E H Megchiche

C Mijouië

M Amarouche

J C Barthelat

First-principles calculation of the vacancy formation energy in nickel

Within the framework of density functional lheory, we investigate with the VASP code three key properties of a vacancy in crystalline nickel: the fonnation, migration and activation energy. We use Blochl's all electron-like pro j ector augmented wave method (PAW) to perform self-consistent total-energy structure calculations in the supercell approximation and pseudopotentials to describe the computationally expensive core-valence interaction.

Ali calculations have been carried out using either the local spin density approximation ( LSDA) or the generalized gradient approximations (GGA ) . In a first step we show that the inclusion of non local e f fects improves signifi cantly the detem1ination of various weU known experimental properties of the bulk ( lattice parameter, rnagnetic moment, bulk modulus, cohesion energy ... ). Thus, we conclude that the non local approacb Jeads to a be tter estimation of vacancy formation (] .4 eV ) , migration ( L.28 eV) and activation (2.65 eV) energies al O K.

Introduction

The existence of vacancies in metals and intermetallic compounds plays an important role for the kinetic and thermodynamique properties of materials. In this connection the vacancy formation energy is a key concept in the understanding of the processes which occur in alloys during mechanical àeformation or heat treatment. Certainly large defects like grain boundaries and long-ranged dislocations are important, but even one of the simplest defect, the lattice vacancy, can be responsible for a variety of phenomena such as trapping of impurities and vacancy-mediated dopant diffusion. Botb the structure and the energitics of the vacan cy are expected to play a si gn ificant raie in these phenomena, therefore, it is important to have tborough understanding of the vacancy in order to make a detailed study of other phenomena involving the vacancy. The fom,ation energies of point defects in metals [START_REF] Deder I Chs | Application of Multiple Scattering theory to Material Science[END_REF][2] and compounds [3-5) can nowadays be obtained from ab initio electronic structure calculations based on density functional theory [6]. These energies may be used to evaluate the equilibrium concen�ations of point defects as a function of temperature and composition within the frarfiework of statistical mechanical approaches. Many theoretical [7][8] and experimental works [9][10][11][12] are available conceming the vacancy formation energy in nickel. The range of the data is very large even for experimental and theoretical approaches between nearly 1.4 and 1.8 eV. The goal of this paper is to provide a converged ab initio density functional theory within the pseudopotential approximation results for the energitic of monovacancy in crystalline nickel.

Methodology

The first-princilpes calculation are based on the density functional theory [13] calculations were perfonned with the VASP [14] Code which uses Blochl's all electron-like projector augmented wave method (PA W) [ 15] to describe the electron-ion interaction.

The PA W method is a frozen core all-elect r on method and, in many aspects, resembles the ultraso f t pseudopotential method of Vanderbilt [16]. lt has the advantage that the exact shape of the valence wave fi.mction is taken into account, and this in turn improves the description of the magnetic materials considerably [ 17]. Brillouin zone sampling in the reciprocal space was perfonned using a

MonkhorstPack grid [ 18] of special k points, and electronics occupancies were determined according to

MethfesselPaxton scheme [19] with an energy smearing of 0. Conceming the computational parameters, the plane wave energy eut-o f f is fixed to 400 eV for ail calculations. Supercells containing For the pseudoptentials, we have taken the Paw-GGA and Paw-LSDA in V A SP library similar to those proposed and tested in Ref. [17].

The determination of the cohesive energy is obtained as being the difference between the energy of the isolated nickel and that of the bul.k per unit atom. Thus, we had to detennine the energy of the nickel atom and more particularly the polarisation and broken symmetry contribution, since the reforence energy used in V A SP concerns the nonpolarized case.

Unfortunately, DFT approach is unavailable to describe the ground triplet state of 4s 2 3d 8 because this latter has a multireferential character, while DFT works only with a single determinant. ln fact the mono re f erential 4s�3d 8 triplet corresponding to total kinetic moment J= 2, 3, 4 whose energies are 0, O.

165 and 0.275 eV. Thus, we can expect that the DFT calculated cohesive energy will be overestimated by 0.1 to 0.3 eV.

Results

Structurai and elastic properties of the bulk

A s a first test of our method we calculated the properties of bulk Ni.

Formation and migration eoergy of vacancy

We study the vacancy in nickel by supercell method;

that is, we begin with the periodic FCC nickel lattice and remove atoms in periodic array so that each supercell of the lattice contains one vacancy.

The fotmation enthalpy of a vacancy is given by: Wherè E,' is the formation energy of monovacancy and p is the extemal pressure applied to the solid. This latter term is signi f icative only for pressure or several KBar; thus, in the following H ,' will be equivalent to E( because ail calculations are done for zero pressure.

ff N is the number of lattice sites of the supercell containing N -1 atoms and one monovacancy, H ! can be calculated as:

where the E(l,0 1 ) is d1e total energy of the supercell with N -J atoms and one vacancy at a relaxed unit cell volume 0 1 and E( 0, 0 0 ) the total energy of the supercell with N atom al a relaxed unit cell volumeQ 0 .

The mi gr ation enthalpy of the vacancy can be expressed as:

where E 111 is the total energy of the su perce li with N -1 atoms and one monovacancy located at the saddle site at relaxed volume 0 2 • Finally, the diffusion enthalpy Q 1 is calculated as: ln table II the results for the vacancy parameters ( f ormation, migration and f01mation volume) are presented at both the LSDA and GGA levels.

Conclusions

Firstly, we discuss the accuracy of the present calculation, for the bulk properties, which serve as guide to . access to accuracy for other properties.

The lattice parameter ao, and bulk modulus Bo.

The LSDA calculaLion underestimate a 0 and overestimat� Bo strongly as already know. il seen that the GGA calculation considerably improve the LSDA values and reproduce the measured values almost completely. We also found that the presents results agree very well the results of full potential-linearised-augmented-plane-wave calculation [23]. SecondJy, we discuss the calculated values for vacancy formation energy in nickel which as the central result of the present work. If we take the experimental lattice parameter, the vacancy formation energy will be nearly 2.1 eV, strongly overestimated (~20%) compared with the measured values. Detail results will be published elsewhere. We conclude that the GGA con-ects not only the deficiencies of LSDA for lattice parameter and buJk moduius, but also the deficiency (overestimation) of LSDA for vacancy formation energy.

  conducted for each distinct supercell geometry, ranging from the primitive cell of bulk Ni up to the largest 108-atom cells. For exchange-correlation functional. we use both the generalized gradient approximation (GGA) with the exchange-correlation functional of Perdew en Wang (PW9 I) [20], and local spin density (LSDA) with the exchange-correlation functional of Ceperly and Adler [21] as parameterized by Perdew and Zunger 122].

Table 1 .

 1 Comparison of calculated Ni properties to experimental and others ab initio calculations. LSDA refers to spin-polarized LDA

	Using primitive cell geometry,
	128 irreducible k points and plane wave eut-off energy
	of 400 eV were needed to converge the total .energy per
	atom to witbin I me V.
	ln table l, computational results conceming the bulk
	parameter a 0 the cohesive energy E coh . the magnelic
	moment M 0 and the bulk modulus B 0 • The GOA
	calculated bulk parameter is closed to the experimental
	data (Jess than 0.1 %) while the local approach is
	underestimate a 0 by 3%. The cohesive energies in each
	:::ase are overestimated more particularly at the local
	level. The contribution of both spi11 polarization and

broken symmetry enhance of 0.58 eV and' 0.63 eV the local and GGA approximation. Finally, the discrepancy wilh experiment is signifïcanl in GGA (8%) and dramallc in local (33%). As explained above, the ... " Reforence 23, b Reference 24, 'Reference 25

Table Il .

 Il Fonnation, migralion and activation energies of the monovacancy at the local and non-local levels.

	NR	FR	NR	FR	Experiment
	Hf 1. 71 1.62	l.43	l.38	1.39 11, 1.79 h
	H "' ,. l.80 1.51	1.58	1.28	l.72 C 1.04 h
	Q, 3.51 3.13	3.02	2.65	2.77 h

LSDA for PA W-LSDA calculation, GGA for PA W GGA calculation. NR and FR refers !o the 11011-relaxed and füU relaxed calculations. LSDA OGA a Reference 26, b Reference 12, 'Relèrence 10

  Table JI] lists the results bath for local and non-locaJ calcuJations. The non-local result agrees very well with the experimental values. It' s also seen that the LSDA reproduce well the measured values. However, we must be careful for the comparison in the case of LSDA results, because ail calculations was done at the relaxed lattice parameter listed in table I which are shorter 3% than the experimental values.

Acknowledgements

Computer resources for this job were provided by CALMJP (Toulouse, France). Sincere gratitude to the AMRS Committee of the CIGM4 to prepare this instruction.


Daniel Aletaha

Tuhina Neogi

Dr Alan J Silman
email: a.silman@arthritisresearchuk.org.

Julia Funovits

David T Felson

Clifton O Bingham

Neal S Birnbaum

Gerd R Burmester

Vivian P Bykerk

Marc D Cohen

Bernard Combe

Karen H Costenbader

Maxime Dougados

Paul Emery

Gianfranco Ferraccioli

Johanna Mw Hazes

Kathryn Hobbs

Tom Wj Huizinga

Arthur Kavanaugh

Jonathan Kay

Tore K Kvien

Timothy Laing

Philip Mease

Henri A Ménard

Larry W Moreland

Raymond L Naden

Theodore Pincus

Josef S Smolen

Ewa Stanislawska-Biernat

Deborah Symmons

Paul P Tak

Katherine S Upchurch

Jiří Vencovský

Frederick Wolfe

Gillian Hawker

2010 Rheumatoid arthritis classifi cation criteria: an American College of Rheumatology/European League Against Rheumatism collaborative initiative

Objective The 1987 American College of Rheumatology (ACR; formerly the American Rheumatism Association) classifi cation criteria for rheumatoid arthritis (RA) have been criticised for their lack of sensitivity in early disease. This work was undertaken to develop new classifi cation criteria for RA. Methods A joint working group from the ACR and the European League Against Rheumatism developed, in three phases, a new approach to classifying RA. The work focused on identifying, among patients newly presenting with undifferentiated infl ammatory synovitis, factors that best discriminated between those who were and those who were not at high risk for persistent and/ or erosive disease-this being the appropriate current paradigm underlying the disease construct 'RA'. Results In the new criteria set, classifi cation as 'defi nite RA' is based on the confi rmed presence of synovitis in at least one joint, absence of an alternative diagnosis better explaining the synovitis, and achievement of a total score of 6 or greater (of a possible 10) from the individual scores in four domains: number and site of involved joints (range 0-5), serological abnormality (range 0-3), elevated acute-phase response (range 0-1) and symptom duration (two levels; range 0-1). Conclusion This new classifi cation system redefi nes the current paradigm of RA by focusing on features at earlier stages of disease that are associated with persistent and/or erosive disease, rather than defi ning the disease by its late-stage features. This will refocus attention on the important need for earlier diagnosis and institution of effective disease-suppressing therapy to prevent or minimise the occurrence of the undesirable sequelae that currently comprise the paradigm underlying the disease construct 'RA'.

Rheumatoid arthritis (RA) is a chronic infl ammatory disease characterised by joint swelling, joint tenderness and destruction of synovial joints, leading to severe disability and premature mortality. 1 -5 Given the presence of autoantibodies, such as rheumatoid factor (RF) and anti-citrullinated protein antibody (ACPA) (tested as anti-cyclic citrullinated peptide), which can precede the clinical manifestation of RA by many years, 6 -9 RA is considered an autoimmune disease. 10 11 Autoimmunity and the overall systemic and articular infl ammatory load drive the destructive progression of the disease. However, although structural changes, which can be visualised by conventional radiography or other imaging techniques, best distinguish RA from other arthritic disorders, 12 joint damage is rarely apparent in the very early stages of disease, but rather accumulates consistently over time. 13 -16 Over the last decade, the optimal use of diseasemodifying antirheumatic drugs (DMARD), in particular the anchor DMARD methotrexate, 17 -19 and the availability of new biological agents, 11 20 have dramatically enhanced the success of RA management. Moreover, it has been recognised that early therapeutic intervention improves clinical outcomes and reduces the accrual of joint damage and disability. 21 -23 Undoubtedly, treating patients at a stage at which evolution of joint destruction can still be prevented would be ideal. However, at

Criteria

these factors and their weights using a series of 'paper patients', as well as to identify any other factors that may be of relevance based on current clinical thinking. Phase 3, which is the focus of this report, describes the derivation, from the previous two phases, of the fi nal classifi cation criteria set. The details of the methods and results from Phases 1 and 2 are provided elsewhere, 25 26 and are briefl y summarised below.

Phase 1

The aim of Phase 1 was to identify the contributions of clinical and laboratory variables that in practice were the most predictive of the decision to initiate DMARD therapy in a population of patients with early undifferentiated synovitis. Initiation of DMARD therapy was used as an indicator of the physician's opinion that the patient was at risk of developing persistent and/or erosive arthritis that we would currently consider to be RA. Data on 3115 patients from nine early arthritis cohorts who were considered not to have evidence of another possible diagnosis explaining their presentation were obtained. Between July 2007 and November 2008 an expert working group developed an analysis strategy that related an agreed-upon list of standardised clinical and laboratory variables collected at baseline to the initiation of DMARD treatment within the next 12 months. Methotrexate initiation was used as the gold standard for this purpose. The analytical process aimed to identify the independent contribution of each variable on this list and included univariate regression modelling, a subsequent principal components analysis, and a multivariate regression model that included all identifi ed components. [START_REF] Funovits | The 2010 American College of Rheumatology/ European League Against Rheumatism classifi cation criteria for rheumatoid arthritis: methodological report Phase l[END_REF] The resulting list of informative variables identifi ed during that process and the weights based on the odds ratios are shown in table 1 .

Phase 2

Phase 2 consisted of a consensus-based, decision scienceinformed approach, which took place between November 2008 and June 2009. The purpose of this phase was to derive a clinician-based judgement on the relative contribution of clinical and laboratory factors deemed to be important in infl uencing the probability of developing 'persistent infl ammatory and/or erosive arthritis that is currently considered to be RA' (hereinafter referred to as 'developing RA').

An expert panel was assembled, comprising 12 rheumatologists from Europe and 12 from North America with extensive experience in the diagnosis and management of RA. They provided real-life case scenarios of patients with early undifferentiated infl ammatory arthritis representing low to high present, clinical trials of RA treatments are hampered by lack of criteria allowing for study enrolment of patients at early stages of disease. Thus, to date it has not been possible effectively to investigate the effi cacy of early interventions in terms of their ability to prevent later-stage RA, since there are no validated or accepted uniform criteria to classify such individuals with early disease.

The standard and accepted means of defi ning RA is by use of classifi cation criteria. Classifi cation criteria enable the stratifi cation of groups of individuals into those with and those without RA in order to standardise recruitment into clinical trials and related studies, and provide the basis for a common approach to disease defi nition that can be used to compare across studies and centres. The classifi cation criteria set that is in widespread international use to defi ne RA are the 1987 American College of Rheumatology (ACR; formerly the American Rheumatism Association) criteria. [START_REF] Arnett | The American Rheumatism Association 1987 revised criteria for the classifi cation of rheumatoid arthritis[END_REF] These criteria are well accepted as providing the benchmark for disease defi nition, but have a signifi cant limitation in that they were derived by trying to discriminate patients with established RA from those with a combination of other defi nite rheumatological diagnoses. They are therefore not helpful in achieving the goal of identifying patients who would benefi t from early effective intervention, as discussed above. Indeed, with modern therapies, the goal is to prevent individuals from reaching the chronic, erosive disease state that is exemplifi ed in the 1987 criteria for RA.

A joint working group of the ACR and the European League Against Rheumatism (EULAR) was therefore formed to develop a new approach for classifi cation of RA. While classifi cation criteria are potentially adopted for use as aids for diagnosis, the focus of this endeavour was not on developing diagnostic criteria or providing a referral tool for primary care physicians. Indeed, a separate body of work is needed to develop such tools, which may be informed by classifi cation criteria. Thus, the specifi c charge was to develop new classifi cation criteria for RA to facilitate the study of persons at earlier stages of the disease. It was with this framework in mind that the working group developed the 2010 ACR/EULAR classifi cation criteria for RA.

OVERVIEW ON HYPOTHESIS AND METHODS OF PHASES 1 AND 2

A priori, the working group focused on developing an approach that would be appropriate for newly presenting patients with undifferentiated infl ammatory synovitis, in order to identify that subset of patients who are at suffi ciently high risk of persistent and/or erosive disease-this being the appropriate current paradigm underlying the disease construct 'RA'-to be classifi ed as having RA. It was recognised that such a scheme should not be developed using existing criteria sets as the 'gold standard,' because of the inherent circularity. The goal set forth was to develop a set of rules to be applied to newly presenting patients with undifferentiated synovitis that would: (1) identify the subset at high risk of chronicity and erosive damage;

(2) be used as a basis for initiating disease-modifying therapy; and (3) not exclude the capture of patients later in the disease course.

To achieve these goals, the working group devised a three phase programme. Phase 1 was a data-driven approach based on cohorts of real-world patients with early arthritis, to identify factors and their relative weights, which were associated with the subsequent decision by a physician to start methotrexate treatment. Phase 2 was a consensus-driven, decision sciencebased approach, informed by the data from Phase 1, to refi ne Low-positive versus negative 2.2 High-positive versus negative 3.9 †Derived from OR from the multivariate regression model, and interpreted as the increase in the odds of having RA with as opposed to without the respective feature (eg, weight of 1.5 for swelling of PIP joints means that the odds of having RA is 1.5-fold in patients with as opposed to patients without swelling of a PIP joint). ACPA, anti-citrullinated protein antibody; MCP, metacarpophalangeal; PIP, proximal interphalangeal; RA, rheumatoid arthritis; RF, rheumatoid factor.

Criteria

inactive. However, the working group recognised that patients may present for the fi rst time with disease that is at a later stage and being treated. Thus, although it was not the explicit charge of the working group to provide rules for the classifi cation of such patients, it is appropriate to have a single criteria system that could be applied to all patients; these issues were addressed by the expert panel during Phase 3.

Determination of the optimal cut point for defi nite RA

Determination of the optimal cut point to classify an individual as having defi nite RA was achieved using two complementary approaches, mirroring the approaches used in the fi rst two phases: data informed and consensus based. From the consensus-based approach, the expert panel was asked to examine the rankings of case scenarios based on the new scoring system and to indicate, in their opinion, the point at which the cases changed from 'probable' to 'defi nite' RA. Four cases were excluded due to missing domain information (n=2) or ineligibility (two cases were more likely another diagnosis). For the remaining 50 cases, the mean cut point defi ning defi nite RA was 65.7 (median 66.1; range 60.0-70.3) of a total possible score of 100.

A data-driven verifi cation of that cut point was then attempted, in which the new scoring system was applied to three of the existing cohorts used for Phase 1 (the Etude et Suivi des Polyarthrites Indifferenciees Recentes dataset from France, the Norwegian dataset and the Rotterdam Early Arthritis Cohort dataset from Rotterdam). [START_REF] Funovits | The 2010 American College of Rheumatology/ European League Against Rheumatism classifi cation criteria for rheumatoid arthritis: methodological report Phase l[END_REF] These cohorts were chosen based on the completeness of data and the collected variables, enabling calculation of the patients' probability scores at baseline. The disease characteristics of these cohorts were not substantively different from those of the remaining cohorts (data not shown). probability of developing RA. A 2-day workshop was held in May 2009 in which domains (factors) and categories within those domains that were important in determining the probability of developing RA were identifi ed. When appropriate, these judgements were informed by the results of Phase 1 and other available literature. The relative importance or weights of these domains and their categories were determined by means of decision science theory and conjoint adaptive technology, using the computerised 1000Minds program (http:// www.1000minds.com) in an interactive and iterative process. [START_REF] Neogi | The 2010 American College of Rheumatology/ European League Against Rheumatism Classifi cation Criteria for Rheumatoid Arthritis: Phase 2 methodological report[END_REF] This analysis permitted the calculation of an individual's score of the likelihood of developing RA from 0 to 100, in which a higher score indicated greater likelihood of RA development. The domains, categories and weights derived during that initial process are shown in table 2 .

OBJECTIVES, METHODS AND RESULTS OF PHASE 3 Objectives of Phase 3

In Phase 3 the working group integrated the fi ndings of the fi rst two phases, refi ned the scoring system, and determined the optimal cut point to defi ne 'defi nite RA'. The goal of this fi nal phase was to utilise the results of Phases 1 and 2 to develop a scoring system that would be applicable to newly presenting patients with undifferentiated infl ammatory arthritis to permit identification of those with a high probability of developing persistent and/or erosive RA. Being intended for use with newly presenting patients, the scoring system should be robust enough that it could be applied repeatedly during the early course of disease, such that a patient identifi ed as not classifi able as having definite RA at initial presentation might be classifi ed as having definite RA at a subsequent time point. The work was not aimed at classifying individuals with established disease, either active or In the category '>10 joints,' at least one of the involved joints must be a small joint; the other joints can include any combination of large and additional small joints, as well as other joints not specifi cally listed elsewhere (eg, temporomandibular, acromioclavicular, sternoclavicular, etc). †Negative refers to international unit (IU) values that are less than or equal to the upper limit of normal (ULN) for the laboratory and assay; low-positive refers to IU values that are higher than the ULN but three of less times the ULN for the laboratory and assay; high-positive refers to IU values that are more than three times the ULN for the laboratory and assay. When rheumatoid factor (RF) information is only available as positive or negative, a positive result should be scored as low-positive for RF. ‡Normal/abnormal is determined by local laboratory standards. §Duration of symptoms refers to patient self-report of the duration of signs or symptoms of synovitis (eg, pain, swelling, tenderness) of joints that are clinically involved at the time of assessment, regardless of treatment status. ACPA, anti-citrullinated protein antibody; CRP, C-reactive protein; ESR, erythrocyte sedimentation rate.

Criteria

(Spearman's r=0.82, p<0.0001). The correlation of score with the proportion of experts who would refer the patient to a trial of a new biological agent with inherent risks was similarly strong (Spearman's r=0.0.85, p<0.0001).

As a further validation, three cohorts that were not used in the identifi cation of factors from Phase 1 were studied (Leiden (The Netherlands), Leeds (UK) and Toronto (Canada)); their characteristics were not substantively different from those of the remaining cohorts. [START_REF] Funovits | The 2010 American College of Rheumatology/ European League Against Rheumatism classifi cation criteria for rheumatoid arthritis: methodological report Phase l[END_REF] Among cohort participants who received methotrexate within a year from symptom onset, the proportions with a score of 6/10 or greater were 96.8% (Leiden), 90.5% (Leeds) and 87.2% (Toronto).

The area under the curve (AUC) for the three receiver operating characteristic curves (which plot sensitivity against 1-specifi city for the range of scores) indicated good discrimination of those who did versus those who did not receive methotrexate (or another DMARD/biological agent) within a year (AUC 0.82 for Norway, 0.66 for France, and AUC 0.69 for Rotterdam; p<0.0001 for all). The probability scores similarly discriminated between those who fulfi lled the 1987 ACR criteria at 12 months and those who did not (AUC for the receiver operating characteristic curves 0.88 (Norway), 0.67 (France) and 0.72 (Rotterdam)). Visual inspection of the diagnostic test parameters associated with curves that used methotrexate initiation as the outcome showed a maximum slope for both the positive and negative likelihood ratios between a score of 60/100 and 70/100, with fl attening thereafter (67 in the Norway cohort, 66 in the French cohort and 66 in the Rotterdam cohort). The cut point of 60-70 that was derived from expert consensus was therefore supported by these data. Given the consistency with the consensus-based approach, and to maximise sensitivity of the criteria, a cut point of 60 was deemed to be most appropriate.

Rationale for the composition and weight of the fi nal criteria

For development of the fi nal criteria set, the results and weights from the comprehensive Phase 2 process [START_REF] Neogi | The 2010 American College of Rheumatology/ European League Against Rheumatism Classifi cation Criteria for Rheumatoid Arthritis: Phase 2 methodological report[END_REF] were used as a starting point. Based on these categories and weights, we aimed in the fi nal steps of the project to simplify the criteria in order to ensure that they were user friendly. We used the results of the data-driven Phase 1 as a guide for these adaptations, and verifi ed at each step that the main properties of the criteria were not altered and that classifi cation of patients remained unchanged.

The general steps towards simplifi cation are shown in table 2 , and included rescaling the scoring system to a scale of 0-10. Then, each of the category weights was rounded to multiples of 0.5. We tested the rounded scoring system in the case scenarios that had been used in Phase 2, and found no changes in the ranking of the cases compared with the exact scale. According to the rescaling, the cut point for defi nite RA would be 6/10 or greater.

Despite their slightly different weights, the categories of asymmetric and symmetric oligoarthritis of large joints were merged for several reasons: fi rst, symmetry was not found to be signifi cantly important in the data analysis during Phase 1, [START_REF] Funovits | The 2010 American College of Rheumatology/ European League Against Rheumatism classifi cation criteria for rheumatoid arthritis: methodological report Phase l[END_REF] and second, the impact of merging was minimal when all possible presentations of patients with symmetric or asymmetric large joint infl ammation were explored. For simplicity and ease of use, an integer scale was sought for all components of the scoring system. Thus, the high positive serology category was rounded from 3.5 (originally 3.39) to an integer of 3 because in no instance would classifi cation status be altered by this change. In addition, the weight for abnormal acute-phase response was rounded from 0.5 (originally 0.59) to 1, based on the stronger weight of acute-phase response (and of high-level acute-phase response) in Phase 1. [START_REF] Funovits | The 2010 American College of Rheumatology/ European League Against Rheumatism classifi cation criteria for rheumatoid arthritis: methodological report Phase l[END_REF]

Validation of the fi nal criteria set

The fi nal criteria set with its simplifi ed scoring system was further validated. Using the Phase 2 patient case scenarios, the correlation between cases' mean derived probability scores (0-100) and the proportion of expert panel members who indicated that they would initiate treatment with methotrexate out of concern about risk for persistence and/or erosive damage was strong ‡Although patients with a score of less than 6/10 are not classifi able as having RA, their status can be reassessed and the criteria might be fulfi lled cumulatively over time.

§ Joint involvement refers to any swollen or tender joint on examination, which may be confi rmed by imaging evidence of synovitis. Distal interphalangeal joints, fi rst carpometacarpal joints and fi rst metatarsophalangeal joints are excluded from assessment. Categories of joint distribution are classifi ed according to the location and number of involved joints, with placement into the highest category possible based on the pattern of joint involvement. ¶'Large joints' refers to shoulders, elbows, hips, knees and ankles. **'Small joints' refers to the metacarpophalangeal joints, proximal interphalangeal joints, second to fi fth metatarsophalangeal joints, thumb interphalangeal joints and wrists. † †In this category, at least one of the involved joints must be a small joint; the other joints can include any combination of large and additional small joints, as well as other joints not specifi cally listed elsewhere (eg, temporomandibular, acromioclavicular, sternoclavicular, etc.). ‡ ‡Negative refers to international unit (IU) values that are less than or equal to the upper limit of normal (ULN) for the laboratory and assay; low-positive refers to IU values that are higher than the ULN but three of less times the ULN for the laboratory and assay; high-positive refers to IU values that are more than three times the ULN for the laboratory and assay.

Criteria

that incorporates the weights of each domain and the cut point of 6 for classifi cation as defi nite RA. To classify a patient as having or not having defi nite RA, a history of symptom duration, a thorough joint evaluation, and at least one serological test (RF or ACPA) and one acute-phase response measure (erythrocyte sedimentation rate (ESR) or C-reactive protein (CRP)) must be obtained. It is acknowledged that an individual patient may meet the defi nition of RA without requiring that all tests be performed. For example, patients with a suffi cient number of joints involved and longer duration of symptoms will achieve six points regardless of their serological or acute-phase response status. However, for the purposes of clinical research and trial enrolment, documentation of each domain will be necessary for phenotyping.

Other clinical presentations: erosions and late disease

Because the aim of the new classifi cation criteria is to enable diagnosis and treatment earlier in the course of disease to prevent disease complications, erosions were not considered for inclusion in the scoring system. However, as stated above, the working group recognised that patients may present at later stages of disease. In addition, a single criteria system that could be applied to all patients was desired. Therefore, in addition to those who are newly presenting, three other groups of patients had to be considered: (1) those with erosions typical of RA were deemed to have prima facie evidence of RA and can be classifi ed as such;

(2) those with longstanding disease, either active inactive, who, based on retrospectively available data, can be determined to have previously satisfi ed the classifi cation criteria

Eligibility for testing with the new criteria

The classifi cation criteria can be applied to any patient or otherwise healthy individual, as long as two mandatory requirements are met: fi rst, there must be evidence of currently active clinical synovitis (ie, swelling) in at least one joint as determined by an expert assessor ( table 3 ). All joints of a full joint count may be assessed for this purpose with the exception of the distal interphalangeal (DIP) joints, the fi rst metatarsophalangeal (MTP) joint and the fi rst carpometacarpal (CMC) joint, since these joints are typically involved in osteoarthritis. Although currently no methods other than clinical examination should be used to evaluate the presence of synovitis in this determination of eligibility, this may change in the future as validated imaging techniques become more widely available. Second, the criteria may be applied only to those patients in whom the observed synovitis is not better explained by another diagnosis ( table 3 ). For example, conditions that should be considered and excluded include systemic lupus erythematosus, psoriatic arthritis and gout, among others.

Classifi cation criteria for RA

Four additional criteria can then be applied to eligible patients, as defi ned above, to identify those with 'defi nite RA' these are shown in table 3 . Application of these criteria provides a score of 0-10, with a score of 6 or greater being indicative of the presence of defi nite RA. This fi nal scoring system was derived from both Phase 1 and Phase 2 data. A patient with a score below 6 cannot be classifi ed as having defi nite RA, but might fulfi l the criteria at a later time point. Figure 1 depicts a tree algorithm Figure 1 Tree algorithm to classify defi nite rheumatoid arthritis (RA) (green circles) or to exclude its current presence (red circles) among those who are eligible to be assessed by the new criteria. APR, acute-phase response. See footnotes to table 3 for defi nitions of categories (eg, serology + or ++, or joint regions).

Criteria

or negative, patients with a positive level should be scored as 'low-level positive' for RF. When a value for a serological test is not available or the normal range is not available for the reported test value, the result for that test should be considered 'negative/normal'. Patients should be scored only if information from at least one serological test is available.

Defi nition of abnormal acute-phase response

The acute-phase response measures CRP or ESR are scored as normal or abnormal based on the local laboratory standards.

If results of at least one of these two tests are abnormal, the patient should be scored as having an abnormal acute-phase response. If a value for an acute-phase reactant is not available or information on the normal range for the reported test value is not available, the result of that test should be considered 'negative/normal'. For ESR, a standard approach that considers age and sex differences would be valuable. Patients should be scored only if at least one acute-phase response test is available for scoring.

Defi nition of duration of symptoms

The 'duration of symptoms' domain refers to the patient's selfreport of the maximum duration of signs or symptoms of synovitis (pain, swelling and tenderness) of any joint that is clinically involved at the time of assessment (ie, the day the criteria are applied). Thus, joints that are reported to have been previously symptomatic but are not involved at the time of assessment, whether due to treatment or not, should not be considered in estimating symptom duration.

DISCUSSION

We present here new classifi cation criteria for RA, representing the culmination of an international collaborative effort supported by both a data-driven and a consensus-based approach. This classifi cation scheme is designed to present a standardised approach to identifying that subset of individuals who present with an otherwise unexplained infl ammatory arthritis of a peripheral joint(s), for whom the risk of symptom persistence or structural damage is suffi cient to be considered for intervention with DMARD. This is thus the new proposed paradigm for the entity 'RA', importantly, not criteria for 'early' RA. If there was an intervention that was both infi nitely effective and safe and could be provided at no cost and no discomfort, then there would be no requirement for such a subset to be identifi ed, as every patient with infl ammatory arthritis would be treated. Given that such an intervention does not exist, the search for appropriate classifi cation rules is justifi ed, and will also be helpful in guiding clinical diagnosis.

It is important, however, to stress that the criteria are meant to be applied only to eligible patients, in whom the presence of obvious clinical synovitis in at least one joint is central. They should not be applied to patients with mere arthralgia or to normal individuals. However, once defi nite clinical synovitis has been determined (or historical documentation of such has been obtained), as indicated in the glossary, a more liberal approach is allowed for determining the number and distribution of involved joints, which permits the inclusion of tender or swollen joints.

Symmetry is not a feature of the new criteria since it did not carry an independent weight in any phase of the work. In practice, symmetry is diffi cult to operationalise. Inevitably, though, the greater the number of involved joints the higher the likelihood of bilateral involvement.

can similarly be classifi ed as having defi nite RA; and (3) in the setting of early disease that is being treated, individuals may not fulfi l the new criteria at initial presentation, but may do so as their condition evolves over time.

GLOSSARY OF DEFINITIONS

In this section, we provide the detailed defi nitions necessary correctly and accurately to apply the new classifi cation criteria for RA. A brief version of the glossary is included in the footnotes to table 3 .

Defi nition of an 'involved' joint

Joint involvement, as used for the determination of the pattern of joint distribution, differs from the defi nition of synovitis in one joint needed for eligibility in the eligibility criteria (see above): here it refers to any joint with swelling or tenderness on examination that is indicative of active synovitis. Tenderness is included as an equally important feature as swelling for the determination of joint involvement, particularly for the second to fi fth MTP joints, in order to maximise sensitivity. Again, the DIP joints, the fi rst MTP joint and the fi rst CMC joint should not be considered, given their prevalent involvement in osteoarthritis. Furthermore, any joints with known recent injury that could contribute to swelling or tenderness should not be considered. Additional evidence of joint activity from other imaging techniques (such as MRI or ultrasound) may be used for confi rmation of the clinical fi ndings.

Defi nition of small joints

Small joints include the MTP, proximal interphalangeal, second to fi fth MTP and thumb interphalangeal joints and the wrists. They do not include the fi rst CMC, fi rst MTP, or DIP joints, which are often affected by osteoarthritis.

Defi nition of large joints

The term 'large joints' refers to the shoulders, elbows, hips, knees and ankles.

Determination of the joint pattern category

Patients are categorised according to the number and location of involved joints by placing them into the category with the highest possible score. For example, a patient with involvement of two large joints and two small joints is placed in the category 'one to three small joints', as this category has the higher score. Patients should be scored for their joint involvement assuming that all of the peripheral joints indicated above have been assessed. For the highest category of joint involvement, in which >10 joints must be involved (including at least one small joint), additional joints that can be considered for inclusion in this count include the temporomandibular joint, sternoclavicular joint, acromioclavicular joint and others that may reasonably be expected to be involved in RA.

Defi nition of the serological categories

ACPA and IgM-RF levels are usually reported in IU. Based on the upper limit of normal (ULN) for the respective laboratory test and assay the following defi nitions can be made: negative=less than or equal to the ULN for the laboratory test and assay; lowlevel positive = higher than the ULN but ≤3 times the ULN for the laboratory test and assay; highlevel positive=>3 times the ULN for the laboratory test and assay. When RF information is available only qualitatively or as a level, and thus positive on November 16, 2023 by guest. Protected by copyright. http://ard.bmj.com/

Criteria

have been developed such that they can be applied to patients at more than one time point in the evolution of their symptoms and signs. Thus, a patient who does not fulfi l criteria for defi nite RA at fi rst presentation might be classifi ed as having defi nite RA at a subsequent time point.

Rather than developing a parallel system for 'established' disease or continuing to use the 1987 criteria for that purpose, the working group recommends that, when patient records allow, application of the criteria and assignment as defi nite RA may be made retrospectively. If there is a history compatible with defi nite RA as defi ned by these new criteria but no records of such, signifi cant erosive disease seen on radiographs, typical of destructive RA, can be used as prima facie evidence of RA, precluding the need for applying additional criteria. Such individuals would need to be included as part of the total population of individuals affected by RA. It was not part of the working group's mission to defi ne what is meant by signifi cant erosive disease either in terms of the size, site, or number of erosions. Such agreement could be the task for further consensus, although current evidence suggests that such a defi nition should be highly specifi c. [START_REF] Thabet | The prognostic value of baseline erosions in undifferentiated arthritis[END_REF] Thus, future work will be needed to defi ne what evidence of erosions is acceptable to be considered as 'typical' of RA. This was a similar issue faced with the 1987 ACR criteria, but it did not detract from the usability of those criteria.

The use of the new criteria should be limited to a target population in whom there is otherwise no explanation (ie, defi nite diagnosis) for their synovitis. The working group has deliberately not provided an exhaustive list of diagnoses or tests that should be performed to exclude these, since that is not the purpose of classifi cation criteria. Differential diagnosis is a physician responsibility and will be infl uenced by patient age and sex, practice type, as well as variations in the background population in terms of the incidence of 'competing' disorders. Thus, Lyme arthritis may be a frequent cause of synovitis in endemic areas, but testing for Borrelia burgdorferi would not be appropriate elsewhere. Differential diagnosis is, in this context, inevitably subject to differences in interpretation, as best exemplifi ed by the confl ict between those who would exclude patients with psoriasis from further classifi cation. In this respect, the important utility of appropriate exclusion assessment is the avoidance of misclassifying patients as having RA who might otherwise not need to be treated or have self-limiting disease.

It was not the charge of the working group to create a referral tool for primary care physicians. Indeed, the original 1987 ACR criteria were also not designed for such use. Primary care physicians and other specialists need an easy-to-use tool to facilitate identifi cation of individuals who have an infl ammatory arthritic syndrome and for whom referral to a rheumatologist for further evaluation and diagnosis is appropriate. Support for such an endeavour is already being undertaken as a joint effort by ACR/ EULAR and other important stakeholders.

One limitation of the new criteria is that they are based on current knowledge. Genetic, proteomic, serological, or imaging biomarkers that provide a more robust basis for risk stratifi cation may emerge, and this would necessarily lead to a modifi cation or amendment of the 2010 criteria. Similarly, biomarkers, including imaging modalities, that more robustly identify highrisk subgroups of patients with synovitis may one day be available and validated. A pertinent example of a new biomarker is ACPA (typically, with testing for anti-cyclic citrullinated peptide). Based on a detailed literature review [START_REF] Aggarwal | Anti-citrullinated peptide antibody assays and their role in the diagnosis of rheumatoid arthritis[END_REF] and our analytical approach to physician decision-making, ACPA status did not add importantly to the ability to classify an individual as having RA, beyond the information provided by RF when it is The development of these new criteria was based on diverse study cohorts and was performed by RA experts of diverse nationalities, enhancing the criteria set's generalisability. The fi nal criteria also refl ect consistency in domains of importance through the use of two independent methodologies. Previous classifi cation schemes centered on a pre-identifi ed clinical concept, typifi ed by the occurrence of a predominantly symmetric small joint polyarthritis associated with autoantibody production and a high prevalence of erosions. [START_REF] Van Der Helm-Van Mil | Validation of a prediction rule for disease outcome in patients with recent-onset undifferentiated arthritis: moving toward individualized treatment decision-making[END_REF] This was the 'gold standard' used to devise the rules that could be applied in a repeatable manner to identify homogeneous groups for observation and study. The new criteria redefi ne RA, refl ecting our collective hope that in the future, RA will no longer be characterised by erosive joint disease and persistence of symptoms, although these characteristics will continue to defi ne established or longstanding untreated disease. This refl ects several conceptual issues that are relevant in all areas of research and also in clinical practice.

Once the disease entity is redefi ned, existing epidemiological data on prevalence will have less relevance. Generally, this should not be a major concern, since there are well recognised diffi culties in gathering and interpreting epidemiological data regarding the occurrence of RA: prevalence estimates are infl uenced by the effects of therapy and therefore are inherently unstable. The prevalence of RA could variously be described as the proportion of the population who have satisfi ed the new criteria at some relevant point in time.

The greater problem is the extrapolation of current literature on clinical trials and the design of future trials. The working group recommends that clinical trials should henceforth apply these new criteria; however, results from studies using the new criteria cannot necessarily be directly compared with the extensive body of existing work. In reality, in most trials, even of recent-onset RA, subjects with much higher levels of disease activity than is needed for fulfi lment of the new criteria, and often those who have been treated unsuccessfully with multiple previous therapies, are selected. [START_REF] Hyrich | OMERACT 7 Special Interest Group . Reconciling subject differences in recruitment to clinical trials and clinical practice[END_REF] Thus, comparison between trials will be based much more on the distribution of disease activity at entry, for example, than the assumption that criteria satisfaction leads to homogeneity in patients recruited. Still, it might be useful, over this transition to the new criteria, for researchers to document the proportions of study subjects who fulfi l the previous1987 and the new RA classifi cation criteria, to enable comparisons.

There is a potential problem related to the pursuit of basic research. For example, genetic association studies have relied on a standardised approach to phenotypic assessment based on the 1987 ACR criteria. Such association studies still hold and, as stated above, the new criteria are likely to be easily satisfi ed by the participants in such studies. RA is always considered a heterogeneous disorder, and the new criteria scheme will probably increase that heterogeneity. Thus, basic scientists should be aware and, when appropriate, restrict recruitment or stratify results based on clinically meaningful phenotypes. For example, even among patients fulfi lling the 1987 criteria, those who are ACPA positive and those who are ACPA negative have been shown to differ from a pathogenetic, clinical and prognostic perspective. [START_REF] Huizinga | Refi ning the complex rheumatoid arthritis phenotype based on specifi city of the HLA-DRB1 shared epitope for antibodies to citrullinated proteins[END_REF] The criteria have been intentionally derived from paper patient cases and cohorts of newly presenting individuals with undifferentiated infl ammatory synovitis. Once classifi ed, unless an alternative explanation for the synovitis becomes apparent over time, the individual is labelled as having 'defi nite RA'. However, acknowledging that RA is not a static disease, the new criteria

Criteria

Mjaavatten, Valerie Nell, Nathalie Rincheval, Marleen van de Sande and Annette van der Helm-van Mil, who were involved in development, data management, or maintenance of their respective datasets used in Phase 1 of the project, and to Rohit Aggarwal, Dinesh Khanna, Katherine Liao, Raj Nair and Sarah Ringold, who were involved in the design or implementation of Phase 2. The authors also wish to thank the ACR and the EULAR for their fi nancial support of this project. The authors are especially grateful to Amy Miller and Regina Parker from the ACR and Heinz Marchesi and Anja Schönbächler from the EULAR, for their administrative support of the project. Funding Dr. Aletaha has received consulting fees, speaking fees, and/or honoraria from Abbott, Bristol-Myers Squibb, UCB, Schering-Plough, Wyeth, and Roche (less than $10,000 each). Dr. Bingham has received consulting fees, speaking fees, and/or honoraria from UCB, Roche, Genentech, Celgene, and Merck Serono (less than $10,000 each); he has received research and/or educational grant support from Bristol-Myers Squibb, Genentech, UCB, Centocor, Abbott, and Amgen. Dr. Birnbaum has received consulting fees, speaking fees, and/ or honoraria from Amgen, Pfi zer, Centocor, Abbott, and UCB (less than $10,000 each). Dr. Burmester has received consulting fees, speaking fees, and/or honoraria from Abbott, Bristol-Myers Squibb, Pfi zer, UCB, and Roche (less than $10,000 each). Dr. Bykerk has received consulting fees, speaking fees, and/or honoraria from Amgen, Wyeth, Abbott, Schering-Plough, Roche, Bristol-Myers Squibb, and UCB (less than $10,000 each); her spouse is employed by Genzyme and owns stock in the company. Dr. Cohen has received consulting fees, speaking fees, and/ or honoraria from UCB, Genentech, Bristol-Myers Squibb, and Human Genome Sciences (less than $10,000 each). Dr. Combe has received consulting fees, speaking fees, and/or honoraria from Abbott, Bristol-Myers Squibb, Pfi zer, Roche, Schering-Plough, and Merck, Sharpe, and Dohme (less than $10,000 each). Dr. Emery has received consulting fees, speaking fees, and/or honoraria from Pfi zer, Abbott, Centocor, UCB, Roche, Bristol-Myers Squibb, and Merck, Sharpe, and Dohme (less than $10,000 each). Dr. Ferraccioli holds a patent for T cell receptor clonotype analysis (PCT/IB 2008/053152 NP). Dr. Huizinga has received consulting fees, speaking fees, and/or honoraria from Schering-Plough, Bristol-Myers Squibb, UCB, Biotest AG, Wyeth/Pfi zer, Novartis, Roche, Sanofi -Aventis, Abbott, and Axis-Shield (less than $10,000 each). Dr. Kavanaugh has conducted clinical research for Amgen, Abbott, Bristol-Myers Squibb, UCB, Roche, Centocor, Genentech, and Sanofi -Aventis. Dr. Kay has received consulting fees from Array BioPharma, Bristol-Myers Squibb, Celgene, Centocor, Genentech, Roche, UCB, and Sanofi -Aventis (less than $10,000 each). Dr. Mease has received consulting fees, speaking fees, and/or honoraria from Abbott, Amgen, Biogen Idec, Bristol-Myers Squibb, Centocor, Roche, Genentech, UCB, Pfi zer, Novartis, and Eli Lilly (less than $10,000 each). Dr. Ménard has received unrestricted educational and research grants as well as consulting and speaking fees from Abbott, Amgen, Inova, Merck, Pfi zer, Roche, Schering-Plough, UCB, and Wyeth (less than $10,000 each) and investigator-initiated research grants from Bristol-Myers Squibb, EuroImmun AG, and Roche (more than $10,000 each); he owns stock or stock options in Merck; and he has a license agreement with EuroImmun AG for an anti-Sa enzyme-linked immunosorbent assay. Dr. Moreland has received consulting fees, speaking fees, and/or honoraria from Biogen Idec, Centocor, Pfi zer, Takeda, KaloBios, ChemoCentryx, UCB, Genentech, Incyte, and Eli Lilly (less than $10,000 each). Dr. Naden has received consulting fees from the American College of Rheumatology in regard to the methodology of developing weighted scoring systems (more than $10,000). Dr. Pincus has received consulting fees, speaking fees, and/or honoraria from Amgen, Abbott, Bristol-Myers Squibb, Centocor, UCB, Wyeth, and Genentech (less than $10,000 each) and investigator-initiated research grants from Amgen, Bristol-Myers Squibb, UCB, and Centocor. Dr. Stanislawska-Biernat has received speaking fees from Abbott and Pfi zer (less than $10,000 each). Dr. Vencovský has received speaking fees from Pfi zer, UCB, Abbott, Roche, and Merck, Sharpe, and Dohme (less than $10,000 each).

Competing interests None.

Contributors All authors were involved in drafting the article or revising it critically for important intellectual content, and all authors approved the fi nal version to be published. Dr Silman had full access to all of the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. positive. The working group has therefore included both markers (ACPA and RF) equally in the criteria. However, as ACPA testing becomes more standardised, further refi nement may be needed.

Because there is no gold standard for a diagnosis of RA, the cut-off score of 6 or greater is the best estimate from the current approaches used; testing in other cohorts will provide further evidence regarding its validity. Since the classifi cation scheme actually provides a continuum of 'risk for developing persistent and/or erosive RA' (ie, it assigns the risk or probability of developing RA on a continuous score (from 0 to 100%)), there is scope for investigators to use other cut points-or multiple cut points-for different purposes. For example, in a clinical trial of a new potentially toxic agent, a higher, more conservative, cut point might be more appropriate; this is akin to clinical trials enrolling patients who meet criteria but also have evidence of a certain degree of severity or extent of involvement. In contrast, a population study of familial aggregation might use a less restrictive cut point. As such, there is information derived from scores across the range from 0 to 10 that may be utilised for different purposes in the future.

The working group has deliberately labelled these criteria as 'classifi cation criteria' as opposed to 'diagnostic criteria'. The aim is to provide a standardised approach for discriminating, from a population of individuals presenting with undifferentiated synovitis, the subgroup with the highest probability of persistent or erosive RA, who may be enrolled into clinical trials and other studies through the use of uniform criteria. These individuals are also the ones who may therefore benefi t from DMARD intervention. The criteria do not remove the onus on individual physicians, especially in the face of unusual presentations, to reach a diagnostic opinion that might be at variance from the assignment obtained using the criteria. Nonetheless, it is recognised that the new criteria will likely also be used as a diagnostic aid and be required to be satisfi ed, for example, by healthcare providers to enable access to particular interventions. However, much like other classifi cation criteria, clinicians may be able to diagnose an individual who has not met the classifi cation criteria defi nition or who has features that are not a component of the classifi cation criteria. Diseases often present a much wider spectrum in clinical medicine that can be expected to be captured by classifi cation criteria, the purpose of which is simply to provide a uniform set of standards by which an individual can be classifi ed as having a clinical entity or not.

The new criteria need to be tested in several clinical situations and settings. Physicians need to report particularly if there is an important proportion of newly presenting patients who do not satisfy these criteria but for whom there is a compelling reason to treat with a DMARD, or who on follow-up, without a change in their classifi cation status develop persistent or erosive disease. Validation in three of the cohorts available to us showed that the criteria were satisfi ed in 87-97% of the patients in whom the physicians chose to initiate methotrexate treatment.

In summary, the new ACR/EULAR classifi cation criteria for RA present a new approach with a specifi c emphasis on identifying patients with a relatively short duration of symptoms who may benefi t from early institution of DMARD therapy or entry into clinical trials of promising new agents that may halt the development of disease that currently fulfi ls the 1987 ACR criteria.

on 16 ,

 16 2023 by guest. Protected by copyright. http://ard.bmj.com/

  Study conception and design. Aletaha, Neogi, Silman, Felson, Birnbaum, Bykerk, Combe, Costenbader, Dougados, Emery, Hazes, Huizinga, Kay, Kvien, Moreland, Naden, Smolen, Stanislawska-Biernat, Vencovský, Hawker. Acquisition of data. Aletaha, Neogi, Silman, Bingham, Birnbaum, Burmester, Bykerk, Combe, Costenbader, Dougados, Emery, Hazes, Huizinga, Kavanaugh, Kay, Kvien, Laing, Ménard, Naden, Smolen, Stanislawska-Biernat, Tak, Upchurch, Vencovský, Hawker. Analysis and interpretation of data. Aletaha, Neogi, Silman, Funovits, Bingham, Birnbaum, Burmester, Bykerk, Cohen, Combe, Dougados, Emery, Ferraccioli, Hazes, Hobbs, Huizinga, Kay, Laing, Mease, Ménard, Moreland, Naden, Pincus, Smolen, Stanislawska-Biernat, Symmons, Tak, Upchurch, Vencovský, Wolfe, Hawker. . Provenance and peer review Not commissioned; externally peer reviewed.

Table 1

 1 Summary of Phase 1 results

	Variable	Comparison	Relative weight †
	Swollen MCP joint	Present versus absent	1.5
	Swollen PIP joint	Present versus absent	1.5
	Swollen wrist	Present versus absent	1.6
	Hand tenderness	Present versus absent	1.8
	Acute-phase response	Lowly abnormal versus normal	1.2
		Highly abnormal versus normal	1.7
	Serology (RF or ACPA)

Table 2

 2 Summary of Phase 2 results and subsequent modifi cations

Exact (0-100) Rescaled (0-10) Rounded to 0.5 (0-10)

	Joint involvement*
	1 large	0	0	0
	>1-10 large, asymmetric	10.2	1.02	1
	>1-10 large, symmetric	16.1	1.61	1.5
	1-3 small	21.2	2.12	2
	4-10 small	28.8	2.88	3
	>10, including at least 1 small joint	50.8	5.08	5
	Serology †
	Negative RF and negative ACPA	0	0	0
	Low-positive RF or low-positive ACPA	22.0	2.20	2
	High-positive RF or high-positive ACPA	33.9	3.39	3.5
	Acute-phase reactants ‡
	Normal CRP and normal ESR	0	0	0
	Abnormal CRP or abnormal ESR	5.9	0.59	0.5
	Duration of symptoms §
	<6 weeks	0	0	0
	≥6 weeks	9.3	0.93	1
	*Joint involvement refers to any swollen or tender joint on examination. Distal interphalangeal joints, fi rst carpometacarpal joints
	and fi rst metatarsophalangeal (MTP) joints are excluded from assessment. Categories of joint distribution are classifi ed according
	to the location and number of the involved joints, with placement into the highest category possible based on the pattern of joint
	involvement. 'Large joints' refers to shoulders, elbows, hips, knees and ankles. 'Small joints' refers to the MTP joints, proximal
	interphalangeal joints, second to fi fth MTP joints, thumb interphalangeal joints and wrists. 'Symmetric' is defi ned as bilateral
	involvement of at least one region.

Table 3

 3 The 2010 American College of Rheumatology/European League Against Rheumatism classifi cation criteria for RA

	Score

*The criteria are aimed at classifi cation of newly presenting patients. In addition, patients with erosive disease typical of rheumatoid arthritis (RA) with a history compatible with prior fulfi lment of the 2010 criteria should be classifi ed as having RA. Patients with long-standing disease, including those whose disease is inactive (with or without treatment) who, based on retrospectively available data, have previously fulfi lled the 2010 criteria should be classifi ed as having RA. †Differential diagnoses differ in patients with different presentations, but may include conditions such as systemic lupus erythematosus, psoriatic arthritis and gout. If it is unclear about the relevant differential diagnoses to consider, an expert rheumatologist should be consulted.

on November 16, 2023 by guest. Protected by copyright. http://ard.bmj.com/ Ann Rheum Dis: first published as 10.1136/ard.2010.138461 on 10 August 2010. Downloaded from

Ann Rheum Dis: first published as 10.1136/ard.2010.138461 on 10 August 2010. Downloaded from

Acknowledgements The authors are grateful to Celina Alves, Carly Cheng, Tracey Farragher, Elisabeth Hensor, Jolanda Luime, Klaus Machold, Maria Dahl on November 16, 2023 by guest. Protected by copyright. http://ard.bmj.com/ Ann Rheum Dis: first published as 10.1136/ard.2010.138461 on 10 August 2010. Downloaded from

▶ This criteria set has been approved by the American College of Rheumatology (ACR) Board of Directors and the European League Against Rheumatism (EULAR) Executive Committee.

the criteria set has been quantitatively validated using patient data, and it has undergone validation based on an external data set. ▶ All ACR/EULARapproved criteria sets are expected to undergo intermittent updates.

Criteria


Vincent Minerbe

RIGIDITY FOR MULTI-TAUB-NUT METRICS

published or not. The documents may come

Rigidity for Multi-Taub-NUT metrics

Introduction.

Since the late seventies, there has been considerable interest in the so-called gravitational instantons, namely complete non-compact hyperkähler four-manifolds with decaying curvature at infinity. They were introduced by S. Hawking [Haw] as building blocks for his Euclidean quantum gravity theory. Beside their natural link with gauge theory, they have also appeared relevant in string theory. Their mathematical beauty, including the nice twistorial point of view [Bes], is definitely a good motivation to understand them.

All known examples fall into four families -ALE, ALF, ALG, ALH -which differ by their asymptotic geometry. The ALE gravitational instantons are Asymptotically Locally Euclidean, in that their asymptotic geometry is that of R 4 , up to a finite covering. In the other families, the topology outside a compact set and up to finite covering is that of a T 4-k -fibration over R k , with k = 3 for the ALF family, k = 2 for the ALG family, k = 1 for the ALH family ; moreover, the geometry is asymptotically adapted to these fibrations. Much more details will be given below, in the ALF case. It is useful to keep in mind that ALE gravitational instantons are exactly those gravitational instantons that have Euclidean volume growth -vol B(x, t) ≍ t 4 - [BKN], while ALF gravitational instantons are characterized by their cubic volume growthvol B(x, t) ≍ t 3 [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF].

In 1989, P. Kronheimer classified ALE gravitational instantons [START_REF] Kronheimer | The construction of ALE spaces as hyper-Kähler quotients[END_REF][START_REF] Kronheimer | A Torelli-type theorem for gravitational instantons[END_REF]. The possible topologies are given by the minimal resolutions of the Kleinian singularities C 2 /Γ. Here, Γ is a finite subgroup of SU (2): cyclic, binary dihedral, tetrahedral, octahedral or icosahedral. For every such manifold M , the different hyperkähler structures are parameterized by three classes in H 1,1 (M, C), with a non-degeneracy condition [START_REF] Kronheimer | A Torelli-type theorem for gravitational instantons[END_REF]. The simplest situation corresponds to "A k+1 " ALE gravitational instantons, namely Γ = Z k , with k ≥ 1, since they are given by the explicit multi-Eguchi-Hanson metrics. When k = 1, the manifold is T * CP 1 and the metric is also known as Calabi's metric.

ALF gravitational instantons should be classified in a similar way. What are the examples ? The trivial one is C × C/Z, with fundamental group at infinity Z. More sophisticated examples are given by multi-Taub-NUT metrics (cf. section 1), which live exactly on the same manifolds as the multi-Eguchi-Hanson metrics; the fundamental group at infinity is then Z k . Some more mysterious examples were built recently by S. Cherkis and A. Kapustin, with a dihedral fundamental group at infinity. It is conjectured that these are the only ALF gravitational instantons.

Indeed, [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF] already ensures the topology outside a compact set is like in these examples. Let us recall the precise statement. We will denote by ρ the distance to some distinguished point.

Theorem 0.1 -[Mi2]
Let M be a four-dimensional complete hyperkähler manifold such that Rm = O(ρ -3 ) and vol B(x, t) ≍ t 3 . Then M is ALF in that there is a compact subset K of M such that M \K is the total space of a circle fibration π over R 3 or R 3 /{± id} minus a ball and the metric g can be written

g = π * g R 3 + η 2 + O(ρ -τ ) for any τ < 1,
where η is a (local) connection one-form for π; moreover, the length of the fibers goes to a finite positive limit at infinity.

It shows that the topology of the ALF gravitational instanton M can be described outside a compact subset by : M \K = E × R * + , where E is the total space of a circle fibration over S 2 or RP 2 . When the base of the circle fibration at infinity is S 2 , we will say that M is ALF of cyclic type : in this case, E is either S 2 × S 1 or S 3 /Z k (where Z k is seen as the group of kth roots of 1, acting by scalar multiplication in C 2 ), so the fundamental group of the end is Z or Z k , hence the denotation. When the base of the circle fibration at infinity is RP 2 , we say that M is ALF of dihedral type : E is then a quotient of S 3 by a binary dihedral group D k (of order 4k), so the fundamental group is dihedral (the trivial bundle is excluded for orientability reasons). In particular, this topological classification for the end rules out any tetrahedral, octahedral or icosahedral fundamental group at infinity. Note that D 1 is indeed Z 4 , so "ALF of cyclic type" means a little bit more than just "with cyclic fundamental group at infinity".

The aim of this paper is to establish a complete classification for ALF gravitation instantons of cyclic type. The precise statement is as follows.

Theorem 0.2 -An ALF gravitational instanton of cyclic type is either the flat C × C/Z or a multi-Taub-NUT manifold.

This classification is up to triholomorphic isometry. In view of Theorem 0.1, it means that a complete hyperkähler manifold asymptotic to C × C/Z (resp. a multi-Taub-NUT manifold) is necessarily C × C/Z (resp. a multi-Taub-NUT manifold). In this statement, "hyperkähler" cannot be relaxed to "Ricci-flat". For instance, there is a non-flat Ricci-flat manifold asymptotic to C × C/Z : the Schwarzschild metric [START_REF] Minerbe | A mass for ALF manifolds[END_REF], for instance).

The strategy of the proof is as follows. [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF] provides an asymptotic model for the manifolds under interest. It involves as key ingredients three functions and a one-form.

In the examples, this data extends harmonically to the interior of the manifold and determines the metric. We will prove the existence of such an harmonic extension and then recover the metric from these. In particular, we will extend a "Killing vector field at infinity" into a Killing vector field on the whole manifold.

This work is related to R. Bielawski's paper [Bie], which (in particular) classifies simply-connected hyperkähler four-manifolds endowed with a trihamiltonian action of S 1 (see also [KS]). In our context, we are not given a global action of S 1 but we build it, thanks to the Killing vector field mentionned above.

In a first section, we will describe the multi-Taub-NUT examples, since their properties are crucial for the proof. In a second section, we will start our construction, extending the fibration at infnity into a harmonic map on the whole manifold. In a third section, we will use the complex structures to construct the promised Killing vector field and then finish the proof.

1. Multi-Taub-NUT metrics.

Given an integer k ≥ 1 and k points a 1 , . . . , a k in R 3 , we let M k * be the total space of the principal S 1 bundle over R 3 \ {a 1 , . . . , a k } whose Chern class integrates to -1 over the small two-spheres around each point a i -this definition makes sense for these two-spheres form a basis for the homology of R 3 \ {a 1 , . . . , a k } in degree two. Note that the Chern class consequently integrates to -k over the large two-spheres at infinity (Stokes). We need to endow this bundle with a connection. To fix conventions, we recall a connection on a S 1 bundle is merely a S 1 invariant one-form η 0 , normalized so that its value on the generator of the action is 1 (we identify the Lie algebra of S 1 with R and not iR). It follows that dη 0 is the pull back of a ("curvature") two-form Ω 0 on the base, whose cohomology class is the Chern class of the bundle up to a factor 2π.

Denoting by x = (x 1 , x 2 , x 3 ) the coordinates on R 3 , we pick a positive number m and introduce:

V = 1 + k i=1 2m |x -a i | .
This a harmonic function so * R 3 dV is a closed two-form. It follows that * R 3 dV integrates to -8mπ over the small two-spheres around each a i , so that * R 3 dV 4m represents the Chern class of the bundle, hence is the curvature two-form of a connection one-form η 4m (when k = 1, this is basically the standard contact form on S 3 ). The multi-Taub-NUT metrics are then given by the Gibbons-Hawking ansatz:

g = V dx 2 + 1 V η 2 with dη = * R 3 dV.
This extends as a complete metric on the manifold M k obtained by adding one point p i over each point a i . The p i 's should be thought of as the fixed points of the action of the circle. The ambiguity resulting resulting from the choice of the form η only produces isometric metrics (two convenient one-forms η only differ by the pull-back of an exact one-form df on the base, which makes them gauge-equivalent: the automorphism x → e if (x) • x carries one onto the other). This metric g turns out be hyperkähler ! The Kähler structures are easy to describe. We simply define a complex structure I on T * M k * by the relations Idx 1 = η V and Idx 2 = dx 3 . One can check that this is indeed Kähler [Leb] (hence extends to the whole M k ). The other Kähler structures are obtained by rotating the roles of dx 1 , dx 2 and dx 3 . For any of these complex structures, the complex manifold M k is biholomorphic to the minimal resolution of C 2 /Z k [Leb].

At infinity, the curvature decays as |x| -3 and the length of the fibers goes to 8mπ.

2.

A refined asymptotic model.

The aim of this section is to improve the asymptotic model provided by [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF] for the manifolds we are interested in. In a first step, we recall the rough model from [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF], which is basically a circle fibration at infinity, plus a connection on it. In a second time, we improve the fibration and then the connection.

2.1. The rough model. Let us give a precise definition for the class of manifolds we are interested in. Given a Riemannian manifold, we denote by Rm the curvature tensor and by ρ the distance function to some distinguished point o. We can define an ALF gravitational instanton as follows.

Definition 2.1 -An ALF gravitational instanton is a complete hyperkähler fourmanifold with cubic curvature decay -Rm = O(ρ -3 ) -and cubic volume growth -

∃ c > 0, ∀ x, ∀ t ≥ 1, c -1 t 3 ≤ vol B(x, t) ≤ ct 3 .
In view of [START_REF] Minerbe | Weighted Sobolev inequalities and Ricci flat manifolds[END_REF], under the other assumptions, the cubic curvature decay is indeed automatic as soon as the curvature decays faster than quadratically. And it implies the covariant derivatives obey ∇ i Rm = O(ρ -3-i ) [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF]. These facts follow from the Ricci-flatness.

The paper [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF] describes the geometry at infinity of such manifolds. Theorem 0.1 in the introduction sums up what we need. It ensures the existence of a circle fibration over R 3 or R 3 /Z 2 minus a ball at infinity. We assume in this paper that the base of this fibration at infinity is R 3 . minus a ball : the ALF gravitational instantons satisfying this property are called "ALF of cyclic type". In what follows, we consider an ALF gravitational instanton of cyclic type (M, g). Let us describe precisely the geometry at infinity, relying on [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF], where every detail is given.

Basically, M minus a compact subset K is the total space of a circle fibration π over R 3 minus a ball B 3 . Furthermore, this fibration encodes the asymptotic geometry as follows. First, the length of the fibers of π goes to some positive and finite value L ∞ at infinity and a g-unit π-vertical vector field U obeys

∇ g U = O(ρ -2 ) and ∀ i ≥ 2, ∇ g,i U = O(ρ -i ).
Second, if we average the metric g into g along the fibers of π (i.e. along the flow of U ), then

g = g + O(ρ -2 ) and ∀ i ∈ N * , ∇ g,i g = O(ρ -1-i ).
Third, if we push g down into a metric ǧ on R 3 \B 3 , then ǧ is asymptotically Euclidean of order τ for any τ ∈]0, 1[ in the sense of [BKN]. It implies the existence of ǧ-harmonic

coordinates x k on R 3 \B such that ǧ = dx 2 + O(|x| -τ ) and ∇ ǧdx k = O(|x| -τ -1 ).
It turns out we can strengthen a little bit the statement about the asymptotic of ǧ. In [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF], we proved a bound on the curvature tensor of ǧ that ensured this metric was asymptotically Euclidean in C 1,α topology, via [BKN]. It turns out we will need a C 2 decay of ǧ to the flat metric. This is only a minor technicality, which we fix now.

Lemma 2.2 -∇ ǧ,2 dx k = O(|x| -τ -2 ).
Proof. In view of [BKN] (p. 314-315), if we prove ∇ ǧ Rm ǧ = O(|x| -4 ), then the asymptotically Euclidean behaviour is true with order τ in C 2,α topology and we are done. To prove this estimate, we choose exponential coordinates at the running point on the base and lift the coordinate vector fields ∂ i into vector fields X i that are g-orthogonal to the fibers of π. O'Neill's formula ( [Bes]) expresses the quantity ǧ(Rm

ǧ(∂ i , ∂ j )∂ k , ∂ l ) as g(Rm e g (X i , X j )X k , X l ) plus a linear combination of terms like g([X i , X j ], U ) g([X k , X l ], U ).
The formula can be differentiated to get

∇ ǧ Rm ǧ ≤ c ∇ e g Rm e g + c ∇ e g U Rm e g + ∇ e g,2 U + ∇ e g U 2 .
Using the estimates recalled above, we find ∇ e g,i Rm e g = O(ρ -3-i ), ∇ e g U = O(ρ -2 ) and ∇ e g,2 U = O(ρ -2 ), hence the result.

2.2.

A best fibration at infinity. The estimates above make it possible to find gharmonic functions that approach the (pullback of the) functions x k at infinity in C 1 topology, with appropriate C ∞ estimates ; a "best" fibration π will stem from these harmonic functions. Beware we will often use the same notation for a function on the base R 3 and its pullback by the fibration.

Lemma 2.3 -For every index k, one can find a g-harmonic function x k on M such that for any ǫ > 0,

x k = x k + O(ρ ǫ ) and dx k = dx k + O(ρ ǫ-1 ), with moreover: ∀i ≥ 2, ∇ g,i x k = O(ρ ǫ-i ).
Proof. Let us extend x k as a smooth function on the whole M and observe

∆ g x k -∆ e g x k ≤ c |g -g| ∇ e g dx k + c ∇ g -∇ e g |dx k | ≤ cρ -2 .
Since the functions x k are harmonic with respect to ǧ or g, we deduce ∆ g x k = O(ρ -2 ). This lies in ρ δ-2 L 2 for any δ > 3 2 , so we can apply the analysis of [START_REF] Minerbe | A mass for ALF manifolds[END_REF] to find a solution u k for the equation ∆

g u k = -∆ g x k , with ∇ i u k ∈ ρ δ-i L 2 , 0 ≤ i ≤ 2. As explained in the appendix of [Mi2], a Moser iteration yields u k L ∞ (A R ) ≤ cR -3 2 u k L 2 (A ′ R ) + cR 2 ∆ g u k L ∞ (A ′ R ) , where A R = {R ≤ ρ ≤ 2R} and A ′ R = {R/2 ≤ ρ ≤ 4R}. Since u k is in ρ 3 2 +ǫ L 2 and ∆ g u k = -∆ g x k = O(ρ -2
), we get u k = O(ρ ǫ ) for any positive ǫ. Since Ric g = 0, the Hodge Laplacian and the Bochner Laplacian (which we denote by ∆ g on the whole tensor algebra) coincide on one-forms. With lemma 2.2, we can apply the same argument to du k , with this Laplacian (cf. [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF]) and find du k = O(ρ ǫ-1 ). As a result, the function x k := x k + u k is g-harmonic, with

x k = x k + O(ρ ǫ ) and dx k = dx k + O(ρ ǫ-1 ).
The equation ∆ g dx k = 0 can be used together with [∆ g , ∇ g ] = Rm g ∇ g + ∇ g Rm g (here, denotes any bilinear pairing depending only on g) to obtain:

∆ g ∇ g,i dx k = i j=0 ∇ g,j Rm g ∇ g,i-j dx k .
Since ∇ g,j Rm g = O(ρ -3-j ), the estimates follow from an induction argument based on the following inequality (Moser iteration and cutoff argument, cf. [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF]):

∇ g,i dx k L ∞ (A R ) + R -1 2 ∇ g,i+1 dx k L 2 (A R ) ≤ cR -3 2 ∇ g,i dx k L 2 (A ′ R ) + cR 1 2 ∆ g ∇ g,i dx k L 2 (A ′ R ) .
Enlarging K if necessary, these g-harmonic functions x 1 , x 2 , x 3 provide a new S 1fibration π = (x 1 , x 2 , x 3 ) : M \K -→ R 3 \B. As a consequence of (2.3), vector fields X that are g-orthogonal to the fibers of π obey ( 1)

|dπ(X)| g R 3 |X| g = 1 + O(ρ ǫ-1 )
and π satisfies the following estimates

(2) ∀i ≥ 2, ∇ g,i π = O(ρ ǫ-i ),
with respect to the metric g on M \K and the Euclidean metric on the base. Let us denote by U a g-unit π-vertical vector field. Differentiating the relation dπ(U ) = 0 and using (1), (2), we get as in [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF]:

(3)

∀ i ∈ N * , ∇ g,i U = O(ρ ǫ-1-i ).
We also introduce the function L assigning to each point p the length of the π-fiber through p and the flow ψ t of U . With ψ L = id, one finds dL = g -ψ L * g (U , .). Since the Lie derivative of g along U is twice the symmetrization of ∇ g U , we have g -ψ t * g ≤ ctρ ǫ-2 and therefore |dL| ≤ cLρ ǫ-2 .

So d log L = O(ρ ǫ-2 ). From Cauchy's criterion, we deduce L goes to a limit L ∞ at infinity (and L ∞ = L ∞ , indeed). Using the arguments of [START_REF] Minerbe | On some asymptotically flat manifolds with non maximal volume growth[END_REF], we can then estimate the metric g obtained by averaging g along the flow of U by ( 4)

∀ i ∈ N, ∇ g,i ( g -g) = O(ρ ǫ-2-i )
and the derivatives of L by

∀ i ∈ N * , ∇ g,i (L) = O(ρ ǫ-1-i ).
We introduce the vector field

T := L L ∞ U . on M \K and see L ∞
2π T as the infinitesimal generator of a S 1 action, which makes π into a principal S 1 -fibration. The one-form η := e g(T ,.)

e g(T ,T ) is S 1 invariant and satisfies η(T ) = 1. It is L ∞
2π times a connection on the S 1 -bundle. The metric g induces a metric ǧ on the base and we have

∀ i ∈ N, ∇ ǧ,i ǧ -dx 2 = O( xǫ-1-i ).
The outcome of all this is a "model" metric h := dx 2 + η 2 such that dη = π * Ω with

∀ i ∈ N, ∇ h,i (g -h) = O(ρ ǫ-1-i ) and D i (Ω) = O(|x| ǫ-2-i ).
For the sake of simplicity, we will forget the underlining and denote |x| by r. The letter ǫ will refer to any small positive number and we basically use the convention "ǫ = 2ǫ" (finitely many times, hopefully).

What have we gained ? We chose to sacrifice an ǫ in the lower order estimates of π and U for two benefits. The first one is technical : we are provided with better estimates for the higher order derivatives. The second one is essential: the fibration at infinity extends as a harmonic function on the whole manifold, as in the examples. It is encouraging to notice that such a function π is unique up to the addition of a constant: this is certainly the good object to look at! 3. Using the hyperkähler structure 3.1. A Killing vector field. We can rely on the hyperkähler structure (g, I, J, K) to build a Killing vertical vector field. To fix ideas, let us proceed to some normalization. Since d(dx k (IT )) = O(r ǫ-2 ), the functions dx k (IT ) go to some constants at infinity (Cauchy criterion). Indeed, since moreover η(IT ) = O(r ǫ-1 ), we can rotate the coordinates x k so that dx k (IT ) = δ 1k + O(r ǫ-1 ). Similarly, up to a second rotation (in the plane x 1 = 0 in R 3 ), we can assume dx k (JT ) = δ 2k + O(r ǫ-1 ), and consequently

dx k (KT ) = δ 3k + O(r ǫ-1 ).
The following proposition is a key step. Its proof again uses the fact that a hyperkähler metric has vanishing Ricci curvature, so that the Hodge Laplacian on 1-forms coincides with the Bochner Laplacian.

Proposition 3.1 -There is a unique g-Killing vector field W such that

ι W ω I = -dx 1 , ι W ω J = -dx 2 , ι W ω K = -dx 3 .
It is π-vertical, preserves the complex structures I, J, K and obeys the estimate

∀ k ∈ N, ∇ g,k (W -T ) = O(r ǫ-1-k ).
Proof. One can define three vector fields W 1 , W 2 , W 3 by the relations ι W 1 ω I = -dx 1 , ι W 2 ω J = -dx 2 and ι W 3 ω K = -dx 3 . Since the one-forms dx l are g-harmonic and the Kähler forms are parallel, the vector fields W l are g-harmonic. Now, for l = 1, 2, 3, our previous estimates ensure

∇ k (W l -T ) = O(r ǫ-1-k ),
for every positive integer k. Our choice of coordinates in R 3 moreover implies that W l -T goes to zero at infinity. One can therefore integrate the estimate for k = 1 into the corresponding estimate for k = 0. In particular, the difference X between any two of these vector fields W l is harmonic and goes to zero at infinity. It follows that the function |X| 2 goes to zero at infinity and satisfies :

∆ |X| 2 = 2g(∆X, X) -2 |∇X| 2 ≤ 0.
From the maximum principle, we deduce |X| 2 = 0, namely X = 0. We conclude: W 1 = W 2 = W 3 =: W . By definition, we have dx 1 (W ) = -ι W ω I (W ) = 0 as well as dx 2 (W ) = dx 3 (W ) = 0 for the same reason. So W is vertical. Since ω I and ι W ω I = -dx 1 are closed, Cartan's magic formula yields L W ω I = 0; similarly, L W ω J = L W ω K = 0. Finally, since ω I is parallel, we have: ι ∇W ω I = -∇dx 1 . The right-hand side is a symmetric bilinear form, so for any two vector fields X, Y , we are given the identity: g(∇ g X W, Y ) = -g(∇ g IY W, IX). We can of course play the same game with K and then J, so as to find

g(∇ g X W, Y ) = g(∇ g KIX W, KIY ) = -g(∇ g JKIY W, JKIX) = -g(∇ g Y W, X).
The tensor ∇ g W is therefore skew-symmetric, which exactly means W is Killing. Since the Kähler forms and the metric are preserved by W , the complex structures are also preserved.

Let us set α := g(W, .). The definition of W means dx 1 = Iα, dx 2 = Jα and dx 3 = Kα. In particular, the covectors dx k are everywhere g-orthogonal and have the same g-norm as α -notice these covectors thus vanish only all at the same time. With V := |W | -2 , we might keep in mind the following formulas, on the open set M * where W does not vanish:

g = V (dx 2 + α 2 ) and ω I = V (dx 1 ∧ α + dx 2 ∧ dx 3 ) . 3.2. The map π. Lemma 3.2 -The set M \M * is finite : M \M * = {p 1 , . . . , p k }.
Proof. M \M * is the where the vector field W vanishes. Pick a point p such that W p = 0. The flow φ t of W preserves the hyperkähler structure so the differential T p φ t acts on T p M by a transformation in SU (2), reading e iλt 0 0 e -iλt in some basis, for some λ ∈ R * ; this is indeed a special case of [GH]. It follows that p is an isolated fixed point of the flow. This ensures M \M * is discrete. Since V goes to 1 at infinity, M \M * is compact. It is therefore a finite set.

Lemma 3.3 -π : M -→ R 3 is onto.

Proof. Since π| M * is a submersion, the set π(M * ) is open in R 3 . Let y be a point of its boundary and let y n = π(x n ) denote a sequence in π(M * ) and converging to y. Since π is asymptotic to a circle fibration with fibers of bounded length, π is proper, so we may assume x n goes to some point x in M , with π

(x) = y. Since π(M * ) is open, x cannot belong to M * : x is one of the p i 's. To sum up, π(M * ) is an open set of R 3 whose boundary is contained in {π(p 1 ), . . . , π(p k )}. In particular, π(M ) is dense in R 3 . It is also closed, for π is proper : π(M ) = R 3 .
Let us set a i = π(p i ).

Lemma 3.4 -The map π : M * -→ R 3 \ {a 1 , . . . , a k } is a circle fibration.

Proof. This submersion π : M * -→ R 3 \ {a 1 , . . . , a k } is surjective and proper (it can be proved exactly as in the lemma above), hence a fibration by Ehresmann's theorem. Since it is a circle fibration outside a compact set, it is a circle fibration.

Remark. π can be interpreted as the hyperkähler moment map for the Hamiltonian action given by the flow of W . This kind of situation is studied in [Bie], which inspired us.

The connection.

Let us redefine the one-form η := V α, so that η(W ) = 1, L W η = 0 and thus ι W dη = 0 ; it follows that dη = π * Ω for some two-form Ω on the base. We wish to interpret η as a connection form (up to some normalization).

The fibers of π are also the orbits of W , so that the flow of W is periodic. In M * , the period P x of the orbit π -1 (x) is equal to the integral π -1 (x) η. Given nearby points x and y, it follows from Stokes theorem that the difference P x -P y is the integral of dη over the cylinder π -1 ([x, y]), which vanishes because dη is basic! The period is therefore constant and can be computed at infinity : it is equal to the length L ∞ of the fibers at infinity. In particular, the vector field L∞ 2π W is the infinitesimal generator of an action of S 1 , making π| M \K a principal S 1 bundle over R 3 minus a ball. Furthermore, 2π L∞ η is a connection one-form on this S 1 -bundle and its curvature can be computed through the following lemma.

Lemma 3.5 -Ω = * R 3 dV . Proof. The Kähler form ω I = dx 1 ∧ α + V dx 2 ∧ dx 3 is closed, hence the relation dx 1 ∧ dη = dV ∧ dx 2 ∧ dx 3 , which means: Ω(∂ 2 , ∂ 3 ) = ∂ 1 V .
The other Kähler forms provide the remaining components.

So we know Ω as soon as we know V .

Lemma 3.6 -There are positive numbers m i such that

V = 1 + k i=1 2m i |x -a i | .
Proof. Lemma 3.5 implies V is g R 3 -harmonic outside the a i 's. Moreover, it is positive. A classical result [ABR] then ensures that around each singularity a i , it is the sum of the function 2m i |x-a i | , m i > 0, and of a smooth harmonic function. Globally, this means V = ϕ + k i=1 2m i |x-a i | for some smooth harmonic function ϕ on R 3 . The asymptotic of the metric implies ϕ goes to 1 at infinity, so that ϕ = 1.

We can then identify Ω. If dω i is the volume form of the unit sphere around a i , we have :

Ω = - k i=1 2m i dω i .
As recalled in section 1, this data determines the connection up to gauge (which is enough for a classification up to isometry).

Observe the topology of the circle bundle determines the cohomology class of Ω, seen as L∞ 2π times the curvature of a connection. For large R, this means 1 L∞ r=R Ω = c ∞ 1 , where c ∞ 1 is the Chern number of the fibration over the large spheres. A relation follows:

-8π k i=1 m i = L ∞ c ∞ 1 .
Let us now look at the circle bundle induced on a small sphere near a i . It has Chern number c i 1 = ±1 for there is no orbifold singularity on M and it can be computed by an integral of Ω as above:

-8πm i = L ∞ c i 1 . This has two consequences. First, c i 1 = -1 because m i is nonnegative. Second, 8πm i = L ∞ ! The parameters m i are necessarily all equal, depending only on the length of the fibers at infinity (a similar remark can be found [GH]).

Observe also that the number k of singularities is given by the Chern number at infinity : k = -c ∞ 1 ; the fundamental group at infinity is simply Z k . The topology is completely determined by the parameter k.

Remark. It is interesting to relate the asymptotic of V to the mass m defined in this paper. It is a nonnegative Riemannian invariant, given by

m := - 1 12πL ∞ lim R-→∞ ∂B R * h div h g + d Tr h g - 1 2 d g(W, W ) ,
where h = dx 2 + η 2 in our context (this definition indeed differs by a factor 3 from that in [START_REF] Minerbe | A mass for ALF manifolds[END_REF]). Here, a slight computation (cf. [START_REF] Minerbe | A mass for ALF manifolds[END_REF]) ensures this mass is m = k i=1 m i , hence -8πm = L ∞ c 1 . The classification result when k = 0 is an immediate application of [START_REF] Minerbe | A mass for ALF manifolds[END_REF] for the mass m vanishes iff (M, g) is isometric to the standard R 3 × S 1 .

3.4. The classification. We can conclude by the Theorem 3.7 -When k = 0, (M, g) is isometric to the standard 3 × S 1 , with no holonomy and circles of length L ∞ . When k ≥ 1, (M, g) is isometric to M k endowed with the multi-Taub-NUT metric given by

g = V dx 2 + 1 V η 2 , where V = 1 + k i=1 2m
x-a i and dη = * R 3 dV . The positive parameter m is the mass of (M, g) and it is k 8π times the length of the fibers at infinity. The holomorphic classification follows from the explicit formulas for the Kähler forms.

Date:

  October 29, 2009.