ik ∈ [0, 1], for all i = 1, ..., M and k = 1, ..., N . Formally, a profile graph is defined as follows.

Definition 1 (Dataset Topic Profile Graph). A dataset topic profile graph is a weighted directed bipartite graph P = (S, E, ∆), where S = D ∪ T , E is a set of edges of the form e ik = (D i , T k ) such that D i ∈ D and T k ∈ T and

∆ : E → [0, 1]
e ik → w ik is a function assigning weights to the edges in E.

The bipartite property of P allows to represent a given dataset by a set of topics-its profile. For the purposes of this study, it is worth noting that, inversely, a topic can be represented by a set of weighted datasets-what we will call the signature of a topic (see Figure 1). We will denote by Profile(D i ) the function returning the topic profile of D i , i.e., the set of topics together with their weights with respect to D i . Inversely, we will denote by D T k the set of datasets together with their weights with respect to a topic T k , derived again from the graph P.

Datasets Connectivity

The connectivity behavior of datasets is a central concept within the proposed recommendation framework. We consider the following definition of a measure of the strength of dataset connectedness.

Definition 2 (Dataset inter-connectivity measure). Let D i , D j ∈ D be two datasets. We define a measure of their common degree of connectivity as follows.

C(D i , D j ) = shared(D i , D j ) × [total(D i ) + total(D j )] 2 × total(D i ) × total(D j ) (1)
where shared(., .) returns the number of links between two datasets and total(D i ) returns the total number of links between D i and any other dataset in D.

Note that ( 1) is the symmetric version of the measure of connectivity of D i to D j given by

C (D i , D j ) = shared(D i , D j ) total(D i ) .
Explicitly, ( 1) is obtained by taking the mean

C (D i , D j ) + C (D j , D i ) 2 = C(D i , D j ).
The measure C is in the interval [0, 1] and has the advantage of considering the relative connectivity between datasets instead of simply looking at the number of links. In our experimental setting, shared(D i , D j ) is taken as the sum of the links between two datasets in both directions: D i → D j and D j → D i , resulting in the number of incoming and outgoing links between the datasets. A specific version of the measure C can be defined by taking only certain types of links (or predicates) in consideration (in our application scenario, we have considered LOD datasets, therefore an example of a specific predicate can be owl:sameAs).

In a more general manner, it is possible to use any dataset connectivity measure of our choice. The measure given above is one that worked well in our experiments (see Section 3). In addition, one can define in a broader sense a measure of dataset relatedness incorporating semantic elements such as vocabulary and keywords overlap. Dataset complementarity can be of interest in certain scenarios, as well. However, in the current study we have focused on connectivity only, leaving the other possibilities out for future work.

The Preprocessing/Learning Step

In many cases the number of indexed elements (e.g., datasets in our case) is much lower than the entire number of elements of interest. In that respect, it is interesting to consider a procedure that allows inexpensively to include novel elements in the index. As a preprocessing step of our recommendation workflow, we adopt a learning approach that consists of assigning topics to datasets by linking them into the dataset-topic-graph after computing their connectivity with already indexed (profiled) datasets. This step is useful in order to include in the recommendation pipeline datasets that have not been initially indexed, in an inexpensive manner, keeping in mind that the indexing process can be often quite costly and time-consuming.

Let P be a topic profile graph and let D j ∈ D be a random dataset, which is not necessarily included in the original topic profile graph P. We assign topic weights to D j considering its degree of connectivity with respect to datasets from the topic profile graph by using the following measure of relatedness between linked datasets and topics (see Fig. 2, steps 1 and 2).

Definition 3 (Connectivity-based dataset and topic relatedness measure). Let D j ∈ D and T k ∈ T . We define the following dataset and topic relatedness measure.

σ(D j , T k ) = max Di∈D C(D i , D j ) * w ik . (2)
Recall that w ik is the weight of the topic T k with respect to D i as given in Def. 1, taking a zero value in case T k is not in Profile(D i ). C(D i , D j ) is the connectivity measure between two datasets, as defined in [START_REF] Bizer | Linked data -the story so far[END_REF]. The dataset and topic relatedness measure σ is a way to measure the datasets connectivity behavior using their profiles. We will use the notation

σ jk = σ(D j , T k ) as a shortcut. Note that σ is in the [0, 1] interval.
This new weighting scheme allows to propagate inexpensively the profile of D i to datasets that are connected to it. Hence, a new graph is created between target datasets and source datasets topics. Precisely, a topic T k ∈ Profile(D i ) will be assigned to a dataset D j that has a non-zero value of C(D i , D j ). The weight of this novel topic-dataset relation is now based on the connectivity order of D j with respect to D i , scaled by the weight w ik of T k with respect to D i . In that sense, w ik plays a penalization role: the novel weight σ jk of T k with respect to D j is penalized by the weight of T k in the original topic graph, i.e., datasets with high degree of connectivity to D i will get relatively low weights with respect to a topic, if that topic has a relatively low weight with respect to D i . We consider the maximum value over all datasets in D, the set of the originally profiled datasets. In this way, we avoid ambiguity when a non-indexed dataset D j is connected to a single topic T k via multiple already indexed datasets, assuring that the highest value of relation between T k and D j is preserved. Thus, the choice of a topic to be assigned to a dataset is not influenced, only its weight is, and no connectivity information is lost.

The topic-dataset relatedness measure (3) allows to construct a novel profile graph by computing σ jk for all possible values of j and k (j = 1, ..., M and k = 1, ..., N ). The novel graph, that we call the Linked Dataset Topic Profile Graph (LDPG), includes new datasets and the original topics as its nodes and is defined as follows (see Fig. 2, step 2). Definition 4 (Linked Dataset Topic Profiles Graph (LDPG)). The LDPG is a weighted directed bipartite graph P l = (S l , E l , ∆ l ), where S l = D ∪ T , E l is a As this was the case within the original profiling scheme, the inherently bipartite nature of the graph P l allows for a two-fold interpretation -either a dataset is modeled as a set of topics (a dataset's profile), or, inversely, a topic is modeled as a set of datasets assigned to it (a topic's signature). Therefore, it is easy to define a set of significant datasets with respect to a given topic, by thresholding on their weights in the Linked profiles graph with respect to the topic of interest. Note again that for the purposes of the recommendation task, we will be interested in keeping the weights of every dataset in the resulting topic representations and thus model every topic by a set of (dataset, weight) couples.

Definition 5 (Dataset significance for a topic. Topic signature). A dataset D j ∈ D is significant with respect to a topic T k ∈ T if its weight in the LDPG σ jk = σ(D j , T k ) is greater than a given value θ ∈ (0, 1).

A topic T k is modeled by the set of its significant datasets together with their respective weights, given as

D * T k = {(D j , σ jk )|σ jk > θ} j=1,...,M , (3)
for k = 1, ..., N. We will call D * T k the signature of the topic T k .

With this definition, the profile of a given dataset, Profile(D i ), is modeled as a number of sets of significant datasets -one per topic in Profile(D i ) coupled with their weights with respect to each topic (see Fig. 2, step 3), or otherwisea set topic signatures.

For sake of generality, we draw the readers attention to the fact that the learning approach resulting in index extension applies to any dataset profile definition that one might like to consider and not exclusively to the one based on the topic modeling paradigm.

Profile-Based Dataset Ranking

Let D 0 be a new dataset to be linked. The aim of the recommendation task is to provide the user with an ordered list of datasets, potential candidates for interlinking with D 0 , which narrows down considerably the original search space i.e., the web of data. Thus the dataset recommendation can be seen as the problem of computing a rank score for each D j ∈ D that indicates the likelihood of D j to be relevant to a dataset D 0 . In the context of using topic-based dataset profiles for linking recommendation, we restate the problem in the following manner.

For a given non-linked dataset D 0 , profile-based dataset recommendation is the problem of computing a rank score r 0j for each D j ∈ D based on topic overlap between D j and D 0 , so that r 0j indicates the relevance of D j to D 0 for the entity linking task.

We start by generating the topic profile of D 0 , Profile(D 0 ) = {(T 1 , w 01 ), ..., (T m , w 0m )}. Then, we extract from the result of the learning step the set of target datasets for each topic in Profile(D 0 ) together with their corresponding relevance values σ, namely the set of m topic signatures {D * T k } m k=1 . These datasets constitute the pool, from which we will recommend interlinking candidates to D 0 .

We will use n to denote their number, that is n = m j=1 |D * Tj |, or the sum of the numbers of datasets in each topic signature. The aim is to serve the user with the most highly ranked datasets from that pool. There are two ranking criteria to consider: the weight w of each topic in Profile(D 0 ) and the weight σ of each dataset in each of the topic signatures in {D * T k } m k=1 (step 4 in Fig. 2). Since the ranking score in our setting depends on topic overlap, we define the interlinking relevance of a dataset D j with respect to D 0 in the following manner.

Definition 6 (Dataset interlinking relevance). For all j = 1, ..., n, the relevance of a dataset D j ∈ D to a dataset D 0 via the topic T k is given by

r 0 j = w 0k * σ jk , (4)
with k = 1, ..., m.

Note that j covers the total number of datasets in the set of m topic signatures, therefore the relevance value depends on j only (i.e., a single relevance value per dataset from the pool of candidates). Similarly to the definition of σ in Def. 6, w has a penalization function, decreasing the ranks of datasets that have high values of their σ weights, but are found in topic signatures of a low relevance to D 0 (expressed by a low value of w).

It is easy to define a mapping f : R → N from a space of interlinking relevance values R to the natural numbers such that f (r 0 j1 ) > f (r 0 j2 ) ⇐⇒ r 0 j1 ≤ r 0 j2 , for any j 1 , j 2 ∈ [1, n] and 1 = max j f (r 0 j ). With this definition, since there is a relevance value r 0 j per dataset D j ∈ D, f (r 0 j ) returns the rank of the dataset D j with respect to D 0 . The results of the recommendation process are given in a descending order with respect to these ranks.

Experiments and Results

In this section, we start by a discussion on the evaluation setting then we proceed to report on the experiments conducted in support of the proposed recommendation method.

Evaluation Framework

The quality of the outcome of a recommendation process can be evaluated along a number of dimensions. Ricci et al. [START_REF] Ricci | Recommender systems handbook[END_REF] provide a large review of recommender system evaluation techniques and cite three common types of experiments: (i) offline setting, where recommendation approaches are compared without user interaction, (ii) user studies, where a small group of subjects experiment with the system and report on the experience, and (iii) online experiments, where real user populations interact with the system.

In our approach, we assume that the dataset connectivity behavior when data were collected (i.e., steps 1, 2 and 3 in Fig. 2) is similar enough to the profile connectivity behavior when the recommender system is deployed (i.e., step 4 in Fig. 2), so that we can make reliable decisions based on an offline evaluation. The offline experiment is performed by using pre-collected data as evaluation data (ED). Using these data, we can simulate the profiles connectivity behavior that impacts the recommendation results.

The most straightforward, although not unproblematic (see the discussion that follows below) choice of ED for the entity linking recommendation task is the existing link topology of the current version of links between web datasets. Since this evaluation data are the only available data that we have for both training (our preprocessing steps 1, 2 and 3 in Fig. 2) and testing (the actual recommendation in step 4 of Fig. 2), we opted for a 5-fold cross-validation [START_REF] Weiss | Computer Systems That Learn: Classification and Prediction Methods from Statistics, Neural Nets, Machine Learning, and Expert Systems[END_REF] to evaluate the effectiveness of the recommendation system. In 5-fold crossvalidation, the ED was randomly split into two subsets: the first one, containing random 80% of the linked datasets in the ED, was used as training set while the second one, containing the remaining linked datasets (i.e., random 20% of the ED), was retained as the validation data for tests (i.e., the test set). We repeated these experiments five times changing at each time the 20% representing the test set in order to cover 100% of the whole data space. The evaluation is based on the capacity of our system to reconstruct the links from the ED in the recommendation process.

The most common measures of the efficiency of a recommendation system are Precision, Recall and F1-Score, formalized as functions of the true positives (TP), false positives (FP), true negatives (TN) and false negatives (FN) as follows.

P r = T P T P + F P

; Re = T P T P + F N

;

F 1 = 2T P 2T P + F N + F P . (5)
In addition, [START_REF] Ricci | Recommender systems handbook[END_REF] present a measure of the false positive overestimation, particularly important in our offline evaluation case:

F alseP ositiveRate = F P F P + T N . (6)
A small value of the false positive rate means that every time you call a positive, you have a high probability of being right. Conversely, a high value of the false positive rate means that every time you call a positive, you have a high probability of being wrong.

Experimental Setup

Since our recommendation is based on the connectivity of the D graph as well as the topic profiles of indexed datasets, the Linked Open Data (LOD) [START_REF] Bizer | Linked data -the story so far[END_REF] is clearly our best use case to experiment our recommendation. As a topic modeling approach, we adopt the dataset profiles index provided by [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF] since it generates accurate profiles and outperforms established topic modeling approaches such as ones based on the well known Latent Dirichlet Allocation3 . Using the topic profiles approach from [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF] it is easy to produce a weighted bipartite graph as described in 1, where datasets and topics represent the nodes, related through weighted edges, indicating the relevance of a topic for a specific dataset. A dataset profile is represented through topics, which in this case are DBpedia categories4 derived through a processing pipeline5 analyzing representative resource samples from specific datasets. For example, the T profile of the Semantic Web Dog Food Corpus6 dataset includes the following DBpedia categories: Data management, Semantic Web, Information retrieval, World Wide Web Consortium standards, etc.

In the following, we distinguish between the set D of datasets in the entire LOD and the datasets indexed by the profiling approach [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF], denoted by D . Explicitly, we consider: We trained our system as described in steps 1, 2 and 3 in Fig. 2. We started by extracting the topic profiles graph from the available endpoint of Data Observatory 9 . Then we extracted VoID descriptions of all LOD datasets, using the datahub2void tool 10 . The constituted evaluation data (ED) corresponds to the outgoing and incoming links extracted from the generated VoID file (it is made available on http://www.lirmm.fr/benellefi/void.ttl).

Note that in the training set we used the actual values of VoID:triples (see Section 1) to compute dataset connectivity, while in the test set we considered binary values (two datasets from the evaluation data are either linked or not).

For example, shared(tip, linkedgeodata) = 6, so in the training set we considered 6 as the number of links in Eq. ( 1), while in the test set we only consider the information that tip is connected to linkedgeodata and vice versa. Training is performed only once.

Results and Analysis

We ran our recommendation workflow as described in step 4 in Fig. 2. Using 5-fold cross-validation, for each dataset in D , we recommended an ordered list of datasets from D. The results are given in Fig. 3.

The results show a high average recall of up to 81%. Note that the recommendation results for 59% of the source datasets have a recall of 100% and two of them have an F1-score of 100%. As mentioned in Section 3.2, we considered only the binary information of the existence of a link in the LOD as evaluation data in the testing set. This simplification has been adopted due to the difficulty of retrieving all actual links in the LOD graph (implying the application of heavy instance matching or data linking algorithms on a very large scale). Certainly, the explicit currently existing links are only a useful measure for recall, but not for precision. In our experiments, we measured an average precision of 19%. We explain that by the fact that the amount of explicitly declared links in the LOD cloud as ED is certain but far from being complete to be considered as ground truth. Subsequently, we are forced to assume that the false positive items would have not been used even if they had been recommended, i.e., that they are uninteresting or useless to the user. For this reason, based on our evaluation data, a large amount of false positives occur, which in reality are likely to be relevant recommendations. In order to rate this error, we calculated the false positive rate over all results, shown in the Fig. 4. The small values of this rate indicate that every time you call a positive, you have a probability of being right, which provide support to our hypothesis with an average FP-Rate of 13%.

To further illustrate the effect of false positives overestimation, we included in the ED new dataset links based on the shared keywords of the datasets. Precisely, if two datasets share more then 80% of their VoID tags, they are considered as linked, and are added to the ED. For example, linkedgeodata is connected to 4 datasets in the main ED: osm-semantic-network, dbpedia, tip et dbpedia-el. However, we found that linkedgeodata shared more than 80% of its tags with fu-berlin-eurostat and twarql 11 . By adding both links to the original ED, we noted a gain in precision of 5% for the linkedgeodata dataset with no impact on recall. Thus, we believe that our system can perform much better on more complete ED.

The current version of the topic dataset profile graph from [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF] contains 76 datasets and 185 392 topics. Working with this already annotated subset of existing datasets is not sufficient and limited the scope of our recommendations significantly. In addition, the number of the profiled datasets, compared to the number of topics is very small, which in turn appeared to be problematic in the recommendation process due to the high degree of topic diversity leading to a lack of discriminability. One way of approaching this problem would be to index all LOD datasets by applying the original profiling algorithm [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF]. However, given the complexity of this processing pipeline-consisting of resource sampling, analysis, entity and topic extraction for a large amount of resources-it is not efficient enough, specifically given the constant evolution of Web data, calling for frequent re-annotation of datasets. Also we note that this profiles propagation technique can be of interest for any dataset profile-based application, providing an inexpensive way of computing a profile of an arbitrary dataset. As one of the original contributions of this paper, the preprocessing step in our recommendation pipeline can be seen as an efficient method for automatic expansion of the initial profiles index over the entire linked open data space based on dataset connectivity measures. The main goal of a recommender system is to reduce the cost of identifying candidate datasets for the interlinking task. Some systems may provide recommendations with high quality in terms of both precision and recall, but only over a small portion of datasets (as is the case in [START_REF] Leme | Identifying candidate datasets for data interlinking[END_REF]). We obtain high recall values for the majority of datasets over the entire set of LOD datasets with a price to pay of having relatively low precision. Here, low precision/high recall systems still offer significant contributions by narrowing the size of the search space. Therefore, we highlight the efficiency of our system in reducing the search space size. Fig. 5 depicts the reduction of the original search space size (258 datasets) in percentage over all source datasets to be linked. The average space size reduction is of up to 86%. As mentioned previously, our system can cover 100% of the available linked datasets, since the topics-datasets profiling approach [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF] as well as our profile expansion approach presented in Section 2.2 are able to profile any arbitrary dataset. Our system is also capable of dealing with the well-known cold-start problem (handling correctly newly published and unlinked datasets), since not linked datasets are handled by using the indexing technique in [START_REF] Fetahu | A scalable approach for efficiently generating structured dataset topic profiles[END_REF], which does not rely on dataset connectivity. Based on the learning step, our system is able to recommend a sorted list of candidate datasets to these not linked datasets. To the best of our knowledge, there does not exist a common benchmark for dataset interlinking recommendation. One of the contributions of this paper is the provision of three simple baseline approaches for this problem. Given two datasets, D 0 and D j , we define the following baseline recommendation methods. 1. Average precision, recall and F1-score of our system versus the baselines over all D datasets based on the ED. Fig. 6 and Fig. 7, respectively, depict detailed comparisons of the F1-Score and the Recall values between our approach and the baselines over all D datasets taken as source datasets. From these figures, it can be seen that our method largely outperforms the baseline approaches, which even fail to provide any results at all for some datasets. The baseline approaches have produced better results than our system in a limited number of cases, especially for source and target datasets having the same publisher. For example, the shared keywords baseline generated an F-Score of 100% on oceandrilling-janusamp, which is connected to * oceandrilling-codices, due to the fact that these two datasets are tagged by the same provenance (data.oceandrilling.org).

Baselines and Comparison

Shared

Table 1 compares the performance of our approach to the three baseline methods in terms of average precision, recall and F1-score.

As a general conclusion, these observations indicate that the collaborative filtering-like recommendation approach, which exploits both existing dataset profiles as well as traditional dataset connectivity measures, shows high performance on identifying candidate datasets for the interlinking task.We make all of the ranking results of our recommendation approach available to the community on http://www.lirmm.fr/benellefi/results.csv.

Related Work

With respect to finding relevant datasets on the Web, we cite briefly several studies on discovering relevant datasets for query answering have been proposed. Based on well-known data mining strategies, the works in [START_REF] Wagner | Discovering related data sources in data-portals[END_REF] and [START_REF] Wagner | Entity-based data source contextualization for searching the web of data[END_REF] present techniques to find relevant datasets, which offer contextual information corresponding to the user queries. A used feedback-based approach to incrementally identify new datasets for domain-specific linked data applications is proposed in [START_REF] De Oliveira | Feedback-based data set recommendation for building linked data applications[END_REF]. User feedback is used as a way to assess the relevance of the candidate datasets.

In the following, we cite approaches that have been devised for the datasets interlinking candidates recommendation task and which are directly relevant to our work.

Nikolov et al. [START_REF] Nikolov | Identifying relevant sources for data linking using a semantic web index[END_REF] propose a keyword-based search approach to identify candidate sources for data linking. The approach consists of two steps: (i) searching for potentially relevant entities in other datasets using as keywords randomly selected instances over the literals in the source dataset, and (ii) filtering out irrelevant datasets by measuring semantic concept similarities obtained by applying ontology matching techniques.

Leme et al. [START_REF] Leme | Identifying candidate datasets for data interlinking[END_REF] present a ranking method for datasets with respect to their relevance for the interlinking task. The ranking is based on Bayesian criteria and on the popularity of the datasets, which affects the generality of the approach (cf. the cold-start problem discussed previously). The authors extend this work and overcome this drawback in [START_REF] Lopes | Two approaches to the dataset interlinking recommendation problem[END_REF] by exploring the correlation between different sets of features-properties, classes and vocabularies-and the links to compute new rank score functions for all the available linked datasets.

Mehdi et al. [START_REF] Mehdi | Discovering domain-specific public SPARQL endpoints: a life-sciences use-case[END_REF] propose a method to automatically identify relevant public SPARQL endpoints from a list of candidates. First, the process needs as input a set of domain-specific keywords which are extracted from a local source or can be provided manually by an expert. Then, using natural languages processing techniques and queries expansion techniques, the system generates a set of queries that seek for exact literal matches between the introduced keywords and the target datasets, i.e., for each term supplied to the algorithm, the system runs a matching with a set of eight queries: {original-case, proper-case, lower-case, upper-case} * {no-lang-tag, @en-tag}. Finally, the produced output consists of a list of potentially relevant SPARQL endpoints of datasets for linking. In addition, an interesting contribution of this technique is the bindings returned for the subject and predicate query variables, which are recorded and logged when a term match is found on some particular SPARQL endpoint. The records are particularly useful in the linking step.

In contrast to the approaches described above, our method is the first to be based on topic overlap and collaborative filtering. To the best of our knowledge, the current paper is also the first study to provide simple baseline recommendation techniques that can serve as a benchmark (cf. Section 3.4).

In comparison with all the work discussed above, our approach has the potential to overcome a series of complexity related problems. Precisely, considering the complexity to generate the matching in [START_REF] Nikolov | Identifying relevant sources for data linking using a semantic web index[END_REF], to produce the set of domainspecific keywords as input in [START_REF] Mehdi | Discovering domain-specific public SPARQL endpoints: a life-sciences use-case[END_REF] and to explore the set of features of all the network datasets in [START_REF] Lopes | Two approaches to the dataset interlinking recommendation problem[END_REF], our recommendation results are much easier to obtain since we only manipulate the already generated topic profiles (or inexpensively propagated profiles).

Since the majority of used datasets in the papers discussed above were not yet indexed by the topic profiles graph, a direct comparison of the performance of the different recommendation methods to our system seems unfair. In addition, none of the authors have shared data, except for [START_REF] Lopes | Two approaches to the dataset interlinking recommendation problem[END_REF]. However, the evaluation of this approach in its first version [START_REF] Leme | Identifying candidate datasets for data interlinking[END_REF] concludes on one dataset only. In its second version [START_REF] Lopes | Two approaches to the dataset interlinking recommendation problem[END_REF], while they provide the data used in their experiments, the authors do not give details on the resulting rank scores.

Conclusion and Future Work

We have presented an interlinking candidate dataset recommendation approach, based on the connectivity behavior learning of topic-profiles datasets. We demonstrate that our technique allows to reduce considerably the original search space and that it outperforms the results obtained by three baseline recommendation approaches, developed for the purposes of this study and made available to the community. Since dataset topic profiling is a key component in the recommendation pipeline, we show a simple way of propagating dataset profiles to the entire set of linked open datasets, starting off with a limited number of profiled datasets.

In the future, we plan to improve the evaluation framework by developing a more reliable and complete ground truth for dataset recommendation, possibly by using crowdsourcing techniques, in order to deal with the false positives overestimation problem. Our method could potentially benefit from combining it with machine learning techniques. We plan to conduct a thorough evaluation of the efficiency of our profiles propagation technique for the dataset recommendation task.

Fig. 1 .

 1 Fig. 1. (a) An example of a bipartite profile graph with topics and datasets linked by weighted edges. (b) Representing a dataset, Di, as a set of topics. (c) Representing a topic, T k , as a set of datasets.

Fig. 2 .

 2 Fig. 2. The four main steps of the profile-based dataset recommendation framework.

-

  D: All datasets in the LOD cloud group on the Data Hub 7 , which will be considered as target datasets (to be recommended) in the testing set, |D| = 258. -D : All the datasets indexed by the topics profiles graph 8 , which will be considered as source datasets (to be linked) in the testing set, |D | = 76 and |D | ⊂ |D|.

Fig. 3 .

 3 Fig. 3. Recall/Precision/F1-Score over all recommendation lists for all source datasets in D and all target datasets in D.

Fig. 4 .

 4 Fig. 4. False Positive Rates over all recommendation lists over all D datasets.

Fig. 5 .

 5 Fig. 5. Search space reduction in percent over all recommended sets and over all D datasets.

Fig. 6 .

 6 Fig. 6. F1-Score values of our approach versus the baselines overall D datasets

Fig. 7 .

 7 Fig. 7. Recall values of our approach versus the baselines overall D datasets

  Keywords Recommendation: if D 0 and D j share N tags of VoID:Tags extracted from http://www.lirmm.fr/benellefi/void.ttl with N tags > 0, then we recommend (D j , N tags ) to D i , where N tags acts as a rank score. Shared Links Recommendation: if D 0 and D j have N links connected datasets in common from http://www.lirmm.fr/benellefi/void.ttl with N linksets > 0, then we recommend (D j , N links ) to D 0 , where N links acts as a rank score. Shared Topics Recommendation: if D 0 and D j share N topics topics extracted from http://data-observatory.org/lod-profiles/sparql with N topics > 0, then we recommend (D j , N topics ) to D 0 , where N topics acts as a rank score.The recommendation results for all LOD datasets (D 0 covering D) of the three baseline approaches are made available on http://www.lirmm.fr/benellefi/ Baselines.rar.

		Our approach Shared Keywords Shared linksets Shared Topics Profiles
	AVG Precision	19%	9%	9%	3%
	AVG Recall	81%	47%	11%	13%
	AVG F1-Score	24%	10%	8%	4%
	Table

http://mallet.cs.umass.edu/

dbpedia.org/page/Category

http://data-observatory.org/lod-profiles/profiling.htm

http://data.semanticweb.org/

http://datahub.io/group/lodcloud

http://data-observatory.org/lod-profiles/profile-explorer/

http://data-observatory.org/lod-profiles/sparql

https://github.com/lod-cloud/datahub2void

Example: linkedgeodata has 11 tags and twarql has 9 tags. We considered as connected since they shared 8 tags which is higher than the 80% of the average amount, i.e., 8 < (0.8 * (11 + 9)/2).

http://www.datalyse.fr/

Acknowledgements

This research has been partially funded under the Datalyse project 12 and by the European Commission as part of the DURAARK project, FP7 Grant Agreement No. 600908.


B Ammann

E Humbert

THE SECOND YAMABE INVARIANT

Keywords: 53A30, 35J60(Primary) 35P30, 58J50, 58C40 (Secondary)

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. We define the second Yamabe invariant as the infimum of the second eigenvalue of the Yamabe operator over the metrics conformal to g and of volume 1. We study when it is attained. As an application, we find nodal solutions of the Yamabe equation.

Introduction

Let (M, g) be an n-dimensional compact Riemannian manifold (n ≥ 3). In [START_REF] Yamabe | On a deformation of Riemannian structures on compact manifolds[END_REF] Yamabe attempted to show that there is a metric g conformal to g such that the scalar curvature S g of g is constant. However, Trudinger [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF] realized that Yamabe's proof contained a serious gap. The problem is now solved, but it took a very long time to find the good approach. The problem of finding a metric g with constant scalar curvature in the conformal class [g] is called the Yamabe problem. The first step towards a rigorous solution of this problem was achieved by Trudinger [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF] who was able to repair the gap of Yamabe's article in the case that the scalar curvature of g is non-positive. Eight years later, Aubin [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] solved the problem for arbitrary non locally conformally flat manifolds of dimension n ≥ 6. The problem was completely solved another eight years later in an article of Schoen [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF] in which the proof was reduced to the positive-mass theorem which had previously been proved by Schoen and Yau [START_REF] Schoen | On the proof of the positive mass conjecture in general relativity[END_REF][START_REF] Schoen | Conformally flat manifolds, Kleinian groups and scalar curvature[END_REF]. The reader can refer to [START_REF] Lee | The Yamabe problem[END_REF], [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] or [START_REF] Hebey | Introduction à l'analyse non-linéaire sur les variétés[END_REF] for more information on this subject. The method to solve 1 bernd.ammann@gmx.net, humbert@iecn.u-nancy.fr 1 the Yamabe problem was the following. Let u ∈ C ∞ (M ), u > 0 be a smooth function and g = u N -2 g where N = 2n n-2 . Then, multiplying u by a constant, the following equation is satisfied: L g (u) = S g|u| N -2 u.

where

L g = c n ∆ g + S g = 4(n -1) n -2 ∆ g + S g
is called the Yamabe operator. As a consequence, solving the Yamabe problem is equivalent to finding a positive smooth solution u of

L g (u) = C 0 |u| N -2 u. (1)
where C 0 is a constant. In order to obtain solutions of this equation Yamabe defined the quantity

µ(M, g) = inf u =0,u∈C ∞ (M) Y (u)
where

Y (u) = M c n |∇u| 2 + S g u 2 dv g M |u| N dv g 2 N .
Nowadays, µ(M, g) is called the Yamabe invariant, and Y the Yamabe functional. Writing the Euler-Lagrange equation associated to Y , we see that there exists a one to one correspondence between critical points of Y and solutions of equation (1). In particular, if u is a positive smooth function such that Y (u) = µ(M, g), then u is a solution of (1) and g = u N -2 g is the desired metric of constant scalar curvature. The key point of the resolution of the Yamabe problem is the following theorem due to Aubin [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF]. In the theorem and in the whole article, S N will always denote the sphere S n with the standard Riemannian structure.

Theorem 1.1. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. If µ(M, g) < µ(S n ), then there exists a positive smooth function u such that Y (u) = µ(M, g).

This strict inequality is used to show that a minimizing sequence does not concentrate in any point.

Aubin [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] and Schoen [START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF] proved the following.

Theorem 1.2. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Then µ(M, g) ≤ µ(S n ) = n(n -1)ω

2 n n where ω n stands for the volume of the standard sphere S n . Moreover, we have equality in this inequality if and only if (M, g) is conformally diffeomorphic to the sphere.

These theorems solves the Yamabe problem.

In this paper, we introduce and study an invariant that we will call the second Yamabe invariant. It is well known that the operator L g has discrete spectrum Spec(L g ) = {λ 1 (g), λ 2 (g), • • • } where the eigenvalues

λ 1 (g) < λ 2 (g) ≤ λ 3 (g) ≤ • • • ≤ λ k (g) • • • → +∞ appear with their multiplicities. The variational characterization of λ 1 (g) is given by λ 1 (g) = inf u =0,u∈C ∞ (M) M c n |∇u| 2 + S g u 2 dv g M |u| 2 dv g . Let [g] be the conformal class of g. Assume now that the Yamabe invariant µ(M, g) ≥ 0. It is easy to check that µ(M, g) = inf g∈[g] λ 1 (g)Vol(M, g) 2 n ,
where [g] is the conformal class of g. We then enlarge this definition.

Definition 1.3. Let k ∈ N * . Then, the k th Yamabe invariant is defined by

µ k (M, g) = inf g∈[g] λ k (g)Vol(M, g) 2 n .
With these notations, µ 1 (M, g) equals to Yamabe invariant µ(M, g) in the case µ(M, g) ≥ 0, and

µ 1 (M, g) = -∞ in the case µ(M, g) < 0.
The goal of this article is to study the second Yamabe invariant µ 2 (M, g) for manifolds whose Yamabe invariant in the case µ(M, g) ≥ 0. As explained in Section 8, the most interesting case is when µ(M, g) > 0.

In particular, we discuss whether µ 2 (M, g) is attained. This question is discussed in Subsection 5.1. In particular, Proposition 5.2 asserts that contrary to the standard Yamabe invariant, µ 2 (M, g) cannot be attained by a metric if M is connected. In other words, there does not exist g

∈ [g] such that µ 2 (M, g) = λ 2 (g)Vol(M, g) 2 n
. In order to find minimizers, we enlarge the conformal class [g] to what we call the class of generalized metrics conformal to g. A generalized metric is a "metric" of the form g = u N -2 g, where u is no longer necessarily positive and smooth, but u ∈ L N (M ), u ≥ 0, u ≡ 0. The definitions of λ 2 (g) and of Vol(M, g) can be extended to generalized metrics (see section 3). Then, we are able to prove the following result:

Theorem 1.4. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 whose Yamabe invariant is non-negative. Then, µ 2 (M, g) is attained by a generalized metric in the following cases:

• µ 1 (M, g) > 0 and µ 2 (M, g) < µ 1 (M, g) n 2 + µ 1 (S n )) n 2 2 n ; • µ 1 (M, g) = 0 and µ 2 (M, g) < µ 1 (S n ) where µ 1 (S n ) = n(n -1)ω 2 n
n is the Yamabe invariant of the standard sphere.

The result we obtain in the case µ 1 (M, g) = 0 is not surprising. Indeed, when µ 2 (M, g) < µ 1 (S n ), Aubin's methods [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] can be adapted here and allow to avoid concentration of minimizing sequences. However, when µ 1 (M, g) > 0 and µ 2 (M, g) < µ 1 (M, g)

n 2 + µ 1 (S n )) n 2 2
n , the result is much more difficult to obtain (see Subsection 6). A second result is to find explicit examples for which the assumptions of Theorem 1.4 are satisfied. The method consists in finding an appropriate couple of test functions.

Theorem 1.5. The assumptions of Theorem 1.4 are satisfied in the following cases:

• µ 1 (M, g) > 0, (M, g) is not locally conformally flat and n ≥ 11;

• µ 1 (M, g) = 0, (M, g) is not locally conformally flat and n ≥ 9.

One of our motivations is to find solutions of the Yamabe equation (1) with alternating sign, i.e. positive and negative values. If M is connected, alternating sign implies that the zero set u -1 (0) of u is not empty. In the following we will use the standard definition to call the zero set u -1 (0) of a function u the nodal set of u. A solution with a non-empty nodal set is usually called a nodal solution. If M is connected, then the maximum principle implies that a solution of the Yamabe equation is nodal if and only if it has alternating sign. They are called nodal solutions of the Yamabe equation. The articles [START_REF] Hebey | Existence and multiplicity of nodal solutions for nonlinear elliptic equations with critical Sobolev growth[END_REF], [START_REF] Djadli | Nodal solutions for scalar curvature type equations with perturbation terms on compact Riemannian manifolds[END_REF], [START_REF] Jourdain | Solutions nodales pour des équations du type courbure scalaire sur la sphère[END_REF], [START_REF] Holcman | Solutions nodales sur les variétés riemanniennes[END_REF] prove existence of nodal solutions under symmetry assumptions or under some assumptions which allow to use Aubin's methods, as in Theorem 1.4 when µ 1 (M, g) = 0 and µ 2 (M, g) < µ 1 (S n ). If µ(M, g) ≤ 0, another method is given in Section 8. The method we use here is completely different and we obtain solutions on a large class of manifolds. In particular, to our knowledge, there is no work which leads to the existence of such solutions if the Yamabe invariant is positive and if (M, g) is not conformally equivalent to the round sphere. The result we obtain is the following: Theorem 1.6. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Assume that µ 2 (M, g) is attained by a generalized metric u N -2 g where u ∈ L N (M ), u ≥ 0 and u ≡ 0. Let Ω be the nodal set of u. Then, there exists a nodal solution w

∈ C ∞ (M \ Ω) ∩ C 3,α (M ) ( α ≤ N -2) of equation (1) such that |w| = u.
A corollary of Theorems 1.4, 1.5 and 1.6 is then Corollary 1.7. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 whose Yamabe invariant is non-negative. We assume that one of the following assumptions is true:

• µ 1 (M, g) > 0, (M, g) is not locally conformally flat and n ≥ 11;

• µ 1 (M, g) = 0, (M, g) is not locally conformally flat and n ≥ 9.

Then, there exists a nodal solution of Yamabe equation (1).

Acknowledgement

The authors want to thank M. Ould Ahmedou for many interesting conversations about nodal solutions of the Yamabe equation. His large knowledge about such problems was a stimulating inspiration for this article. The author are also extremely obliged to Frédéric Robert for having pointed out a little mistake in the first version of this paper.

2. Variational characterization of µ 2 (M, g) 2.1. Notation. In the whole article we will use the following notations

L N + (M ) := u ∈ L N (M ) | u ≥ 0, u ≡ 0 .
2.2. Grassmannians and the min-max principle. Let Gr k (C ∞ (M )) be the k-dimensional Grassmannian in C ∞ (M ), i.e. the set of all k-dimensional subspaces of C ∞ (M ). The Grassmannian is an important ingredient in the min-max characterization of λ k (g)

λ k (L g) := inf V ∈Gr k (C ∞ (M)) sup v∈V \{0} (L g v)v dv g
M v 2 dv g . We will also need a slightly modified Grassmannian. For any u ∈ L N + (M ) we define Gr u k (C ∞ ) to be the set of all k-dimensional subspaces of C ∞ (M ), such that the restriction operator to M \ u -1 (0) is injective. More explicitly, we have span(v 1 , . . . , v k ) ∈ Gr u k (C ∞ (M )) if and only if v 1 | M\u -1 (0) , . . . , v k | M\u -1 (0) are linearly independent. Sometimes it will be convenient to use the equivalent statement that the functions u

N -2 2 v 1 , . . . , u N -2 2 v k are linearly independent.
Similarly, by replacing C ∞ (M ) by H 2 1 (M ) we obtain the definitions of Gr k (H 2 1 (M )) and Gr u k (H 2 1 (M )).

The functionals. For all

u ∈ L N + (M ), v ∈ H 2 1 (M ) such that u N -2 2 v ≡ 0, we set F (u, v) = M c n |∇v| 2 + S g v 2 dv g M v 2 u N -2 dv g M u N dv g 2 n .
2.4. Variational characterization of µ 2 (M, g). The following characterization will be of central importance for our article.

Proposition 2.1. We have

µ k (M, g) = inf u∈L N + (M) V ∈Gr u k (H 2 1 (M)) sup v∈V \{0} F (u, v) (2)
Proof. Let u be a smooth positive function on M . For all smooth functions f , f ≡ 0, we set g = u N -2 g (N = 2n n-2 ) and

F ′ (u, f ) = M f L gf dv g M f 2 dv g .
The operator L g is conformally invariant (see [START_REF] Hebey | Introduction à l'analyse non-linéaire sur les variétés[END_REF]) in the following sense:

u N -1 L g(u -1 f ) = L g (f ) (3)
Together with the fact that

dv g = u N dv g , (4)
we get that

F ′ (u, f ) = M (uf )L g (uf ) dv g M (uf ) 2 u N -2 dv g .
Using the min-max principle, we can write that

λ k (g) = inf V ∈Gr u k (H 2 1 (M)) sup f ∈V \{0} F ′ (u, f )
Now, replacing uf by v, we obtain that

λ k (g) = inf V ∈Gr k (H 2 1 (M)) sup v∈V \{0} M vL g v dv g M v 2 u N -2 dv g . ( 5
)
Using the definition of µ 2 and Vol g (M ) = M u N dv g , we derive

µ k (M, g) = inf u∈L N + (M) V ∈Gr u k (C ∞ (M)) sup v∈V \{0} F (u, v)
The result follows immediately.

3. Generalized metrics and the Euler-Lagrange equation 3.1. A regularity result. We will need the following result.

Lemma 3.1. Let u ∈ L N (M ) and v ∈ H 2 1 (M ). We assume that L g v = u N -2 v
holds in the sense of distributions. Then, v ∈ L N +ε (M ) for some ε > 0.

This result is well known for the standard Yamabe equation. Proofs for the standard Yamabe equation can be found in [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF] and [START_REF] Hebey | Introduction à l'analyse non-linéaire sur les variétés[END_REF], and the modifications for proving Lemma 3.1 are obvious. Unfortunately, [START_REF] Trudinger | Remarks concerning the conformal deformation of Riemannian structures on compact manifolds[END_REF] contains some typos, and the book [START_REF] Hebey | Introduction à l'analyse non-linéaire sur les variétés[END_REF] is difficult to obtain. This is why we included a proof in the appendix for the convenience of the reader.

3.2.

The k-th eigenvalue of the Yamabe operator for a generalized metric. On a given Riemannian manifold (M, g) we say that g = u N -2 g, u ∈ L N + (M ), is a generalized metric conformal to g. For a generalized metric g, we can define

λ k (g) = inf V ∈Gr u k (H 2 1 (M)) sup v∈V \{0} M vL g v dv g M v 2 u N -2 dv g . ( 6
)
Proposition 3.2. For any u ∈ L N + , g = u N -2 there exist two functions v, w belonging to H 2 1 (M ) with v ≥ 0 and such that in the sense of distributions.

L g v = λ 1 (g)u N -2 v (7)
and

L g w = λ 2 (g)u N -2 w. (8)
Moreover, we can normalize v, w by

M u N -2 v 2 dv g = M u N -2 w 2 dv g = 1 and M u N -2 vw dv g = 0 (9)
For k = 2 the infimum in formula (5) over all subspaces V ∈ Gr u 2 (H 2 1 (M )) is attained by V = span(v, w) and the supremum over the functions in V \ {0} is attained by w. The reader should pay attention to the fact that the space V is in general non unique. As one can check, if w changes the sign then the supremum over all v ∈ V = span(v, w) \ {0} and the supremum over all v ∈ V 1 = span(w, |w|) \ {0} coincide.

From section (2), we get

µ 2 (M, g) = inf g∈[g] λ 2 (g).
Hence, µ 2 (M, g) can be attained by a regular metric, or by a generalized metric or it can be not attained at all. These questions are discussed in Section 5. Let us now prove Proposition 3.2.

Proof of Proposition 3.2: Let (v m ) m be a minimizing sequence for λ 1 (g) i.e. a sequence

v m ∈ H 2 1 (M ) such that lim m→∞ M c n |∇v m | 2 + S g v 2 m dv g M u N -2 v 2 m dv g = λ 1 (g).
It is well known that (|v m |) m is also a minimizing sequence. Hence, we can assume that

v m ≥ 0. If we normalize v m by M u N -2 v 2 m dv g = 1, then (v m ) m is bounded in H 2 1 (M )
and after restriction to a subsequence we may assume that there exists

v ∈ H 2 1 (M ), v ≥ 0 such that v m → v weakly in H 2 1 (M ), strongly in L 2 (M ) and almost everywhere. If u is smooth, then M u N -2 v 2 dv g = lim m M u N -2 v 2 m dv g = 1 (10)
and by standard arguments, v is a non-negative minimizer of the functional associated to λ 1 (g). We must show that (10) still holds if u ∈ L N + (M ). Let A > 0 be a large real number and set u A = inf(u, A). Then, using the Hölder inequality, we write

M u N -2 v 2 m -v 2 dv g ≤ M u N -2 A |v 2 m -v 2 | dv g + M (u N -2 -u N -2 A )(|v m | + |v|) 2 dv g ≤ A M |v 2 m -v 2 | dv g + M (u N -2 -u N -2 A ) N N -2 dv g N -2 N M (|v m | + |v|) N dv g 2 N
.

By Lebesgue's theorem we see that

lim A→+∞ M (u N -2 -u N -2 A ) N N -2 dv g = 0. Since (v m ) m is bounded in H 2 1 (M ), it is bounded in L N (M ) and hence there exists C > 0 such that M (|v m | + |v|) N dv g ≤ C. By strong convergence in L 2 (M ), lim m M |v 2 m -v 2 | dv g = 0.
Equation (10) easily follows and v is a non-negative minimizer of the functional associated to λ 1 (g).

Writing the Euler-Lagrange equation of v, we find that v satisfies equation (7). Now, we define

λ ′ 2 (g) = inf M c n |∇w| 2 + S g w 2 dv g M u N -2 |w| 2 dv g
where the infimum is taken over smooth functions w such that u N -2 2 w ≡ 0 and such that M u N -2 vw dv g = 0. With the same method, we find a minimizer w of this problem that satisfies (8) with λ ′ 2 (g) instead of λ 2 (g). However, it is not difficult to see that λ ′ 2 (g) = λ 2 (g) and Proposition 3.2 easily follows.

Euler-Lagrange equation of a minimizer of λ

2 Vol 2/n . Lemma 3.3. Let u ∈ L N + (M ) with u N = 1. Suppose that w 1 , w 2 ∈ H 2 1 (M ) \ {0}, w 1 , w 2 ≥ 0 satisfy (c n |∇w 1 | 2 + Scal g w 2 1 ) dv g ≤ µ 2 (M, g) u N -2 w 2 1 (11) (c n |∇w 2 | 2 + Scal g w 2 2 ) dv g ≤ µ 2 (M, g) u N -2 w 2 2 (12)
and suppose that (M \ w -1 1 (0)) ∩ (M \ w -1 2 (0)) has measure zero. Then u is a linear combination of w 1 and w 2 and we have equality in (11) and (12).

Proof. We let ū = aw 1 + bw 2 where a, b > 0 are chosen such that

a N -2 b N -2 M u N -2 w 2 1 dv g M u N -2 w 2 2 dv g = M w N 1 dv g M w N 2 dv g (13)
and

M ūN dv g = a N M w N 1 + b N w N 2 = 1. ( 14
)
Because of the variational characterization of µ 2 we have

µ 2 (M, g) ≤ sup (λ,µ)∈R 2 \{(0,0)} F (ū, λw 1 + µw 2 ) (15)
By ( 11),( 12) and ( 14), and since (M \ w -1

1 (0)) ∩ (M \ w -1 2 (0)) has measure zero F (ū, λw 1 + µw 2 ) = λ 2 M c n |∇w 1 | 2 + S g w 2 1 dv g + µ 2 M c n |∇w 2 | 2 + S g w 2 2 dv g λ 2 M |ū| N -2 w 2 1 dv g + µ 2 M |ū| N -2 w 2 2 dv g ≤ µ 2 (M, g) λ 2 M u N -2 w 2 1 dv g + µ 2 M u N -2 w 2 2 dv g λ 2 a N -2 M w N 1 dv g + µ 2 b N -2 M w N 2 dv g . ( 16
)
As one can check, relation (13) implies that this expression does not depend on λ, µ. Hence, setting λ = a and µ = b, the denominator is 1, and we get sup

(λ,µ)∈R 2 \{(0,0)} F (ū, λw 1 + µw 2 ) ≤ µ 2 (M, g) M u N -2 (a 2 w 2 1 + b 2 w 2 2 ) dv g = µ 2 (M, g) M u N -2 ū2 dv g .
By Hölder inequality, sup

(λ,µ)∈R 2 \{(0,0)} F (ū, λw 1 + µw 2 ) ≤ µ 2 (M, g) M u N dv g N -2 N M ūN dv g 2 N = µ 2 (M, g). (17)
Inequality (15) implies that we have both equality in the Hölder inequality of (17) and in (16). The equality in the Hölder inequality implies that there exists a constant c > 0 such that u = cū almost everywhere. Moreover, since u N = ūN = 1, we have u = ū = aw 1 + bw 2 . The equality in (16) implies inequality in (11) and (12).

Theorem 3.4 (Euler-Lagrange equation). Assume that µ 2 (M, g) = 0 and that µ 2 (M, g) is attained by a generalized metric g = u N -2 g with u ∈ L N + (M ). Let v, w be as in Proposition 3.2. Then, u = |w|. In particular,

L g w = µ 2 (M, g)|w| N -2 w (18)
Moreover, w has alternating sign and w ∈ C 3,α (M ) (α ≤ N -2).

Remark 3.5. Assume that µ 2 (M, g) is equal to 0 and is attained by a generalized metric g ′ , then, using the conformal invariance of the Yamabe operator, it is easy to check that for all generalized metrics g conformal to g ′ , we have λ 2 (g) = 0. Consequently, each metric conformal to g is a minimizer for µ 2 (M, g) and Theorem 3.4 is always false in this case. However, we will still get a nodal solution of (1) if µ 2 (M, g) = 0. Indeed, by Theorem 1.4 and the remark above, λ 2 (g) = 0. Let w be an eigenfunction associated to λ 2 (g). We have L g w = 0. Then, we have a solution of (18).

Remark 3.6. Assume that µ 2 (M, g) = 0 and that µ 2 (M, g) is attained by a generalized metric. Let w be the solution of equation ( 18) given by Theorem 3.4. We let Ω + = {x ∈ M s.t. w(x) > 0} and Ω -= {x ∈ M s.t. w(x) -0}. Then, a immediate consequence of Lemma 3.3 is that Ω + and Ω -have exactly one connex component.

Proof. Without loss of generality, we can assume that M u N dv g = 1. By assumption we have λ 2 (g) = µ 2 (M, g). Let v, w ∈ H 2 1 (M ) be some functions satisfying equations ( 7), (8) and relation (9).

Step 1. We have λ 1 (g) < λ 2 (g).

We assume that λ 1 (g) = λ 2 (g). Then, after possibly replacing w by a linear combination of v and w, we can assume that the function u N -2 2 w changes the sign. We apply Lemma 3.3 for w 1 := sup(w, 0) and w 2 := sup(-w, 0). We obtain the existence of a, b > 0 with u = aw 1 + bw 2 . Now, by Lemma 3.1, w ∈ L N +ε (M ). By a standard bootstrap argument, equation (8) shows that w ∈ C 2,α (M ) for all α ∈]0, 1[. It follows that u ∈ C 0,α (M ) for all α ∈]0, 1[. Now, since λ 1 (g) = λ 2 (g) and by definition of λ 1 (g), w is a minimizer of the functional w → F (u, w) among the functions belonging to H 2 1 (M ) and such that u

N -2 2 w ≡ 0. Since F (u, w) = F (u, |w|),
we see that |w| is a minimizer for the functional associated to λ 1 (g) and hence, writing the Euler-Lagrange equation of the problem, w satisfies the same equation as w. As a consequence, |w| is C 2 (M ). By the maximum principle, we get |w| > 0 everywhere. This is false. Hence, the step is proved.

Step 2. The function w changes the sign.

Assume that w does not change the sign, i.e. after possibly replacing w by -w, we have w ≥ 0. Using (9) we see that (M \ v -1 (0)) ∩ (M \ w -1 (0)) has measure zero. Setting w 1 := v and w 2 := w we have (11) and (12). While we have equality in (12), Step 1 implies that inequality (11) is strict. However using Lemma 3.3 we can derive equality in (11). Hence we obtain a contradiction, and the step is proved.

Step 3. There exists a, b > 0 such that u = a sup(w, 0) + b sup(-w, 0). Moreover, w ∈ C 2,α (M ) and u ∈ C 0,α (M ) for all α ∈]0, 1[. As in the proof of Step 1 we apply Lemma 3.3 for w 1 := sup(w, 0) and w 2 := sup(-w, 0). We obtain the existence of a, b > 0 such that u = aw 1 + bw 2 . As in Step 1 we get that w ∈ C 2,α (M ) and u ∈ C 0,α (M ) for all α ∈]0, 1[. This proves the present step.

Step 4. Conclusion.

Let h ∈ C ∞ (M ) whose support is contained in M \ {u -1 (0)}. For t close to 0, set u t = |u + th|. Since u > 0 on the support of h and since u is continuous (see last step), we have for t close to 0, u t = u + th. As span(v, w) ∈ Gr u 2 (H 2 1 (M )) we obtain using (2) for all t µ 2 (M, g) ≤ sup

(λ,µ)∈R 2 \{(0,0)} F (u t , λv + µw).

Equations ( 7), (8), and relation (9) yield

F (u t , λv + µw) = λ 2 λ 1 (g) M u N -2 v 2 dv g + µ 2 λ 2 (g) M u N -2 w 2 dv g λ 2 M u N -2 t v 2 dv g + 2λµ M u N -2 t vw dv g + µ 2 M u N -2 t w 2 dv g M u N t dv g 2 n = λ 2 λ 1 (g) + µ 2 λ 2 (g) λ 2 a t + λµb t + µ 2 c t M |u t | N dv g 2 n
.

where

a t = M u N -2 t v 2 dv g , b t = 2 u N -2 t vw dv g and c t = M u N -2 t w 2 dv g .
The functions a t , b t and c t are smooth for t close to 0, furthermore a 0 = c 0 = 1 and b 0 = 0. The function f (t, α) := F (u t , sin(α)v + cos(α)w) is smooth for small t. Using λ 1 (g) < λ 2 (g) one calculates

∂ ∂α f (0, α) = 0 ⇔ α ∈ π 2 Z ∂ 2 ∂α 2 f (0, α) < 0 for α ∈ πZ ∂ 2 ∂α 2 f (0, α) > 0 for α ∈ πZ + π 2
Applying the implicit function theorem to ∂f ∂α at the point (0, 0), we see that there is a smooth function t → α(t), defined on a neighborhood of 0 with α(0) = 0 and

f (t, α(t)) = sup α∈R f (t, α) = sup (λ,µ)∈R 2 \{(0,0)} F (u t , λv + µw). As a consequence d dt | t=0 sin 2 α(t) = d dt | t=0 cos 2 α(t) = d dt | t=0 (sin 2 α(t)a t ) = d dt | t=0 (sin α(t) cos α(t)b t ) = 0.
Hence, d dt | t=0 f (t, α(t)) exists and we have

d dt | t=0 f (t, α(t)) = λ 2 (M, g) - d dt | t=0 c t + d dt | t=0 M |u t | N dv g 2 n = λ 2 (M, g)(N -2) - M u N -3 hw 2 dv g + M u N -1 h dv g .
By definition of µ 2 (M, g), f admits a minimum in t = 0. As λ 2 (M, g) = µ 2 (M, g) = 0 we obtain

M u N -3 hw 2 dv g = M u N -1 h dv g .
Since h is arbitrary (we just have to ensure that its support is contained in M \ {u -1 (0)}), we get that u N -3 w 2 = u N -1 on M \ {u -1 (0)}, hence u = |w| on M \ {u -1 (0)}. Together with Step 3, we get u = |w| everywhere. This proves theorem 3.4.

4.

A sharp Sobolev inequality related to µ 2 (M, g) 4.1. Statement of the results. For any compact Riemannian manifold (M, g) of dimension n ≥ 3, Hebey and Vaugon have shown in ( [START_REF] Hebey | Meilleures constantes dans le théorème d'inclusion de Sobolev[END_REF]) that there exists B 0 (M, g) > 0 such that

µ 1 (S n ) = n(n -1) ω 2 n n = inf u∈H 2 1 (M)\{0} M c n |∇u| 2 + B 0 M u 2 dv g M u N dv g 2 n (S)
where ω n stands for the volume of the standard n-dimensional sphere S n and where µ 1 (S n ) is the Yamabe invariant of S n .

This inequality is strongly related to the resolution of the Yamabe problem. It allows to avoid concentration for the minimizing sequence of µ 1 (M, g). For the minimization of µ 2 (M, g), this inequality is not sufficient and another one must be constructed. The following result is adapted to the problem of minimizing µ 2 (M, g).

Theorem 4.1. On a compact connected Riemannian manifold (M, g) of dimension n ≥ 3 we have

2 2 n µ 1 (S n ) = inf u∈L N + (M) V ∈Gr u k (H 2 1 (M)) sup v∈V \{0} M c n |∇v| 2 dv g + B 0 (M, g) M v 2 dv g M u N dv g 2 N M u N -2 v 2 dv g (S 1 )
where B 0 (M, g) is given by inequality (S).

We present now two corollaries of Theorem 4.1.

Corollary 4.2. For the standard n-dimensional sphere we have

µ 2 (S n ) = 2 2 n µ 1 (S n ). Corollary 4.3. For all u ∈ C ∞ c (R n ) and V ∈ Gr u 2 (C ∞ c (R n )) we have 2 2/n µ 1 (S n ) ≤ sup v∈V \{0} R n c n |∇v| 2 dv g R n |u| N dv g 2 N R n |u| N -2 v 2 dv g 4.2. Proof of theorem 4.1. The functional G(u, v) := M c n |∇v| 2 dv g + B 0 (M, g) M v 2 dv g M u N dv g 2 N M u N -2 v 2 dv g is continuous on L N + (M ) × (H 2 1 (M ) \ {0}). As a consequence I(u, V ) := sup v∈V \{0} G(u, v) depends continuously on u ∈ L N + (M ) and V ∈ Gr u 2 (H 2 1 (M ))
. Thus, in order to show the theorem it is sufficient to show that I(u, V ) ≥ 2 2/n µ 1 (S n ) for all smooth u > 0 and V ∈ Gr 2 (C ∞ (M )). Without loss of generality, we can assume

M u N dv g = 1. ( 19
)
The operator

v → P (v) := c n u 2-N 2 ∆(u 2-N 2 v) + B 0 (M, g)u 2-N
v is an elliptic operator on M , and P is self-adjoint with respect to the L 2 -scalar product. Hence, P has discrete spectrum λ 1 ≤ λ 2 ≤ . . . and the corresponding eigenfunctions ϕ 1 , ϕ 2 , . . . are smooth. Setting

v i := u 2-N 2 ϕ i we obtain (c n ∆ + B 0 ) (v i ) = λ i u N -2 v i u N -2 v i v j dv g = 0 if λ i = λ j .
The maximum principle implies that an eigenfunction to the smallest eigenvalue λ 1 has no zeroes. Hence λ 1 < λ 2 , and we can assume v 1 > 0.

We define w + := a + sup(0, v 2 ) and w -:= a -sup(0, -v 2 ), where we choose a + , a -> 0 such that

M u N -2 w 2 -dv g = M u N -2 w 2 + dv g = 1.
We let Ω -= {w < 0} and Ω + = {w ≥ 0}. By Hölder inequality,

2 = M u N -2 w 2 -dv g + M u N -2 w 2 + dv g ≤ Ω-u N dv g N -2 N M w N -dv g 2 N + Ω+ u N dv g N -2 N M w N + dv g 2 N .
Using the sharp Sobolev inequality (S), we get that

2µ 1 (S n ) ≤ Ω- u N dv g N -2 N M w -u N -2 2 P u N -2 2 w -dv g (20) + Ω+ u N dv g N -2 N M w + u N -2 2 P u N -2 2 w + dv g ( 21
)
Question 1 is solved by the following result.

Proposition 5.2. If M is connected, then µ 2 (M, g) cannot be attained by a metric.

Indeed, otherwise by Theorem 3.4, we would have that u = |w| and hence u cannot be positive. Theorem 1.4 and the following result answer Question 2.

Proposition 5.3. The invariant µ 2 (S n ) is not attained by a generalized metric.

This proposition immediately follows from Proposition 5.6.

5.2. Some bounds of µ 2 (M, g). At first, we give an upper bound for µ 2 (M, g).

Theorem 5.4. Let (M, g) be an n-dimensional compact Riemannian manifold with µ 1 (M, g) ≥ 00. Then,

µ 2 (M, g) ≤ (µ 1 (M, g) n 2 + µ 1 (S n ) n 2 ) 2 n . ( 22
)
This inequality is strict in the following cases:

• µ 1 (M, g) > 0, (M, g) is not locally conformally flat and n ≥ 11;

• µ 1 (M, g) = 0, (M, g) is not locally conformally flat and n ≥ 9.

From the solution of the Yamabe problem by Aubin and Schoen [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF][START_REF] Schoen | Conformal deformation of a Riemannian metric to constant scalar curvature[END_REF] we know that if (M, g) is not conformally equivalent to S n , then µ 1 (M, g) < µ 1 (S N ). Hence, (22) implies the following corollary.

Corollary 5.5. Let (M, g) be an n-dimensional compact connected Riemannian manifold whose Yamabe invariant is non-negative. Then µ 2 (M, g) ≤ µ 2 (S n ) with inequality if and only if (M, g) is conformally diffeomorphic to the sphere S n .

These inequalities are very important, because they can be used to avoid concentration of minimizing sequences for µ 2 , in a way which is similar to the resolution of the Yamabe problem.

The following proposition gives a lower bound for µ 2 .

Proposition 5.6. Let (M, g) be a n-dimensional compact Riemannian manifold whose Yamabe invariant is non-negative. Then,

µ 2 (M, g) ≥ 2 2 n µ 1 (M, g). (23)
Moreover, if M is connected and if µ 2 (M, g) is attained by a generalized metric, then this inequality is strict.

When µ 1 (M, g) = 0, inequality (23) is trivial. If µ 1 (M, g) > 0, by a possible chande of metric in the conformal class, we can assume that the scalar curvature is positive. The proof of inequality ( 23) is exactly the same as the one of Theorem 4.1. We just have to replace B 0 (M, g) by S g . Moreover, if M were connected and if µ 2 (M, g) were attained by a generalized metric, then inequality (20) would be an equality and we would have that w + or w -is a function for which equality in the Sobolev inequality (S) is attained. By the maximum principle, we would get that w + or w -is positive on M which is impossible.

5.2.1. Proof of theorem 5.4.

Lemma 5.7. For any α > 2, there is a C > 0 such that

|a + b| α ≤ a α + b α + C(a α-1 b + ab α-1 )
for all a, b > 0.

Proof of Lemma 5.7. Without loss of generality, we can assume that a = 1. Then we set for x > 0,

f (x) = |1 + x| α -(1 + x α ) x α-1 + x . One checks that lim x→0 f (x) = lim x→+∞ f (x) = α. Since f is continuous, f is bounded by a constant C on R + .
Clearly, this constant is the desired C in inequality of Lemma 5.7.

Proof of Theorem 5.4.

For u ∈ H 2 1 (M ) \ {0} let Y (u) = M c n |∇u| 2 + S g u 2 dv g M |u| N dv g 2 N
be the Yamabe functional of M . The solution of the Yamabe problem provides the existence of a smooth positive minimizer v of Y , and we can assume

M v N dv g = 1. (24)
Then, v satisfies the Yamabe equation

L g v = µ 1 (M, g)v N -1 . (25)
Let x 0 ∈ M be fixed and choose a system (x 1 , • • • , x n ) of normal coordinates at x 0 . We note r = dist g (x 0 , .). If δ > 0 is a small fixed number, let η be a smooth cut-off function such that 0 ≤ η ≤ 1, η(B(x 0 , δ)) = {1} and η(M \ B(x 0 , 2δ) = {0}, |∇η| ≤ 2/δ. Then, we can define for all ε > 0

v ε = C ε η(ε + r 2 ) 2-n n .
where

C ε > 0 is such that M v N ε dv g = 1. (26)
By standard computations (see [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF])

lim ε→0 Y (v ε ) = µ 1 (S n ). (27)
If (M, g) is not locally conformally flat, if g is well chosen in the conformal class and if x 0 is well chosen in M , it was also proven in [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] that there exists a constant C(M ) > 0 such that

Y (v ε ) = µ 1 (S n ) -C(M )ε 2 + o(ε 2 ) if n > 6 µ 1 (S n ) -C(M )ε 2 | ln(ε)| + o(ε 2 | ln(ε)|) if n = 6. (28)
Moreover, it follows from [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] that

aε n-2 4 ≤ C ε ≤ bε n-2 4
where a, b > 0 are independent of ε. If p ≥ 1, standard computations made in [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] show that there exist some constants c, C > 0 independent of ε such that

cα p,ε ≤ M v p ε dv g ≤ Cα p,ε (29)
where

α p,ε = ε 2n-(n-2)p 4 if p > n n-2 ; | ln(ε)|ε n 4 if p = n n-2 ; ε (n-2)p 4 if p < n n-2
Since the large inequality if easier to obtain, we only prove strict inequality. Assume first that µ 1 (M, g) > 0, that (M, g) is not locally conformally flat and that n ≥ 11. We set,

u ε = Y (v ε ) 1 N -2 v ε + µ 1 (M, g) 1 N -2 v.
Let us derive estimates for F u ε , λv ε + µv) . Let (λ, µ) ∈ R 2 \ {(0, 0)}. Using (24), (26) and the equation (25) of v, we get that

F (u ε , λv ε + µv) = λ 2 M v ε L g (v ε ) dv g + µ 2 M vL g (v) dv g + 2λµ M v ε L g v dv g λ 2 M |u ε | N -2 (λv ε + µv) 2 dv g M u N ε dv g 2 n . = λ 2 Y (v ε ) + µ 2 µ 1 (M, g) + 2λµµ 1 (M, g) M |v| N -2 vv ε dv g λ 2 M |u ε | N -2 v 2 ε dv g + µ 2 M |u ε | N -2 v 2 dv g + 2λµ M |u ε | N -2 vv ε dv g M u N ε dv g 2 n . ( 30
)
Using the definition of u ε

λ 2 M |u ε | N -2 v 2 ε dv g + µ 2 M |u ε | N -2 v 2 dv g + 2λµ M |u ε | N -2 vv ε dv g ≥ λ 2 Y (v ε ) M |v ε | N dv g + µ 2 µ 1 (M, g) M |v| N dv g + 2λµ M |u ε | N -2 vv ε dv g = λ 2 Y (v ε ) + µ 2 µ 1 (M, g) + 2λµ M |u ε | N -2 vv ε dv g . If λµ ≥ 0, we have 2λµ M |u ε | N -2 vv ε dv g ≥ 2λµµ 1 (M, g) M v N -2 v ε dv g .
This implies that

λ 2 Y (v ε ) + µ 2 µ 1 (M, g) + 2λµµ 1 (M, g) M |v| N -2 vv ε dv g λ 2 M |u ε | N -2 v 2 ε dv g + µ 2 M |u ε | N -2 v 2 dv g + 2λµ M |u ε | N -2 vv ε dv g ≤ 1.
If λµ < 0 then, we write that since N -2 ∈]0, 1[,

|u ε | N -2 ≤ Y (v ε )v N -2 ε + µ 1 (M, g)v N -2 .
We obtain that

λ 2 M |u ε | N -2 v 2 ε dv g + µ 2 M |u ε | N -2 v 2 dv g + 2λµ M |u ε | N -2 vv ε dv g ≥ λ 2 Y (v ε ) + µ 2 µ 1 (M, g) -C M v N -1 v ε dv g + M v N -1 ε v dv g .
where C > 0 is as in in the following a positive real number independent of ε. Together with (29), we get that

λ 2 M |u ε | N -2 v 2 ε dv g + µ 2 M |u ε | N -2 v 2 dv g + 2λµ M |u ε | N -2 vv ε dv g ≥ λ 2 Y (v ε ) + µ 2 µ 1 (M, g) + O(ε n-2 4 ).
It follows that sup

(λ,µ)∈R 2 \{(0,0)} λ 2 Y (v ε ) + µ 2 µ 1 (M, g) + 2λµµ 1 (M, g) M |v| N -2 vv ε dv g λ 2 M |u ε | N -2 v 2 ε dv g + µ 2 M |u ε | N -2 v 2 dv g + 2λµ M |u ε | N -2 vv ε dv g ≤ 1 + O(ε n-2 4 ). (31) By Lemma 5.7, M u N ε dv g ≤ (Y (u ε )) n 2 M v N ε dv g + µ 1 (M, g) n 2 M v N dv g +C M v N -1 v ε dv g + M v N -1 ε v dv g .
By (24), ( 26), ( 28) and (29), we obtain

M u N ε dv g 2 n ≤ (µ 1 (M, g) n 2 + µ 1 (S n ) n 2 ) 2 n -Cε 2 + O(ε n-2 4 ) + o(ε 2 ). ( 32
)
Since n-2 4 > 2, we get from (31) and (32) that for ε small enough µ 2 (M, g) ≤ sup

(λ,µ)∈R 2 \{(0,0)} F (u ε , λv ε + µv) ≤ (µ 1 (M, g) n 2 + µ 1 (S n ) n 2 ) 2 n -Cε 2 + O(ε n-2 4 ) + o(ε 2 ) < (µ 1 (M, g) n 2 + µ 1 (S n ) n 2 ) 2 n .
This proves Theorem 5.4 if µ 1 (M, g) > 0.

Now, we assume that µ 1 (M, g) = 0, that (M, g) is not locally conformally flat and that n ≥ 9. For more simplicity, We set

u ε = v ε instead of u ε = Y (v ε ) n-2
4 v ε as above. We proceed exactly as in the case µ 1 (M, g) > 0. We obtain that for (λ, µ) ∈ R 2 \ {(0, 0)}

F (u ε , λv ε + µv) = λ 2 Y (v ε ) λ 2 M v N ε dv g + µ 2 M |v ε | N -2 v 2 dv g + 2λµ M |v ε | N -1 v dv g M v N ε dv g 2 n = λ 2 Y (v ε ) λ 2 + µ 2 M |v ε | N -2 v 2 dv g + 2λµ M |v ε | N -1 v dv g . Let λ ε , µ ε be such that λ 2 ε + µ 2 ε = 1 and such that F (u ε , λ ε v ε + µ ε v) = sup (λ,µ)∈R 2 \{(0,0)} (u ε , λv ε + µv).
If λ ε = 0, we obtain that F (u ε , λ ε v ε + µ ε v) = 0 and the theorem would be proven. Then we assume that λ ε = 0 and we write that

F (u ε , λ ε v ε + µ ε v) = Y (v ε ) 1 + 2x ε b ε + x 2
ε a ε where x ε = µε λε and where, using (29)

b ε = M v N -1 ε v dv g ∼ ε→0 Cε n-2 4
and

a ε = M v N -2 ε v 2 dv g ∼ ε→0 Cε.
Maximizing this expression in x ε and using (28), we get that

F (u ε , λ ε v ε + µ ε v) ≤ µ 1 (S n ) -C(M )ε 2 + o(ε 2 ) 1 - b 2 ε aε = µ 1 (S n ) -C(M )ε 2 + o(ε 2 ) 1 -O(ε n-4 2 )
.

Since n ≥ 9, n-4 2 > 2 and we get that for ε small,

F (u ε , λ ε v ε + µ ε v) < µ 1 (S n ).
This proves Theorem 5.4.

6. Existence of a minimum of µ 2 (M, g)

The aim of this section is to prove Theorem 1.4.

We study a sequence of metrics (g

m ) m = (u N -2 m g) m (u m > 0, u m ∈ C ∞ (M )
) which minimizes the infimum in the definition of µ 2 (M, g) i.e. a sequence of metrics such that lim

m λ 2 (g m )Vol(M, g m ) 2 n = µ 2 (M, g).
Without loss of generality, we may assume that Vol(M, g m ) = 1 i.e. that M u N m dv g = 1.

(33)

In particular, the sequence (u m ) m is bounded in L N (M ) and there exists u ∈ L N (M ), u ≥ 0 such that u m ⇀ u weakly in L N (M ). We are going to prove that u = 0 and that the generalized metric u N -2 g minimizes µ 2 (M, g). Proposition 3.2 implies the existence of v m , w m ∈ C ∞ (M ), v m ≥ 0 such that

L g v m = λ 1,m u N -2 m v m (34)
and

L g w m = λ 2,m u N -2 m w m . (35)
where λ i,m = λ i (g m ) and such that

M u N -2 m v 2 m dv g = M u N -2 m w 2 m dv g = 1 and M u N -2 m v m w m dv g = 0 (36)
With these notations and by (33),

lim m λ 2,m = µ 2 (M, g).
Moreover, by the maximum principle, v m > 0. If λ 1,m = λ 2,m then w m would be a minimizer of the functional associated to λ 1,m and by the maximum principle, we would get that w m > 0. This contradicts (36). Hence, λ 1,m < λ 2,m for all m. The sequences (v m ) m and (w m ) m are bounded in H 2 1 (M ). We can find v, w ∈ H 2 1 (M ), v ≥ 0 such that v m (resp. w m ) tends to v (resp. w) weakly in H 2 1 (M ). Together with the weak convergence of the (u m ) m towards u in L N (M ), we get that in the sense of distributions

L g v = µ 1 u N -2 v (37)
and

L g w = µ 2 (M, g) u N -2 w. (38)
where

µ 1 = lim m λ 1,m ≤ µ 2 (M, g).
From what we know until now, it is not clear whether v and w are linearly independent, and even if they are, their restrictions to the set M \ u -1 (0) might be linearly dependent.

It will take a certain effort to prove the following claim. Once the claim is proved, we have span(v, w) ∈ Gr u 2 (H 2 1 (M )), and this implies that sup

(λ,µ) =(0,0) F (u, λv + µw) = µ 2 (M, g).
Hence, by equations (37) and (38), the generalized metric u N -2 g minimizes µ 2 (M, g), i.e. Theorem 1.4 is proved.

The first step in the proof of the claim is an estimate that avoids concentration of w m and v m .

Step

1. Let x ∈ M and ε ∈ ]0, N -2 2 [. We choose a cut-off function η ∈ C ∞ such that 0 ≤ η ≤ 1, η(B x (δ)) ≡ 1 (where δ > 0 is a small number) and η(M \ B x (2δ)) ≡ 0, |∇η| ≤ 2/δ. We define W m = η|w m | ε w m . Then, we have M |W m | N dv g 2 N ≤ µ 2 (M, g)(1 -α ε ) -1 µ 1 (S n ) -1 Bx(2δ) u N m 2 n M |W m | N dv g 2 N + C δ . ( 39
)
where C δ is a constant that may depend on δ but not on ε and where lim ε→0 α ε = 0. Moreover, the same conclusion is true with

V m = η|v m | ε v m instead of W m .
The proof uses classical methods. We will explain the proof for W m . The proof for V m uses exactly the same arguments.

At first, we differentiate the definition of W and obtain

|∇W m | 2 ≥ ∇(|w m | ε w m ) 2 η 2 -2|∇η| |w m | 1+ε ∇(|w m | ε w m ) η + |∇η| 2 |w m | 2+2ε ≥ ∇ |w m | ε w m 2 η 2 - 1 2 ∇(|w m | ε w m )| 2 η 2 + 2|∇η| 2 |w m | 2+2ε + |∇η| 2 |w m | 2+2ε
This leads to

η 2 |∇(|w m | ε w m )| 2 ≤ 2|∇W m | 2 + 2|∇η| 2 |w m | 2+2ε . (40)
Now, we want to derive lower bound for

(∇(η 2 |w m | 2ε w m ), ∇w m ) = |∇W m | 2 -∇(η|w m | ε ) 2 |w m | 2 (41)
For the second summand on the right hand side in (41) we have the bound where α ε → 0 when ε → 0 and where C > 0 is a constant independent of ε. This relations shows that From (39), and the fact µ 2 (M, g) < µ 1 (S n ), we get that for ε small enough, there exists a constant K < 1 such that

|∇(η|w m | ε )| 2 |w m | 2 = |∇η| 2 |w m | 2+2ε + 2(∇η, ∇|w m | ε ) η|w m | 2+ε + η 2 ∇(|w m | ε ) 2 w 2 m ≤ 2|∇η| 2 |w m | 2+2ε + 2η 2 ∇(|w m | ε ) 2 w 2 m ≤ 2|∇η| 2 |w m | 2+2ε + 2η 2 ε 2 (1 + ε) 2 ∇(|w m | ε w m ) 2 ≤ (2 + 4ε 2 (1 + ε) 2 )|∇η| 2 |w m | 2+2ε + 4ε 2 (1 + ε) 2 |∇W m | 2 .
M |W m | N dv g 2 N ≤ K Bx(2δ) u N m 2 n M |W m | N dv g 2 N + C δ .
Since Bx(2δ) u N m ≤ 1, the sequence M |W m | N dv g is bounded. This implies that (w m ) m is bounded in L N +ε (B x (δ)) and since x is arbitrary in L N +ε (M ). Weak convergences w m → w in H 2 1 (M ) implies strong convergence w m → w in L N -ε (M ). The Hölder inequality yields then strong convergence in L N (M ). After passing to a subsequence we obtain that (w m ) m tends to w strongly in L N (M ). This implies that we can pass to the limit in (36) and hence that u N -2 2 v and u N -2 2 w are linearly independent. The claim follows in this case.

In the following, we assume that µ 1 (M, g) > 0 and that µ 2 (M, g) < µ 1 (M, g)

n 2 + µ 1 (S n ) n 2 2 n .
We define the set of concentration points

Ω = x ∈ M ∀δ > 0, lim sup m Bx(δ) u N m dv g > 1 2 .
Since M u N m dv g = 1, we can assume -after passing to a subsequence -that Ω contains at most one point.

We now prove that:

Step 3. Let U be an open set such that U ⊂ M \ Ω. Then, the sequence (v m ) m (and (w m ) m resp.) converges towards v (and w resp.) strongly in H 2 1 (U ).

Without loss of generality, we prove the result only for w. For any x ∈ M \ Ω we can find δ > 0 with lim sup m Bx(2δ)

u N m dv g ≤ 1 2 .

Using µ 2 (M, g) < (µ 1 (M, g)

n 2 + µ 1 (S n ) n 2 ) 2 n ≤ 2 2 n µ 1 (S n
) we obtain for a small ε > 0

µ 2 (M, g)(1 -α ε ) -1 µ 1 (S n ) -1 Bx(2δ) u N m 2 n ≤ K < 1
for almost all m. Together with inequality (39), this proves that M |W m | N dv g is bounded. This implies that (w m ) m is bounded in L N +ε (B x (δ)). As in last step, this proves that up to a subsequence, (w m ) m tends to w strongly in L N (U ). Using equation ( 35 Step 4. There exists a sequence (w m ) m (w m ∈ S m ) and w ∈ S such that w m tends to w strongly in H 2 1 (M ).

By theorem 4.1, there exists λ m , µ m such that λ 2 m + µ 2 m = 1 and such that

Claim 6. 1 .

 1 The functions u N -2 2 v and u N -2 2 w are linearly independent.

  Here, we used (40) in the last line. Coming back to (41), we obtain that(∇(η 2 |w m | 2ε w m ), ∇w m ) ≥ (1α ε )|∇W m | 2 -C|∇η| 2 |w m | 2+2ε .

  ) and (38), we easily obtain thatlim m U |∇w m | 2 dv g = U |∇w| 2 dv g .Together with the weak convergence of (w m ) m to w, this proves the step. Now, we set for all m, S m = {λv m + µw m |λ 2 + µ 2 = 1} and S = {λv + µw|λ 2 + µ 2 = 1}.

Since w -resp. w + are some multiples of w on Ω -resp. Ω + , they satisfy the same equation as w. Hence, we get that

. Now, for any real non-negative numbers a, b ≥ 0, the Hölder inequality yields

We apply this inequality with a = Ω-u N dv g

N . Using (19), we obtain

We obtain λ 2 ≥ 2 2 n µ(S n ). Since λ 2 = I(u, span(v 1 , v 2 )), this ends the proof of Theorem 4.1.

4.3. Proof of Corollaries 4.2 and 4.3. It is well known that B 0 (S n ) equals to the scalar curvature of S n , i.e. B 0 (S n ) = n(n -1). Replacing B 0 (S n ) by its value and taking the infimum over u, V , the right hand term of inequality (S 1 ) is exactly the variational characterization of µ 2 (S n ) (see equation ( 2)). This proves that µ 2 (S n ) ≥ 2 2/n µ 1 (S n ). Corollary 4.2 then follows from Theorem 5.4. Since R n is conformal to S n \ {p} (p is any point of S n ), we can use the conformal invariance to prove Corollary 4.3.

5. Some properties of µ 2 (M, g) 5.1. Is µ 2 (M, g) attained? Let (M, g) be an n-dimensional compact Riemannian manifold. The Yamabe problem shows that µ 1 (M, g) is attained by a metric g conformal to g. Some questions arise naturally concerning µ 2 (M, g):

1-Is µ 2 (M, g) attained by a metric?

2-Is it possible that µ 2 (M, g) is attained by a generalized metric?

In this section, we give answers to these questions. The first result we prove is the following:

Proposition 5.1. Let S n ∪S n be the disjoint union of two copies of the sphere equipped with their standard metric. Then, µ 2 (S n ∪S n ) = 2 2/n µ 1 (S n ) and it is attained by the canonical metric.

Proof. One computes

Now, let g be an arbitrary smooth metric on S n ∪S n . We write S n 1 for the first S n and S n 2 for the second

), and obviously we have the same for λ 2 (S n 2 , g). Summing λ 1 (S n i , g) n/2 ≥ µ 1 (S n ) n/2 Vol(S n i , g) over i ∈ {1, 2}, we obtain the remaining inequality max{λ 1 (S n 1 , g), λ 1 (S n 2 , g)} Vol(S n ∪S n , g) 2/n ≥ 2 2/n µ 1 (S n ), and the proposition is proved.

Let δ > 0 be a small fixed number. In the following, o(1) denotes a sequence of real numbers which tends to 0, however we do not claim that the convergence is uniform in δ. By step 5 and the Hölder inequality,

Applying Sobolev inequality (S), we get that

Using equations ( 34), ( 35), (37), (38) and the fact that

By (48), in the limit δ → 0, this gives

This is false by assumption. Hence, the claim is proved, and Theorem 1.4 follows.

7. The invariant µ k (M ) for k ≥ 3

A natural question is: Can we do the same work for µ k (M ) with k ≥ 3? This problem is still open but seems to be hard. Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Using the variational characterization of

It is natural to conjecture that one has equality if M is the round sphere i.e. that µ k (S n ) = k 2 n µ 1 (S n ). However, the following result shows that is false:

Proof: Let us study S n with its natural embedding into R n+1 . We have L g (1) = n(n -1). Hence,

. This shows that

As one can check, for n ≥ 7

This ends the proof of Proposition 7.1.

The case of manifolds whose Yamabe invariant is negative

We let (M, g) be a compact Riemannian manifold of dimension n ≥ 3. Then, we have:

Proof: After a possible change of metric in the conformal class, we can assume that λ k (g) < 0. This implies that we can find some smooth functions

for all i ∈ {1, • • • , k} and such that M v i v j dv g = 0 for all i, j ∈ {1, • • • , k}, i = j. Let v ε be defined as in the proof of Theorem 5.4. We define u ε = v ε + ε to obtain a positive function. We set

Since λ i < 0, it is then easy to see that sup v∈V F (v ε , v) = -∞. Together with the variational characterization of µ k (M, g), we get that µ k (M, g) = -∞. This result proves for example that if the Yamabe invariant of (M, g) is negative, then µ 1 (M, g) = -∞. This is the reason why we restricted in this article to the case of non-negative Yamabe invariant. Many of our results and proofs remain valid in the case µ 2 (M ) ≥ 0. However, if the Yamabe invariant of (M, g) is non-positive, there are other ways to find nodal solutions of Yamabe equation. Indeed, Aubin's methods [START_REF] Aubin | Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire[END_REF] can be applied to avoid concentration phenomenom. See for example [START_REF] Djadli | Nodal solutions for scalar curvature type equations with perturbation terms on compact Riemannian manifolds[END_REF], [START_REF] Jourdain | Solutions nodales pour des équations du type courbure scalaire sur la sphère[END_REF], [START_REF] Holcman | Solutions nodales sur les variétés riemanniennes[END_REF] for such methods. Here, we present very briefly one new method in this case. We just sketch it since it is not the purpose of our paper to find solutions of Yamabe equation with Aubin's type methods.

At first, for any metric g conformal to g, we let λ + 1 (g) be the first positive eigenvalue of Yamabe operator. We then define λ

where the infimum is taken over the conformal class of g. Then, proceeding in a way analogous to [START_REF] Ammann | A variational problem in conformal spin Geometry[END_REF][START_REF] Ammann | The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions[END_REF], one shows that

M uL g u dv g where the infimum is taken over the smooth functions u such that M uL g u dv g > 0.

Then, one shows using test functions that λ + ≤ µ 1 (S n ). If the inequality is strict, then we can find a minimizer for the functional above which is a solution of the Yamabe equation. If the Yamabe invariant is positive, this solution is a Yamabe metric and hence is positive. However, if the Yamabe invariant is non-positive, this solution has an alternating sign.

A. Appendix: Proof of Lemma 3.1 Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 and let v ∈ H 2 1 (M ), v ≡ 0 and u ∈ L N + (M ) be two functions which satisfy in the sense of distributions

We define v + = sup(v, 0). We let q ∈]1, n n-2 ] be a fixed number and l > 0 be a large real number which will tend to +∞. We let β = 2q -1. We then define the following functions for x ∈ R:

x ≥ l and

It is easy to check that for all x ∈ R,

and

Since F l and G l are uniformly lipschitz continuous functions, F l (v + ) and G l (v + ) belong to H 2 1 (M ). Now, let x 0 ∈ M be any point of M . We denote by η a C 2 non-negative function supported in B x0 (2δ) (δ > 0 being a small number to be fixed) such that 0 ≤ η ≤ 1 and such that η(B x0 (δ)) = {1}. Multiply equation (Eq) by η 2 G l (v + ) and integrate over M . Since the supports of v + and G l (v + ) coincide, we get:

Let us deal with the first term of the left hand side of (52). In the following, C will denote a positive constant depending possibly on η, q, β, δ but not on l. We have

Using (49), ( 50) and (51), we get

Using the Sobolev embedding H 2 1 (M ) into L N (M ), there exists a constant A > 0 depending only on (M, g) such that

Together with (53), we obtain

Independently, we choose δ > 0 small enough such that

.

Relation (50) and Hölder inequality then lead to (ηF (v + )) N dv g < +∞.

This proves that v + ∈ L qN (B x0 (δ)). Since x 0 is arbitrary, we get that v + ∈ L qN (M ). Doing the same with sup(-v, 0) instead of v + , we get that v ∈ L qN (M ). This proves Lemma 3.1.

Authors' address:

Bernd Ammann and Emmanuel Humbert Institut Élie Cartan BP 239 Université de Nancy 1 54506 Vandoeuvre-lès -Nancy Cedex France E-Mail: bernd.ammann at gmx.net and humbert at iecn.u-nancy.fr