G Mansvelt

O Lorimier

Glehen

Dominique Elias
email: elias@igr.fr

PhD F Gilly

F Quenet

MD J M Bereder

L Sideris

B Mansvelt

MD G Lorimier

MD O Glehen

MD L Sidéris

Pseudomyxoma peritonei: a French multicentric study of 301 patients treated with cytoreductive surgery and intraperitoneal

Pseudomyxoma peritonei: a French multicentric study of 301 patients treated with cytoreductive surgery and intraperitoneal chemotherapy

M A N U S C R I P T A C C E P T E D

Miniabstract :

The results obtained in 301 patients with pseudomyxoma peritonei treated by cytoreductive surgery combined with perioperative intraperitoneal chemotherapy in 18 French-speaking centers confirms good results and that these procedures should be recommended in specialized centers.

STRUCTURED ABSTRACT Objective :

To analyze a large series of patients with pseudomyxoma peritonei (PMP) treated with cytoreductive surgery associated with perioperative intraperitoneal chemotherapy (PIC) in 18 French-speaking centers.

Patients and methods:

From March 1993 to December 2007, 301 patients with diffuse PMP were treated by cytoreductive surgery with PIC. Complete cytoreductive surgery was achieved in 219 patients (73%), and hyperthermic intraperitoneal chemotherapy (HIPEC) was performed in 255 (85%), mainly during the latter period of the study.

Results:

Postoperative mortality and morbidity were 4.4% and 40%, respectively. The mean follow-up was 88 months. The 5-year overall and disease-free survival rates were 73%% and 56%, respectively. The multivariate analysis identified 5 prognostic factors : the extent of peritoneal seeding (p=0.004), the center (p=0.0004), the pathologic grade (p=0.03), gender (p=0.02), and the use of HIPEC (p=0.04). When only the 206 patients with complete cytoreductive surgery were considered, the extent of peritoneal seeding was the only significant prognostic factor (p=0.004).

Conclusion:

M A N U S C R I P T A C C E P T E D

This large multicentric retrospective study confirms that cytoreductive surgery combined with PIC (with the use of hyperthermia) should be considered as the gold standard treatment of PMP and should be performed in specialized centers. It underlines the prognostic impact of the extent of peritoneal seeding, especially in patients treated by complete cytoreductive surgery. This prognostic impact appears to be greater than that of the pathologic grade. Pseudomyxoma peritonei (PMP) is a clinico-pathologic entity characterized by the production of mucinous ascites (1,2) and mostly originates from appendiceal epithelial neoplasms. The tumor grows, obstructing the narrow lumen of the appendix and subsequently leading to appendiceal perforation [START_REF] Sugarbaker | New standard of care for appendiceal neoplasms and pseudomyxoma peritonei syndrome[END_REF][START_REF] Brendan | The etiology, clinical presentation and management of pseudomyxoma peritonei[END_REF]. Then neoplastic cells progressively and slowly colonize the peritoneal cavity and copious mucin production builds up in the peritoneal cavity. Recently, pathologic enteric bacteria were shown to be present in all specimens and their density was correlated with the prognosis (3).

Key

Appendix tumors causing PMP range from a benign pathologic appearance (disseminated peritoneal adenomucinosis or DPAM), to malignant pathologic findings (peritoneal mucinous carcinomatosis or PMCA), with some intermediate pathologic grades [START_REF] Ronnett | Disseminated peritoneal adenomucinosis and peritoneal carcinomatosis. A clinicopathologic analysis of 109 cases with emphasis on distinguishing pathologic features, site of origin, prognosis and relationship to"pseudomyxoma peritonei[END_REF]. Clinically, this syndrome ranges from early PMP fortuitously discovered on imaging or during a laparotomy performed for another reason, to advanced cases with a distended abdomen, occlusive syndrome and starvation.

The conventional treatment of PMP is surgical debulking repeated, as necessary to alleviate pressure effects [START_REF] Gough | Pseudomyxoma peritonei. Long tern patient survival with an aggressive regional approach[END_REF][START_REF] Miner | Long-term survival following treatment of pseudomyxoma peritonei: an analysis of surgical therapy[END_REF]. However, this treatment is palliative, leaving visible or occult disease in the peritoneal cavity. Repeated debulking surgeries become ever more difficult due to progressively thickened intra-abdominal adhesions [START_REF] Yan | A systematic review of the efficacy of cytoreductive surgery and perioperative intraperitoneal chemotherapy for pseudomyxoma peritonei[END_REF].

A combination of cytoreductive surgery (CRS) with perioperative intraperitoneal chemotherapy (PIC) was proposed by Sugarbaker [START_REF] Sugarbaker | Rationale for perioperative intraperitoneal chemotherapy as a surgical adjuvant for gastrointestinal malignancy[END_REF]. The rationale behind this strategy was to resect visible disease and any occult residual disease would be treated with a high concentration of chemotherapy and hyperthermia which mutually potentiate one another's effects [START_REF] Hettinga | Reduction of cellular cisplatin resistance by hyperthermia. A review[END_REF]10). Fifteen years and 600 publications later [START_REF] Yan | A systematic review of the efficacy of cytoreductive surgery and perioperative intraperitoneal chemotherapy for pseudomyxoma peritonei[END_REF], a systematic review of this novel treatment concluded that this combined approach was very probably superior to conventional therapy, even in the absence of controlled

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 5 5
randomized trials, and even with evidence from only five series of more than 100 cases (11-16). Each series reported the experience of only one team with various experiences, and there was no multicentric series reflecting "reality".

This study aimed to collect patients with PMP treated with this combined approach in 18 different French-speaking centers, to analyze morbidity, mortality and survival results, and to identify the main prognostic factors for better patient selection.

Patients and methods

Patient Population

The AFC (Association Française de Chirurgie) conducted a multicentric retrospective study in French (n=20), Belgian (n=2), Canadian (Quebec, n=2) and Swiss (n=1) centers.

Patients were treated between March 1993 and December 2007. Inclusion criteria were 1) pathologic confirmation of PMP 2) patients treated with the intent of achieving an R-0 resection with cytoreductive surgery combined with hyperthermic intraperitoneal chemotherapy (HIPEC) or early postoperative intraperitoneal chemotherapy (EPIC).

Exclusion criteria were tumors originating from the colon, small bowel and appendiceal adenocarcinoma without a predominant mucinous component.

Treatment

Resection of PMP obeyed principles described elsewhere (17). In short, peritonectomy procedures which included a total anterior and lateral parietal peritonectomy, omentectomy and splenectomy, right and left sub-phrenic peritonectomy, pelvic peritonectomy and lesser omentectomy and cholecystectomy with stripping of the omental bursa were performed exclusively in the macroscopically pathologic areas. More or less extended resections of the colon, rectum, small-bowel and stomach were associated, if invaded by PMP. Intraperitoneal chemotherapy was delivered immediately after cytoreductive surgery. EPIC was performed from day 1 to day 5 after surgery, and was used during the early period of this retrospective study. HIPEC was performed with an opened abdominal cavity (Coliseum technique), or a closed one and was used during the major second period of the study.

Data Forms

A standard data form was created to retrieve information on the primary colorectal tumor, on the status of the patient prior to the combined procedure, and on previous treatment with systemic chemotherapy. The extent of PC is assessed through intraoperative exploration using the Peritoneal Cancer Index (PCI) which scores from 0 to 3 each of the 13 defined areas of the abdominal cavity, and therefore ranges from 1 to 39. Four PCI subgroups were arbitrarily defined for the analysis : 1 to 6, 7 to 12, 13 to 19, and > 19. Information recorded about the combined procedure included the completeness of cytoreductive surgery, any simultaneous resection of the primary tumor and liver metastases, the presence or absence of lymph node metastases, the type of perioperative intraperitoneal chemotherapy (HIPEC or EPIC) and its modalities, and treatment with adjuvant systemic chemotherapy. Assessment of the completeness of the cancer resection (CCR) with cytoreductive surgery was done by the surgeon at the end of the procedure and classified into three categories: CCR-0 signified no macroscopic residual cancer, CCR-1, no residual nodules greater than 2.5 mm, and CCR-2, that the diameter of the residual nodules exceeded 2. Data recorded included the most recent follow-up information, the status of the patient (alive with disease, alive without disease, died with disease, died without disease), and the site of the initial recurrence.

Statistical Analysis

The descriptive analyses of the patients included in the registry, as well as the survival analyses were performed on a per-patient basis (one record per patient). However, all procedures were considered for the analyses of postoperative morbi-mortality events (death or major complications occurring within 30 days after surgery).

Categorical variables were described in terms of frequency and percentages. The distributions of continuous variables were described with the mean, standard error, median, 1 st and 3 rd quartiles. Patient, disease and treatment characteristics were correlated with postoperative morbi-mortality events using univariate and multivariate logistic regression models, adjusted on centers, because of their strong prognostic influence.

The analysis of long-term mortality censored information after the cut-off date of December 31 st 2007 since an active enquiry was performed in all centers to collect the status (dead/alive) of the patients at that date. The analysis of total mortality considered the interval from the first procedure to the date of death, the date of the

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 8 8
last news, or the cut-off date, whichever came first. The analysis of recurrence or death was based on the interval from the first procedure to the date of first recurrence, the date of death, the date of the last news, or the cut-off date, whichever came first. Postoperative deaths were not excluded from the survival analysis. Patients who had CCR-2 resections with residual tumor nodules exceeding 2.5 mm were considered as having immediate relapses. When the date of recurrence was unknown in patients who had died, the date of death was used instead. Kaplan-Meier survival estimates were calculated, and compared between strata with the log-rank test. The influence of baseline risk factors on the hazard of death was assessed using a multiple proportional hazard regression model stratified on centers. Stratification was justified due to considerable heterogeneity in hazards between centers, and to a strong potential confounding effect on other risk factors. In all multivariate analyses, age and PCI were entered as continuous variables. The completeness of cytoreduction was also entered as continuous when justified by a linear trend across categories. Risk factors with a trend of significance (p ≤ 0.10) were retained in the final model. The SAS statistical software (Windows, V9.1) was used for all analyses. PMP originated from the appendix in 91% of the cases (ovary in 7%, and the origin was unknown in 2%). The circumstances leading to the discovery of PMP were : unexplained ascites leading to laparoscopy or laparotomy ( (32%), abdominal pain (27%), increasing abdominal girth (15%), on imaging performed for an another reason (15%), deterioration of general status (5%), wound hernia (4%), and appendicitis (2%).

Results

Patients

Treatment Modalities

The mean peritoneal index was 18.5 (1 st -3 rd quartiles: 11-26, median:18, range:9-39).

Concerning the completeness of cytoreductive surgery, CCR-0 had been achieved in 73%, CCR-1 in 20%, and CCR-2 in 7%. Concerning intraperitoneal chemotherapy, 255 patients (85%) had been treated with HIPEC and 46 patients (15%) with EPIC. Hipec was done even with mitomycin C based-regimens during 60-120 min. at 41-42°C, even with oxaliplatin-based regimens during 30 min at 43°C. EPIC was done with mitomycinC during day 1 and 5-fluorouracyl from day 2 to day 5. Systemic chemotherapy had been administered before or after this combined treatment in 69 patients (23%), mainly those presenting severity of illness criteria (mean number of courses: 7.1).

Mortality and Morbidity

Thirteen patients (4.4%) had died postoperatively. The causes of death were septic shock (9 patients) linked to abdominal infection, respiratory complications (4 patients), hematological toxicity (2 patients), pulmonary embolism (1 patient), and acute renal insufficiency (1 patient).

Grade 3-4 complications had occurred in 119 patients (40%), and 17.5% (n=52) of all patients had been re-operated on; 8% (n=24) had a digestive fistula, 9% (n=27) a intraperitoneal abscess, 13% (n=38) hemorrhage, 14% (n=42) a pulmonary infection, and The logistic regression analysis of factors which significantly increased the risk of mortality and morbidity identified two main factors, the peritoneal index (p=0.002, correlated with the extent and duration of surgery), and the presence of a high pathologic grade (PMCA) (p= 0.03).

Survival

With a mean follow-up of 88.4 months (1 st -3 rd quartile: 28 -125, median: 73), the overall 1-year, 3-year, and 5-year survival rates were respectively 89.4%, 84.8% and 72.6% (figure 1). The 10-year survival rate was 54.8%. Median survival had not yet been reached, but it will be longer than 100 months. The disease-free survival (DFS) rate was 56% at 5 years (figure 1) and the median duration of DFS was 78 months.

There was a strong center effect (p=0.004) that justified an analysis of the impact of other prognostic factors adjusted on centers (figure 2). Figure 2 shows 7 curves corresponding to the six centers which treated at least 10 patients and one curve (the lowest one) for the 11 remaining centers (which treated fewer than 10 patients per center).

In the univariate analysis (table 1), age did not exert an impact on survival unlike gender (p=0.02, in favor of females), the extent of peritoneal seeding (p<0.001), the completeness of CRS (p<0.001), the pathologic grade (p<0.001) and the presence of invaded lymph nodes (p<0.001). Two other prognostic factors must be underlined: HIPEC seemed more efficient than EPIC (p=0.001) and the results were different The multivariate analysis identified five factors with an independent influence on survival (table 2). First, the center effect (p<0.01), second, the extent of peritoneal seeding (p=0.004) (figure 3) with a significant decline in survival when the peritoneal index was greater than 20. This index was strongly correlated with the completeness of the CRS : CCR1 and CCR-2 were only observed when the peritoneal index was high.

Thirdly, the pathologic grade (p=0.03) was a poor prognostic factor for PMCA. No significant difference in survival was found between DPAM and intermediate grades.

The other two factors were gender (p=0.02) in favor of females, and the use of HIPEC rather then EPIC (p=0.04).

Multivariate analysis of disease-free survival identified two significant factors: the extent of the peritoneal seeding (p<0.001) (strongly correlated with the CCR-status), and the center's experience (p=0.006).

In the group of 206 patients with a CCR-0 resection, the 5-year and 10-year survival rates were 84% was 61% respectively. It is noteworthy that two factors became nonsignificant in the univariate analysis : gender and the pathologic grade. Only one factor continued to exert a prognostic impact : the extent of peritoneal seeding (p=0.004), while HIPEC was marginally better (p=0.07) than EPIC.

Discussion

This study reports the results of the first multicentric retrospective analysis of 301 patients with diffuse PMP treated by cytoreductive surgery combined with Survival. This study reports on the largest series of patients with PMP ever published after the monocentric series of 501 cases reported by Sugarbaker (11). Compared to the exceptional expertise of Sugarbaker, it presents results from 18 French-speaking centers, and adequately reflects the management of PMP in real-life settings where the experience of such treatment is very unequal from one center to another. It is always difficult to extrapolate results from a monocentric experience to other centers, due to considerable heterogeneity in selection criteria, in CRS and intraperitoneal chemotherapy techniques. Our study effectively underlines the importance of the experience of the center in the survival results. Due to a healthy degree of scepticism among some oncologists, any multicentric study collecting data from a large number of centers is particularly useful, even if the experience and the techniques differ considerably from one center to another. This trial concerns a collaborative effort of 25 French-speaking centers, which included 1290 patients with primary or digestive peritoneal carcinomatosis treated by CRS combined with PIC to evaluate the efficiency of this combined approach (18 of them have treated PMP). Due to its retrospective nature, this study suffers from many potential biases in the selection of patients and in treatments. Nonetheless, it has allowed us to present a large number of patients treated in many centers and to draw conclusions. Also, the 5-year overall survival rate of 72.6% for PMP represents the worst results that can be obtained with CRS combined with PIC because it takes into account the learning curves of 18 centers, and because progress in patient selection and in the quality of

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 13 13
surgery increases with experience. This overall survival rate is identical to the 72% reported by Sugarbaker in his cohort of 501 patients (11). Thus, our multicentric study corroborates the results of the largest monocentric study. If we analyze the other four trials which included more than 100 patients treated with this combined approach, our survival results are better than those reported by Zoetmulder (13) and Loggie (14), close to those of Chua et al. ( 16) but worse than those reported by Elias (15) who included only patients whose resection was complete or sub-complete (CCR-0 or CCR-1).

Moreover, a mortality rate of 4% and a morbidity rate of 40% for this combined approach are the sort of rates that would be obtained with any kind of complex oncologic surgery such as oesophagectomy or duodeno-pancreatectomy. These rates are therefore acceptable face to benefits for overall and disease-free survival.

The results of the conventional treatment of PMP mainly based on repeated surgical debulking have been reported in two series [START_REF] Gough | Pseudomyxoma peritonei. Long tern patient survival with an aggressive regional approach[END_REF][START_REF] Miner | Long-term survival following treatment of pseudomyxoma peritonei: an analysis of surgical therapy[END_REF]. Gough et al. ( 5) reported a 5-year survival rate of 53%, with a 97% recurrence rate at 10 years in 56 PMP patients who underwent serial debulking and were selectively treated with intraperitoneal radiotherapy or chemotherapy between 1957 and 1983. Minner et al. [START_REF] Miner | Long-term survival following treatment of pseudomyxoma peritonei: an analysis of surgical therapy[END_REF] reported an 88% recurrence rate at 5 years in 97 PMP patients treated with serial debulking, systemic chemotherapy and/or delayed intraperitoneal 5-fluorouracil over a 22-year period. The 3% and 12% disease-free survival rates are well below the 56% rate we observed in our series. Sugarbaker and Zoetmulder also reported higher rates, 47% and 61%, respectively. All these results are probably due to the superiority of the combined over the conventional approach. In France, this combined approach has become the gold standard treatment when feasible (18). Prognostic factors. The multivariate analysis of our series identified four significant and independent prognostic factors, adjusted on centers: the extent of the disease, the pathologic grade, gender, and the use of hyperthermia. One of these factors, namely the extent of the disease, was not identified in the five earlier series of more than 1OO patients treated with the combined approach. In contrast, the completeness of cytoreduction was statistically significant in three (12,14,16). The pathologic grade was not retained in three series (12-15), but selected in two (13,16) which reported only 31 CCR-0 resections in 103 treated patients and 73 CCR-O resections in 106 treated patients for the first and the second study respectively.

Thus, new information is emerging from our multicentric study.

First, this multicentric study underlines the influence of the center in which the procedure was performed and this led us to perform a multivariate analysis adjusted on centers. Centers with less experience had a lower rate of complete cytoreduction and poorer survival results. The fact that incomplete cytoreduction in the 174 patients reported by Glehen et al [START_REF] Glehen | Incomplete cytoreduction in 174 patients with peritoneal carcinomatosis from appendiceal malignancy[END_REF], resulted in only 15% of patients alive at 5 years argues in favor of exclusively entrusting this approach to experienced, good quality centers in the future.

Second, the extent of peritoneal seeding has a potent prognostic impact, even if this impact is probably lower than for patients with colorectal peritoneal carcinomatosis (18-21). In our series, the 5-year survival rate was only 57% for patients with a peritoneal index exceeding 19, whereas it was 83% for the other patients. We also established that the completeness of cytoreduction was strongly correlated with the disease extent (peritoneal cancer index), and also with the pathologic grades (data not shown). Finally, extensive PMP with high pathologic grades are the most frequently incompletely resected lesions and mainly in inexperienced centers.

M A N U S C R I P T A C C E P T E D

Third, the pathologic grade had a significant prognostic impact on the whole series, but it disappeared when only the 206 patients with a complete CCR-0 resection were considered. Thus complete cytoreductive surgery is able to reduce the negative prognostic impact of the pathologic grade as previously pointed out by Elias et al. (15). Fourth, survival was better among female patients in the whole series, but gender was not statistically significant when only CCR-0 patients were taken into account. Thus, male patients more frequently presented with extended and incompletely resectable PMP than female patients.

Fifth, survival results were better with HIPEC than with EPIC. Glehen et al. [START_REF] Glehen | Incomplete cytoreduction in 174 patients with peritoneal carcinomatosis from appendiceal malignancy[END_REF] already reported the superiority of HIPEC over EPIC in a retrospective study evaluating patients treated with incomplete CRS. Our results could stem from a bias because EPIC was mainly used during the earlier period of the study. Although experimental studies are also in favor of the use of hyperthermia [START_REF] Koga | Treatment of implanted peritoneal cancer in rats by continuous hyperthermic peritoneal perfusion in combination with an anticancer drug[END_REF], its benefit should be confirmed in a prospective study.

Finally, it is noteworthy that the extent of peritoneal seeding was the only prognostic factor identified in the multivariate analysis in the group of patients with a CCR-0 resection.

In conclusion, this multicentric study of 301 patients with PMP confirms the promising results reported earlier by Sugarbaker and a few other experienced teams using the combination of cytoreductive surgery with perioperative intraperitoneal chemotherapy to treat PMP. It also establishes that the extent of peritoneal seeding and the completeness of cytoreduction exert a greater impact on survival than the pathologic grade.

  -words: peritoneal pseudomyxoma, cytoreductive surgery, intraperitoneal chemotherapy, hyperthermia, prognosis;

4 M

 4 5 mm. Information was obtained regarding the postoperative course, including postoperative deaths (within 30 postoperative days) and the cause of death. Major complications (grade 3 and the National Cancer Institute's Common Toxicity Criteria) were considered and the reasons for re-operation were detailed. PMP was classified into 3 pathological grades according to the Ronnett classification system (4): grade 1 corresponded to disseminated peritoneal adenomucinosis (DPAM), grade 2 to peritoneal carcinomatosis with intermediate features, and grade 3 to peritoneal mucinous carcinomatosis (PMCA).

From

  March 1993 to December 2007, 301 patients with PMP were treated in 18 Frenchspeaking centers with CRS plus PIC (5 centers did not treat PMP). The number of patients treated per center ranged from 130 to 1. There were 190 females and 111 males and the mean age was 52 years (1 st -3 rd quartile: 44-59; median: 52, range: 26-79).

  ) grade 3-4 hematological toxicity. The median duration of hospitalization was 21 days (mean duration 26 ±14 days).

  centers (p=0.0004). The prognosis was worse (p=0.01) for patients who had received systemic chemotherapy (because it was given in advanced and high-grade cases).

  chemotherapy. Postoperative mortality was 4%, and grade 3-4 morbidity was 40 %. The 5-year survival rate was 72.6% and median diseasefree survival was 78 months. Median survival has not yet been attained but was longer than 100 months.

  A, Koga S, Maeta M. Thermochemosensitivity: augmentation by hyperthermia of cytotoxicity of anticancer drugs against human colorectal cancer, measured by the human tumor clonogenic assay. Oncology 1988; 45: 236-41. 11. Gonzalez-Moreno S, Sugarbaker P. Right hemicolectomy does not confer a survival advantage in patients with mucinous carcinoma of the appendix and peritoneal seeding. Br J Surg 2004; 91: 304-311 12. Sugarbaker P, Chang D. Results of treatment of 385 patients with peritoneal surface spread of appendiceal malignancy. Ann Surg Oncol 1999; 6: 727-731 13. Smeenk RM, Verwaal V, Antonini N, Zoetmulder F. Survival analysis of pseudomyxoma peritonei patients treated by cytoreductive surgery and hyperthermic intraperitoneal chemotherapy. Ann Surg 2007; 245: 104-109 14. Stewart JH, Shen P, RussellGB, Bradley RF, Hundley JC, Loggie BL, et al. Appendiceal neoplasms with peritoneal dissemination: outcomes after cytoreductive surgery and intraperitoneal hyperthermic chemotherapy. Ann Surg Oncol 2006; 13: 624-634 15. Elias D, Honoré C, Ciuchundea R, Billard V, Raynard B, Lo Dico R, et al. Peritoneal pseudomyxoma: results of a systematic policy of complete cytoreductive surgery and hyperthermic intraperitoneal chemotherapy. Br J Surg 2008; 95: 1164-1171 16. Chua TC, Yan TD, Smigielski ME, Zhu KJ, Ng KM, Zhao J, et al. Long-term survival in patients with pseudomyxoma peritonei treated with cytoreductive surgery and perioperative intraperitoneal chemotherapy: 10 years of experience from a single institution; Ann Surg Oncol 2009; 16: 1903-1911 17. Sugarbaker PH. Peritonectomy procedures. Ann Surg 1995; 221: 29-42 18. Recommandations de la Société Française de Chirurgie Digestive (SFCD) et de l'Association de Chirurgie Hépato-biliaire (ACHBT). Cancérologie digestive : pratiques chirurgicales. J Chir (Paris) 2009 ; 146, supplément 2

Figure 1 :Figure 2 :

 12 Figure 1: Overall and disease-free survival of the 301 patients with PMP treated with CRS + HIPEC

Figure 3 : 4 Figure 3 :

 343 Figure 3: Overall survival according to the peritoneal index (p=0.004)

Acknowledgements: The authors thank Lorna Saint Ange for editing and Nadine Bossard and

Florent Boutitié for statistical analysis.

M A N U S C R I P T A C C E P T E D


Ph M Petropoulos

Spindel

F Bourliot
email: bourliot@cpht.polytechnique.fr

J Estes
email: estes@cpht.polytechnique.fr

Ph Spindel
email: philippe.spindel@umons.ac.

G3-homogeneous gravitational instantons

Introduction

Classical solutions of general relativity in diverse dimensions have been analyzed over the years from various perspectives. Instanton-like configurations have in particular attracted attention because of their potential role in the determination of transition amplitudes in quantum gravity. Similarly, their real-time counterpart turns out to be useful in the Hartle-Hawking formulation of quantum cosmology, even though general relativity would ultimately require a ultra-violet completion, possibly provided by strings. In fact, several classical solutions turn out to be embeddable in string theory, sometimes even as exact backgrounds, and instantons have certainly played a major role in addressing many issues, among which are those related to (super)symmetry breaking.

Following the paradigm of self-dual Yang-Mills instantons [START_REF] Belavin | Pseudoparticle solutions of the Yang-Mills equations[END_REF], self-duality has been successfully implemented in four-dimensional general relativity. In order to be operational, self-duality must be accompanied by some specific ansatz for the geometry M. The usual ansatz is to assume M topologically R × Σ 3 and, further, the leaves Σ 3 to be homogeneous spaces, admitting at least a three-dimensional group of motions G3.

The first solution obtained according to the above pattern is due to Eguchi and Hanson [START_REF] Eguchi | Asymptotically flat self-dual solutions to Euclidean gravity[END_REF][START_REF] Eguchi | Selfdual solutions to Euclidean gravity[END_REF] under the assumption of Bianchi IX geometry i.e. with SU(2) isometry. This gravitational instanton was an alternative to the earlier Taub-NUT metric, constructed from a different perspective though [START_REF] Newman | Empty-space generalization of the Schwarzschild metric[END_REF]. Both were self-dual and SU(2)-homogeneous, with isometry enhancement to SU(2) × U [START_REF] Belavin | Pseudoparticle solutions of the Yang-Mills equations[END_REF]. General extensions of Eguchi-Hanson solution to strict SU(2) were obtained soon after [START_REF] Belinsky | Asymptotically Euclidean Bianchi IX metrics in quantum gravity[END_REF], observing that the equations were a special case of the Euler top -with some peculiar inertia momenta -known as the Lagrange system [START_REF] Gibbons | The positive action conjecture and asymptotically Euclidean metrics In quantum gravity[END_REF]. A similar achievement for the Taub-NUT instanton revealed far more involved, and a particular solution was finally obtained in Ref. [START_REF] Atiyah | Selfduality In four-dimensional Riemannian geometry[END_REF]. The difficulty to solve in full generality the corresponding equations was later understood in terms of non-algebraic integrability properties, as it was realized [START_REF] Takhtajan | A simple example of modular forms as tau functions for integrable equations[END_REF] that the system at hand had already been set by Darboux [9], and solved extensively by Halphen [START_REF] Halphen | Sur un système d'équations différentielles[END_REF][START_REF] Halphen | Sur certains systèmes d'équations différentielles[END_REF], more than a century before, in terms of modular forms 1 .

The above account for the Bianchi IX group raises immediately two questions: do other Bianchi groups possess similar solutions and what is the classification principle behind the appearance of distinguished classes of equations such as La-grange versus Darboux-Halphen systems? Despite the large amount of information accumulated so far (see e.g. [START_REF] Lorenz | Gravitational instanton solutions for Bianchi types I-IX[END_REF][START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF]) and the physical interest of some -even simple -solutions like Kasner's Bianchi I [START_REF] Kasner | Geometrical theorems on Einstein's cosmological equations[END_REF], a precise and definite answer to these questions was still missing. The aim of the present work is to tame the plethora of scattered results under a simple classification principle. Our analysis is performed along the lines announced in [START_REF] Bourliot | Gravitational instantons, self-duality and geometric flows[END_REF]. It is general and exhaustive, and makes no assumptions on the geometry, other than those already quoted above. In particular, the issue of the choice of a diagonal versus non-diagonal metric in a given G3-invariant frame is treated with care, as opposed to some former, more cavalier approaches for non-unimodular Bianchi groups. As a bonus, this enables us to discover a new solution for Bianchi III, completing thereby the existing landscape.

Let us summarize the method and the results. Self-dual vacuum solutions satisfy Ω = Ω where Ω is the Riemann curvature two-form2 . These are second-order equations and are equivalent to the first-order set obtained with the connection one-form: ω = ω + A. The one-form A appears as a "constant of motion" of the second-order system. It stands as the anti-self-dual so(3) part of the Levi-Civita connection and must be flat. The program is thus cast as follows: (i) find all possible flat so(3) connections over G3, and (ii) for each of them, write the corresponding first-order equations, and find the most general solution. The latter can represent a bona fide geometry, but in most cases it is spoiled by naked singularities, if not everywhere degenerate like in all non-unimodular groups, except for Bianchi III.

In Sec. 2 we provide some technical tools, useful to set our philosophy for the subsequent developments. Section 3 contains the core of the classification principle (point (i) above), whereas the exhaustive solution search (point (ii)) is presented in Sec. 4. A last section ( 5) is devoted to the subtle issue of rotating (locally) the frame where ω = ω + A into a frame where the connection is genuinely self-dual. Conclusions follow and a summary of all G3-invariant metrics finally available is presented in the appendix.

Notations and general considerations

Inspired by applications to homogeneous cosmology, we consider Euclidean spaces admitting a G3 group of motion acting transitively3 on 3-dimensional invariant subspaces of the form M = R × Σ 3 . The metric on these spaces can always be locally written as (see for instance Ref. [START_REF] Ph | Gravity before supergravity[END_REF] for a proof):

ds 2 = N (t) 2 dt 2 + δ ab Θ a Θ b . (1)
Let us introduce the co-frame defined by Θ 0 = N (t) dt and Θ a = θ a α (t) σ α , where σ α are invariant 1-forms:

dσ α = 1 2 c α βγ σ β ∧ σ γ . ( 2
)
From the latter we obtain

dΘ a = θa α dt ∧ σ α + 1 2 θ a α c α βγ σ β ∧ σ γ (3) = 1 N θa α θ α b Θ 0 ∧ Θ b + 1 2 θ a α c α βγ θ β b θ γ c Θ b ∧ Θ c (4)
providing the non vanishing structure coefficients:

β a 0b = 1 N θa α θ α b = -β a b0 , (5)
β a bc = θ a α c α βγ θ β b θ γ c = -β a cb , (6)
and connection coefficients

γ 0ab = 1 2 (β ab0 + β ba0 ) = γ 0ba , (7)
γ ab0 = 1 2 (β ab0 -β ba0 ) = -γ ba0 , (8)
γ abc = 1 2 (β abc + β bca -β cab ) = -γ bac . (9)
With respect to the basis {dt, σ α } the metric components are g 00 = N (t) 2 , g 0α = 0 and g αβ = δ ab θ a α θ b β . The Levi-Civita connection one-forms are:

ω 0i = 0iα σ α , ω ij = ij0 dt + ijα σ α .
Finally, we need the expression of the curvature components, which are:

Ω 0i = dω 0i + ω 0k ∧ ω k i = ˙ 0iα dt ∧ σ α + 1 2 0iα c α βγ σ β ∧ σ γ -0kα k i0 dt ∧ σ α + 0kβ k iγ σ β ∧ σ γ , (10)
Ω jk = dω jk + ω j0 ∧ ω 0 k + ω jl ∧ ω l k = ˙ jkα dt ∧ σ α + 1 2 jkα c α βγ σ β ∧ σ γ + j0α 0 kβ σ α ∧ σ β + jl0 l kα dt ∧ σ α -jlα l k0 dt ∧ σ α + jlα l kβ σ α ∧ σ β . (11)
With respect to this basis, the anti-self dual components of the curvature 2form read as:

Ω 0i - 1 2 ijk Ω jk = ˙ 0iα -0kα k i0 - 1 2 jk i ˙ jkα + jl0 l kα -jlα l k0 dt ∧ σ α (12) + 1 2 0iα c α βγ + 0kβ k iγ -0kγ k iβ -jk i 1 2 jkα c α βγ + j0β 0 kγ + jlβ l kγ σ β ∧ σ γ or, introducing Īiα := 0iα - 1 2 jk i jkα , (13)
Ω 0i - 1 2 ijk Ω jk = [ İiα -Īlα l i0 ]dt ∧ σ α + 1 2 [ Īiα c α βγ + jk i Ījβ Īkγ ]σ β ∧ σ γ . ( 14
)
Let us emphasize however that the one-forms θ α k are not completely fixed, but defined up to time-dependent O(3) transformations and time-independent GL(3, R) transformations, we have indeed not yet fixed the basis of invariant forms.

Self-duality equations

Self-duality requires that

Ω 0i - 1 2 ijk Ω jk = 0. ( 15
)
We see that to trivially obtain first integrals from this equation a sufficient (and necessary) condition is to require that

ik0 = 0 ⇔ θiα θ α k -θkα θ α i = 0 ⇔ θiα = -g αβ θβ i , (16)
after which [START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF] becomes equivalent to

İiα = 0 Īiα c α βγ + jk i Ījβ Īkγ = 0 (17)
The gauge freedom allows us to always satisfy this condition. Indeed if we have a solution θ aα , using a time dependent rotation we obtain θaα := O b a (t)θ bα which will fulfill the symmetry condition if

Ȯ b a = 1 2 O c a (θ α c θb α -θcα θ αb ). ( 18
)
This is a kinematic problem aiming to determine a sequence of rotations knowing the angular velocity so that it always admits a solution. Then the only transformations still possible correspond to the choice of the initial conditions of the kinematical problem, denoted from now on by the matrix O b a , and of arbitrary GL(3, R) time independent transformations, denoted from now on by the matrix

Λ β α , θ aα → O b a θ bβ Λ β α . (19)
In the gauge defined by ( 16), the first integrals ( Īiα = const.) furnish the first order equations we need to solve:

1 N θiα -θ -1 θ iγ (n γµ -a ρ ργµ ) g µα - 1 2 δ γ α n µ µ := Īiα (20)
where θ = det(g αβ ) and the symmetric tensor density n γµ = n (γµ) and vector a ρ are defined by the relations

c α βγ βγµ = 2(n αµ + αµβ a β ). ( 21
)
In the following we shall call the t coordinate time and speak about evolution, to describe this flow, though the framework is Euclidean.

Using time re-parameterizations, we may choose a specific N without loss of generality. Equations [START_REF] Atiyah | Low-energy scattering of nonabelian magnetic monopoles[END_REF] strongly suggest to adopt, in a first step at least, the gauge N = θ. 4 Let us notice that in order to preserve this gauge condition when we make a transformation [START_REF] Gibbons | Classification Of Gravitational Instanton Symmetries[END_REF] we have to simultaneously rescale t by a factor 1/ det(Λ β α ); then the time variable may still only be changed by a shift t → t + t 0 . But before trying to integrate equations [START_REF] Atiyah | Low-energy scattering of nonabelian magnetic monopoles[END_REF] it is mandatory to solve the constraint equations:

Īiα

c α βγ + jk i Ījβ Īkγ = 0 . ( 22
)
The solutions of these equations describe homomorphisms from the Lie algebra of the homogeneity group g 3 into the Lie algebra so(3):

Ī : g 3 → so(3). ( 23
)
Depending on the subalgebra of so(3) on which we project g 3 different simplifications can occur. Let us remind that the only subalgebras of so(3) are the trivial ones: so(3) and the null one {0} and the one-dimensional ones : R (all equivalent, i.e. linked by internal conjugation). Thus if the constants of motion Īiα are not identically zero (in which case we have the trivial homomorphism mapping g 3 onto {0}) either g 3 = so(3) or the algebra g 3 has a 2 dimensional ideal.

These remarks lead to the following Bianchi types to be considered, according to the rank of the matrix Īiα :

• rank 3 (maximal) type IX ,
• rank 2 impossible,

• rank 1 types I, II, III, IV, V,VI,VII,

• rank 0 all Bianchi types .

But it remains to examine if all these cases can effectively be obtained i.e. if there are no obstructions coming from [START_REF] Bourliot | Gravitational instantons, self-duality and geometric flows[END_REF]. Let us notice that using [START_REF] Atiyah | Low-energy scattering of nonabelian magnetic monopoles[END_REF], condition [START_REF] Bourliot | Gravitational instantons, self-duality and geometric flows[END_REF] can be written as Īiα θ α j -Ījα

θ α i = 2 a ρ θ ρk kij . (24)
From this equation we see that for rank 0, we must have a ρ = 0, which is the statement that rank 0 solutions must be from Bianchi class A, i.e. types I, II, VI 0 , VII 0 , VIII and IX, see table 1 for our conventions. For the rank 1 cases, as shown in the next section, the only solutions whose metric determinant does not everywhere vanish are those of Bianchi class A and type III. 5 Thus if we also require the metric determinant to not everywhere vanish, the above list is reduced to

• rank 3 (maximal) type IX ,
• rank 2 impossible,

• rank 1 types I, II, III, VI 0 ,VII 0 , 5 To see this, first decompose Īiα as Īiα = λ i I α where I α and λ i are two triples. For all types of Bianchi class B, except type III, the only solution to the constraint ( 22) is to take I α proportional to a ρ . In this case, one may show that (24) requires the determinant of θ α i to vanish.

• rank 0 types I, II, VI 0 , VII 0 , VIII, XI.

Thus, according to the rank of the mapping Ī we obtain the classification of solutions in the next section.

All self-dual solutions

In this section, we obtain all the self-dual solutions categorized by the rank of Ī. Some of them have been discussed in the past in [START_REF] Lorenz | Gravitational instanton solutions for Bianchi types I-IX[END_REF][START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF] 6 . On a similar ground of simplicity, the only rank 3 solution coming from the type IX Bianchi algebra has also been studied in [START_REF] Lorenz | Gravitational instanton solutions for Bianchi types I-IX[END_REF][START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF] and various other places in the past. We finally deal with the new cases, solutions of rank 1. Thanks to residual symmetries and constraints coming from the self-duality requirement, we simplify as most as possible the equations and present the most general solutions as well as some particular simplified solutions when available.

Rank zero

All Īkα are zero. The Bianchi type must be of class A, (a ρ = 0), i.e. types I, II, VI 0 , VII 0 , VIII and IX. Equations [START_REF] Atiyah | Low-energy scattering of nonabelian magnetic monopoles[END_REF] imply that the symmetry condition is always satisfied :

θi α θ α j = θ i γ n γµ - 1 2 n ν ν g γµ θ j µ = θj α θ α i . (25)
At an initial time, one may simultaneously diagonalize g µν and put n µν in its canonical form (see table 1). Thus at initial time the co-frame components θ i γ may be chosen diagonal, and they remain so during their evolution. The explicit integration of the evolution equations has been described a long time ago in Ref. [START_REF] Lorenz | Gravitational instanton solutions for Bianchi types I-IX[END_REF][START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF], here this class of self-dual spaces is presented in appendix A for completeness.

Rank three Of course only Bianchi IX is possible for this case of maximal rank.

In an appropriate frame the matrix n is the identity matrix. The self-duality condition [START_REF] Barrow | Some exact non-vacuum Bianchi VI(0) and VII(0) instantons[END_REF] can be reinterpreted as a relation between three 3-dimensional Euclidean vectors :

I 1 = I 2 × I 3 (26)
and two similar relations obtained by circular permutation of the indices 1, 2, 3. Thus these three vectors are perpendicular to each other and actually define an

A I 0 0 0 0 Translations II 0 1 0 0 Galilean VII 0 0 1 1 0 Euclidean VI 0 0 1 -1 0 Poincaré IX 0 1 1 1 Rotation VIII 0 1 1 -1 Lorentz Class B V 1 0 0 0 IV 1 1 0 0 VII h h > 0 1 1 0 VI h =1 h > 0 1 -1 0 III ≡ VI 1 1 1 -1 0 orthonormal frame of E 3 .
Thus acting with an appropriate element of O(3), on the flat indices i, we may assume that -( Īiα ) is the identity matrix. The integration of (20), under these assumptions, was first discussed in [START_REF] Gibbons | The positive action conjecture and asymptotically Euclidean metrics In quantum gravity[END_REF]9,[START_REF] Halphen | Sur un système d'équations différentielles[END_REF][START_REF] Halphen | Sur certains systèmes d'équations différentielles[END_REF] and the solution is given in appendix A.

Rank one

In that case there exists two triplets I α and λ i such that Īiα = λ i I α . Condition [START_REF] Barrow | Some exact non-vacuum Bianchi VI(0) and VII(0) instantons[END_REF] reduces to

I α c α βγ = 0. ( 27
)
Bianchi class A In this case, condition (27) is equivalent to

I α n αβ = 0, (28)
so the matrix (n αβ ) has to be singular. Consequently the Bianchi type has to be I, II, VI 0 or VII 0 . The symmetry condition (24) becomes then:

λ i ρ j -λ j ρ i = 0 with ρ k = θ α k I α . (29)
It is easy to see that if the initial conditions satisfy this condition, then the equation will automatically be preserved throughout the evolution. This can be seen by computing the evolution of ρ i using [START_REF] Atiyah | Low-energy scattering of nonabelian magnetic monopoles[END_REF] to obtain:

ρi = ( 1 2 n µν g µν -λ k ρ k )ρ i . (30)
Thus we find no restrictions on the existence of solutions to our self-dual problem.

In order to find the rank 1 solutions for Bianchi class A, let us start with the most general expression of the matrix of frame components :

(θ α i ) =   a p q r b s u v c   . (31)
It can be simplified by making use of the transformations [START_REF] Gibbons | Classification Of Gravitational Instanton Symmetries[END_REF]. Using an O(3) transformation we may assume λ = (0, 0, 1) and, without altering the canonical value of the structure constants, using GL(3, R) transformations set

• I α = -(0, 0, 1) for Bianchi I,II • I α = -(0, 0, Z) for Bianchi VI 0 ,VII 0
Moreover, for Bianchi type I, II, VI 0 and VII 0 (Bianchi type of class A), without loss of generality, we may also assume the matrix (31) diagonal. This can be done using the transformations [START_REF] Gibbons | Classification Of Gravitational Instanton Symmetries[END_REF], which now leave I α and λ i invariant as well as the canonical values of the structure constants. The integration of the resulting self-duality equations was given in ref [START_REF] Lorenz | Gravitational instanton solutions for Bianchi types I-IX[END_REF][START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF].

For illustrative purpose let us discuss the Bianchi type VI 0 . The integration proceeds as follows. First we observe that (29) imposes u = v = 0. Equations (30) then insure that if u, v are chosen zero at the initial time, they will remain null. Then we remark that the evolution equations imply that the functions q and s are proportional to c:

q(t) = Q c(t), s(t) = S c(t) ( 3 2 )
and that the transformations [START_REF] Gibbons | Classification Of Gravitational Instanton Symmetries[END_REF] of the frame that preserves the canonical choice of the structure constants, of I α and λ are given by:

O =   ± cos(φ) sin(φ) 0 ∓ sin(φ) cos(φ) 0 0 0 1   and Λ -1 =   α β γ β α δ 0 0 1   . (33)
(2ϕ) = - 2(a 0 r 0 + p 0 b 0 ) (a 2 0 + b 2 0 + p 2 0 + r 2 0 ) (34)
we obtain a new frame such that now a 0 r 0 + b 0 p 0 = 0. Let us notice that (34) makes sense because from (a 0 ± r 0 ) 2 + (b 0 ± p 0 ) 2 > 0 we deduce that its right-hand side member is always between -1 and +1. Then choosing Λ = Id and O with φ such that sin(φ)

p 0 + sin(φ) b 0 = 0 = -sin(φ) a 0 + cos(φ) r 0 ( 35
)
we have obtained a diagonal frame at time t 0 .

It remains then five functions to be determined: a, b, r, p and c. These functions have to satisfy the equations:

ȧ = a ∆ -2κ b 2 κ 2 , ( 36
) ḃ = b ∆ + 2κ a 2 κ 2 , ( 37
) ṗ = p ∆ + 2κ r 2 κ 2 , (38) ṙ
= r ∆ -2κ p 2 κ 2 , (39) ċ
= c ∆ -2 Z κ 2 κ 2 , (40)
It is obvious from these equations that κ is a constant. Moreover, the fact that we may diagonalize the frame at t = t 0 implies that it remains diagonal during all of its evolution: thus r(t) = p(t) = 0.

By considering the combination X = a + i b the main equation to solve is :

Ẋ = -X(X 2 -4 i κ)/(2 κ 2 ); (42)
after which we may finally obtain c thanks to (40). Solving both equations yields:

X = (a + i b) = 2X 0 e i t/κ i κ/(X 2 0 e 2 i t/κ -X2 0 e -2 i t/κ ) 1/2 , ( 43
)
c 2 = c 2 0 e -2 Z t/κ /(X(t) X(t)) . (44)
Moreover the remaining transformation with Λ -1 = diag(ρ, ρ, 1) allows us to fix κ = 1 . After a final translation of the variable t, and if necessary a flip of sign we obtain

a = 1/b = cot(t) , (45)
c = c 0 e Z t sin(2 t) . ( 46
)
Similar expressions can be obtained for Bianchi type VII 0 , but they involve real exponentials of time.

Bianchi class B In this case , condition ( 27) is equivalent to

I β n αβ = ˙αµν I µ a ν ( ˙123 = 1). ( 47
)
For algebras of Bianchi types IV, VI h (with h = 0, 1) and VII h (with h = 0) we found that the only solution to the equations ( 47) is I α = λ a α , but for type III = VI 1 , we obtain a two-parameter solution. This is a reflection of the fact that the derived algebra now is of rank one instead of two.

Equations [START_REF] Saha | Bianchi type I universe with viscous fluid: A qualitative analysis[END_REF] read

λ i ρ j -λ j ρ i = 2 α k kij with α k = a µ θ µ k . ( 48
)
As we require a non singular co-frame, they are only compatible with an algebra of Bianchi type III. By derivation, using (25) we obtain the system of equations:

ρi = ( 1 2 n µν g µν -θ λ k ρ k )ρ i + 2 θ jk i α j ρ k , αi = 1 2 n µν g µν α i -θ λ k α k ρ i . (49)
To check the consistency of ( 48) with [START_REF] Atiyah | Low-energy scattering of nonabelian magnetic monopoles[END_REF], let us rewrite them as relations between vectors of a three dimensional Euclidean space:

2 α = λ × ρ, (50)
˙ ρ = n 2 ρ -θ ( λ • ρ) ρ + 2 θ α × ρ, (51)
˙ α = n 2 α -θ ( λ • α) ρ. ( 52
)
Taking the derivative of (50) and inserting in it (51, 52), we obtain

2 ˙ α -λ × ˙ ρ = n α -2 θ ( λ • α) ρ - n 2 λ × ρ + θ ( λ • ρ) λ × ρ -2 θ λ × ( α × ρ), = n 2 α -θ ( λ • ρ) (2 α -λ × ρ).
Thus if 2 α -λ × ρ = 0 at a period of time, it always vanishes. This insures that the solution obeying this condition constitutes a self-dual solution.

It was shown in [START_REF] Lorenz | Gravitational instanton solutions for Bianchi types I-IX[END_REF][START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF] that there doesn't exist any diagonal self-dual metrics with a Bianchi type III symmetry group. We provide in the following the most general solution of type III, without any a priori (non geometrical) assumption on the co-frame, from which it is easy to see that diagonal metrics don't exist but non diagonal ones do. As previously, assuming a non singular frame, by using rotations and linear transformations we may put λ = (1, 0, 0) and I α = -(1, 1, 0). Then [START_REF] Saha | Bianchi type I universe with viscous fluid: A qualitative analysis[END_REF] implies that the frame must be of the form

(θ α i ) =   a 2 c -b -2 v -s r b s 0 v c   . (53)
This frame can still be transformed, without altering the canonical structure constants and constants of motion, by acting on the right with the GL(3, R) matrix

Λ -1 =   1 -µ µ ν µ 1 -µ -ν 0 0 1   (54)
and on the left with a rotation that leaves λ invariant

O =   1 0 0 0 cos(φ) sin(φ) 0 -sin(φ) cos(φ)   . (55)
The self-duality equations imply that v(t) = V c(t). Thus we may assume that v(t) = 0 after, if necessary, a rotation of angle φ such that tan(φ) = V . Moreover we then see that s(t) = S c(t) and that by choosing an appropriate value of ν we may also assume s(t) = 0. The remaining parameter µ can by fixed by requiring b(0) = 0.

So without loss of generality, the frame we have to consider reduces to:

θ α k =   a 2 c -b 0 r b 0 0 0 c   . (56)
In order to pursue the integration it will be useful to redefine the unknown functions as follows (using the fact that c cannot vanish as we only consider non singular frames):

a(t) = α(t) c(t), b(t) = β(t)c(t), r(t) = ρ(t) c(t). (57)
Then the self-duality equations lead to

α = 2 + α 2 + α ρ (2 ρ -(α + ρ) β) c 2 , (58) β
= ρ -α (2 ρ -(α + ρ) β) c 2 , ( 59
) ρ = 2 + ρ 2 + α ρ (2 ρ -(α + ρ) β) c 2 , ( 60
) ċ = ρ 2 -α 2 + 4 β -4 2 c (2 ρ -(α + ρ) β) 2 . ( 61
)
They imply

d(ρ -α) dβ = (ρ + α) and d(ρ + α) dβ = 4 + (ρ + α) 2 ρ -α . (62)
These equations are easy to solve and lead to

α = sinh β -β * K -K cosh β -β * K , ρ = sinh β -β * K + K cosh β -β * K .
Then we may express, by a quadrature, the function c as a function of β using the equation obtained from the ratio of β and ċ:

c 2 = c 2 0 cosh β-β * K (β -1) sinh β-β * K -K cosh β-β * K (63)
and finally express everything as functions of t by integrating equation ( 59) which leads to the surprisingly simple result (after a flip of the sign of t):

β = K c 2 0 t . ( 64
)
The solution we discussed is generic. It depends on two arbitrary constants, namely c 0 , K and β * , note that we have used a time translation to set β = 0 at time t = 0.

Note that we assumed that a is always different from r. If for a value of t we set a = r, then they become identical for all times. In this special case, the derivative of β vanishes and we obtain that b(t) = B c(t) implying the remaining equations:

ċ c = R a 2 , ȧ a = -2 R c 2 - R a 2 . ( 65
) with R = 1 2(1-B) .
To solve the equations (65) we use an auxiliary function y(t) = log |a(t) c(t)| and obtain, after elementary operations, the first integral:

ẏ2 + 4 R 2 e -2 y = L 2 , ( 66
)
from which we deduce the expression of the product a(t) c(t) and

c(t) 2 = 2 R L coth[L t], (67)
a(t) 2 = R L sinh[2 L t]. (68)
Let us notice that R t has to be positive for the solution to be real. When t = 0 we encounter a curvature singularity.

Self-dual Connections

In the conventions we have used, we naturally find solutions whose curvature is self-dual while the connection is not. However, as we will show below, we may always make a local SO(4) rotation to make the connection self-dual. In fact, the vanishing of the anti self-dual part of the connection can be related to the integrability condition, thereby ensuring the existence of a pure self dual-connection. We shall illustrate this point by evaluating, in general, the gauge transformations which map our canonical co-frame to one whose associated connection is self-dual.

As is well known, the two (families of) parallelisms on the three-sphere S 3 provide a way to decompose any SO(4) rotation into the product of self-dual and anti self-dual SO(3) rotations. If u, x, y and z are the coordinates of a point of S 3 : u 2 + x 2 + y 2 + z 2 = 1, an anti self-dual SO(3) generator can be written as:

O ad :=     u -x -y -z x u z -y y -z u x z y -x u     . ( 69
)
The gauge transformation generated by such a rotation leaves invariant the selfdual part of the connection but transforms the anti self-dual one. Accordingly, when the anti self-dual part of the connection, given by the constants Īiα , is nonzero, such a gauge transformation may set it to zero thanks to a solution of the equation:

dO = O ω a . ( 70
)
More explicitly we obtain (with Īk := Īkα σ α and x i := (x, y, z))

du = 1 2 x k Īk , dx i = 1 2 -u Īi + ijk x j Īk . ( 71
)
As a first immediate consequence we see that this gauge transformation has to be t independent. Moreover we also obtain from Eq. ( 22):

d Īi = 1 2 Īiα c α βγ σ β ∧ σ γ = - 1 2 jk i Īj ∧ Īk , (72)
which indeed ensures the integrability of the equations (71). In the case of a rank one mapping, Īkα = λ k I α (normalized such that k λ 2 k = 1), this equation implies there exists a function χ such that dχ = I α σ α . Correspondingly, the gauge transformation (69) that leads to a frame which has vanishing anti self-dual connection is given by

u = sin(χ/2), x k = λ k cos(χ/2). (73)
In the case of rank three, the only possibility is Bianchi type IX. As discussed above it can be taken diagonal, with the matrix ( Īiα ) = -δ iα . Moreover, it is well known that the components of the invariant one-forms also define parallel vector fields. Using the identities u du + x dx + y dy + z dz = 0 and u 2 + x 2 + y 2 + z 2 = 1, it is easy to check that the invariant one-forms may be expressed as :

σ 1 = -2(x du -u dx + z dy -y dz), (74)
σ 2 = -2(y du -z dx -u dy + x dz), (75)
σ 3 = -2(z du + y dx -x dy -u dz). (76)
One may also check that they verify the invariance relations: dσ i = i j k σ j ∧ σ k and that (71) are identities. In the case where the matrix ( Īiα ) is not diagonal, it must be (in order to satisfy to equations ( 26)) of the form ( Īiα ) = -(O iα ) with O an SO(3) rotation matrix. It is elementary to check that the equations (71) are solved by similar functions involving variables u and (x k ) = (x , y , z ) related to (x k ) = (x, y, z) by the rotation O:

x k = -I kl x l (77)
as it must be.

Conclusion

In this note we have achieved a complete classification of all the self-dual Euclidean spaces admitting a G3 simply transitive homogeneity group. For each Bianchi group, there are as many classes of solutions as homomorphisms Ī : g 3 → so(3).

For each of these homomorphisms, labeled by its rank, the self-duality constraint is described by a specific set of equations -like Lagrange or Darboux-Halphen in Bianchi IX, leading to distinct solutions such as Eguchi-Hanson or Taub-NUT respectively. These equations have non-everywhere vanishing solutions for all class A Bianchi groups (I, II, VI 0 , VII 0 , VIII and IX), whereas in class B solutions exist only for Bianchi III. The latter solutions (Eqs. ( 81), ( 83)) are all rank-1 and non-diagonal, contrary to the lore that it seamed "very unlikely to construct nondiagonal self-dual solutions of Bianchi types I-IX" [START_REF] Lorenz | Gravitational instanton solutions for Bianchi types I-IX[END_REF][START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF]. This expectation was empty for Bianchi A because in the present formulation, the diagonal ansatz is the most general, but not for Bianchi B.

It is fair to stress that the above conclusions have been reached by adopting a metric representation, Eq. ( 1), such that the foliation of M as R × Σ 3 is manifest and adapted to the splitting of the group SO(4) into self-dual and anti-self-dual factors. Other choices for the metric may exist, where the distinction between the various classes of self-dual solutions is less sharp. This happens e.g. for Bianchi IX, in the Gibbons-Hawking representation [START_REF] Gibbons | Classification Of Gravitational Instanton Symmetries[END_REF], where Eguchi-Hanson (rank-0) and Taub-NUT (rank-1) are indeed, to some extent, unified. This is possible, however, at the price of abandoning the G3-invariant frames that we use. Here, this choice has been instrumental, not only for providing the classification in terms of the rank of the g 3 → so(3) homomorphism, but also for scanning all possible solutions.

Our analysis can easily be adapted to the ultra-hyperbolic case (spacetime signature (-, -, +, +)), where the algebra of Bianchi type VIII will play a role analogous to the one of Bianchi IX here. It should however be emphasized that we will have to distinguish, in that case, among the inequivalent one-dimensional subalgebras of so(2, 1). Other extensions concern the addition of a cosmological constant (Weyl self-duality), or the implementation of the method to higher-dimensional set-ups admitting self-duality, as e.g. in seven dimensions. Last but not least, the physical analysis of the new Bianchi III solutions remains to be completed. The large number of moduli in (81), (83) make such an analysis quite involved.

It would be interesting to discover if the techniques and classification presented in this paper can be generalized to cases where we have a more general setup than vacuum Einstein equations. For example, the evolution of certain Bianchi spaces in the presence of a cosmological constant, as well as various matter such as a perfect fluid and/or scalar, spinor, and electro-magnetic fields has been studied recently in [START_REF] Barrow | Some exact non-vacuum Bianchi VI(0) and VII(0) instantons[END_REF][START_REF] Saha | Interacting spinor and scalar fields in Bianchi cosmology[END_REF][START_REF] Saha | Bianchi type I universe with viscous fluid: A qualitative analysis[END_REF]. The authors have derived exact solutions in certain cases, some of which are singularity free, and a complete classification of solutions would be appreciated. In the case of Einstein's equations with a cosmological constant, it is well known that requiring self-duality of the Weyl tensor is sufficient to satisfy the equations of motion. Preliminary investigations suggest that our techniques in this case are applicable. However, the algebraic constraint equation ( 22) is modified and in particular becomes time dependent. Following the logic in [START_REF] Bourliot | Gravitational instantons, self-duality and geometric flows[END_REF], the algebraic constraint, in the case without a cosmological constant, can be interpreted as the constraint for a constant background G3-gauge field to be pure gauge. With the introduction of a cosmological constant, the background gauge field must now evolve in time.

We note that most of the Bianchi classes have solutions which necessarily have singularities. It would be interesting to discover if these singularities could be cured by the introduction of either a cosmological constant and/or additional matter fields. Examples have already been discussed in [START_REF] Barrow | Some exact non-vacuum Bianchi VI(0) and VII(0) instantons[END_REF][START_REF] Saha | Interacting spinor and scalar fields in Bianchi cosmology[END_REF][START_REF] Saha | Bianchi type I universe with viscous fluid: A qualitative analysis[END_REF] as well as [START_REF] Spokoiny | The effect of the electromagnetic fields on the evolution of homogeneous cosmological models near the singularity[END_REF][START_REF] Fay | Sufficient conditions for curvature invariants to avoid divergences in hyperextended scalar tensor theory for Bianchi models[END_REF]. most peculiar: contrary to the other classes, it requires a non-diagonal co-frame (31) (and consequently a non-diagonal metric (1)) and allows exclusively rank-1 solutions. These solutions are new.

In the following presentation, integration constants that can be reabsorbed by coordinate redefinitions have been discarded. Consequently, all remaining parameters are genuine moduli of the solutions.

Bianchi I The solution reads

ds 2 = e -2 t dt 2 + e -2 t (σ 1 ) 2 + (σ 2 ) 2 + (σ 3 ) 2 (78)
and describes flat geometry for both rank-solutions ( = 0, 1). Notice, however, that the rank-0 case appears naturally in Cartesian coordinates, whereas rank-1 emerges in a kind of mixed Cartesian/polar, Euclidean-Rindler-like coordinates.

Bianchi II The rank-solutions are given by

ds 2 = t e 2 t/b 0 b 2 0 c 2 0 dt 2 + 1 t (σ 1 ) 2 + t b 2 0 (σ 2 ) 2 + t e 2 t/b 0 c 2 0 (σ 3 ) 2 . ( 79
)
The Kretschmann scalar is given by

K = R MNP Q R MNP Q = 8 b 2 0 c 4 0 e -4 t /b 0 ( t) 2 + 3 b 0 t + 3 b 2 0 t 6 (80)
from which we can see that at t = 0 the metric has a curvature singularity.

Bianchi III The Bianchi-III self-dual metrics are all rank-one. They are captured by two expressions, depending on whether a = r (Eq. ( 81)), or a = r (Eq. ( 83)), in the general co-frame (31) after a convenient rescaling of t by t → c 2 0 t we obtain:

ds 2 = F (t) 32 c 4 0 cosh 4 t -t * dt 2 + K t -1 tanh t -t * -K c 2 0 (σ 3 ) 2 + g 1 (t) F (t) (σ 1 ) 2 + g 2 (t) F (t) (σ 2 ) 2 + 2 g 3 (t) F (t) σ 1 σ 2 , (81)
where we have introduced the functions F (t), g 1 (t), g 2 (t) and g 3 (t) given by

F (t) = 8 c 2 0 cosh t -t * K t -1 sinh t -t * -K cosh t -t * , g 1 (t) = (K 2 + 1) cosh 2(t -t * ) + 2 K sinh 2(t -t * ) + 2K 2 t 2 + K 2 -1, g 2 (t) = (K 2 + 1) cosh 2(t -t * ) -2 K sinh 2(t -t * ) +2K 2 t 2 -8 K t + K 2 + 7, g 3 (t) = (K 2 -1) cosh 2(t -t * ) + 2 K 2 t 2 -4 K t + K 2 + 1. ( 82
)
Denoting the zeroes of F (t) as t i , we find that the Kretschmann scalar diverges as (t -t i ) 6 as t → t i , indicating a curvature singularity at each of the two zeros of F (t). The second solution with a = r is given by

ds 2 = L (B -1) 4 tanh(t) cosh 2 (t) dt 2 + L (B -1) tanh(t) σ 3 2 + L 4 (B -1) 2 B 2 csch(2 t) + tanh(t) σ 1 2 + -2 [tanh(t) -2 B (B -2)csch(2 t)] σ 1 σ 2 csch(2 t)(cosh(2 t) + 2 (B -2) 2 -1 σ 2 2 . (83)
In this case, the Kretschmann scalar is given by

K = 384 coth 6 (t) (B -1) 2 L 2 (84)
and we find at t = 0 the metric has a curvature singularity. The general analysis of these geometries is involved and provides interesting features, which deserve a separate study.

Bianchi VI 0 The general metric for rank-solutions reads:

ds 2 = sin(2 t) e 2 Z t c 2 0 dt 2 + σ 3 2 + tan(t) σ 1 2 + cot(t) σ 2 2 , ( 85
)
where t ∈ [0, π/2]. A curvature singularity appears at the boundaries t = 0 and t = π/2. The Kretschmann scalar is given by

K = 16 c 4 0 e -4 t Z 6 Z sin(4 t) + [9 + ( Z) 2 ] cos(4 t) + ( Z) 2 + 15 sin 6 (2 t) , (86)
and we see that at t = 0, π/2 the metric has a curvature singularity.

Bianchi VII 0 The general metric for rank-solutions reads:

ds 2 = sinh(2 t) e 2 Z t c 2 0 dt 2 + σ 3 2 + tanh(t) σ 1 2 + coth(t) σ 2 2 (87)
The Kretschmann scalar is given by

K = 16 c 4 0 e -4 t Z 6 Z sinh(4 t) + [9 + ( Z) 2 ] cosh(4 t) -( Z) 2 + 15 sinh 6 (2 t) (88)
and we again find a curvature singularity at t = 0. In the rank-1 case, we may take the large-t limit of the above metric to obtain

ds 2 = c 2 0 e 2(1+Z)t dt 2 + σ 3 2 + σ 1 2 + σ 2 2 (89)
which also solves the self-duality equations but has a vanishing Kretschmann scalar and is actually flat.

Bianchi VIII There is only the rank-0 solution whose simplest writing is:

ds 2 = P -1/2 4 dx 2 + P 1/2 (σ 1 ) 2 x 1 -x + (σ 2 ) 2 x 2 -x + (σ 3 ) 2 x ( 90
) with P = (x 1 -x)(x 2 -x)x where x 1 , x 2 are positive and 0 ≤ x ≤ min{x 1 , x 2 }.
The Krestschmann scalar blows up as x -3 (x -x 1 ) -3 (x -x 2 ) -3 , indicating that the singularity at the boundary of the domain of definition of x is a true one. Let us assume x 1 = α 2 > x 2 = β 2 . We may re-express the metric in the gauge N = θ, using Jacobi elliptic functions of module β/α. Making the substitution x = β 2 sn 2 (α t) we obtain: The t variable belongs to the interval [0, 2 K(β/α)/α], K(k) being the complete elliptic integral of the first kind of module k.

ds 2 =
Bianchi IX We have solutions in case of rank-0 (Lagrange system of equations) and of rank-3 (Darboux-Halphen system of equations). The general rank-0 solution8 was found by Belinsky et al in [START_REF] Belinsky | Asymptotically Euclidean Bianchi IX metrics in quantum gravity[END_REF] as a strict SU(2)-symmetric generalization of the Eguchi-Hanson gravitational instanton.

The latter has enhanced SU(2) × U (1) isometry and is a solution of the Lagrange system. An algebraic expression of the solution, analogous in form to Eq. (90), is

ds 2 = P -1/2 4 dx 2 + P 1/2 (σ 1 ) 2 x -x 1 + (σ 2 ) 2 x -x 2 + (σ 3 ) 2 x ( 92
)
with P = (x -x 1 )(x -x 2 )x where x 1 , x 2 are negative. The general strict SU(2)symmetric solutions have curvature singularities. To obtain the Eguchi-Hanson gravitational instanton, one may take x 1 = x 2 = x 0 and by a translation in x set x 3 = 0. Let us assume x 1 = -α 2 < x 2 = -β 2 . In the gauge N = θ, the metric can still be expressed in terms of standard Jacobi elliptic functions but now of module α 2 -β 2 /α; using x = β 2 sn 2 (α t)/cn 2 (α t) with t ∈ [0, K( α 2 -β 2 /α)/α] we obtain: The issue of the rank-3 solution is more subtle. It is a generalization of Taub-NUT metric, with strict SU(2) isometry, and solves the Darboux-Halphen system. As explained in [START_REF] Takhtajan | A simple example of modular forms as tau functions for integrable equations[END_REF], it requires the use of modular forms. For simplicity we trade a, b, c for Ω α , α = 1, 2, 3, where Ω 1 = 1 b c , Ω 2 = 1 a c , and Ω 3 = 1 a b , and introduce a triplet of weight-two modular forms of Γ(2) ⊂ P SL(2, Z):

ds 2 =
E 1 = dλ/dz λ , E 2 = dλ/dz λ -1 , E 3 = dλ/dz λ(λ -1) , ( 94
)
where λ is solution of Schwartz' equation

and generates a Bianchi-IX, rank-1 gravitational instanton (Eqs. ( 1) and (31)). A generic curvature singularity is present in these geometries. This singularity can be pushed at infinity for one specific choice of λ(z): λ = θ 4 2 /θ 4 3 , where θ 2,3 are the standard Jacobi theta functions 9 . This case corresponds to the Atiyah-Hitchin solution [START_REF] Atiyah | Low-energy scattering of nonabelian magnetic monopoles[END_REF][START_REF] Atiyah | Low-energy scattering of nonabelian monopoles[END_REF]. Similar to the rank-0 solution, in the case that two of the metric factors are equal the solutions has enhanced SU(2) × U (1) isometry and takes the form of the well-known Taub-NUT metric ds 2 = r 2 4(1 + k r 2 ) 2 (σ 1 ) 2 + (σ 2 ) 2 + r 2 4 (σ 3 ) 2 + 1 (1 + k r 2 ) 4 dr 2 .

(97)

Transformations ( 33 )

 33 allow to simplify the frame components, without loss of generality. First we may put Q = S = 0 i.e. q(t) = s(t) = 0 by performing a transformation with O = Id and α = 1, β = 0, γ = -Q and δ = -S. Then we still have the freedom to make transformations (33) with γ = δ = 0. Performing, at an arbitrary time t 0 , a new transformation with still O = Id and α = ρ cos(ϕ), β = ρ sin(ϕ) where ρ > 0 is arbitrary and ϕ determined by7 sin

with ∆ = (b 2 + r 2 )

 22 -(a 2 + p 2 ), and κ = a b -p r.

2 .

 2 (95) Any real solution of the Darboux-Halphen system reads:Ω α (t) = -1 2 d dt log E α (it), α = 1, 2, 3,

Table 1 :

 1 Canonical structure constants for the different Bianchi groups Type a n 1 n 2 n 3 Usual name Class

  α β 2 sn(α t) dn(α t) cn 3 (α t) dt 2 (93) + β 2 sn(α t) α cn(α t) dn(α t) (σ 1 ) 2 + α sn(α t) dn(α t) cn(α t) (σ 2 ) 2 + α dn(α t) sn(α t) cn(α t) (σ 3 ) 2 .

A translation of the general Halphen-Darboux solutions in terms of gravitational instantons can be found in[12]. They are all plagued with naked singularities, except, marginally, for the particular solution of Atiyah-Hitchin[START_REF] Atiyah | Selfduality In four-dimensional Riemannian geometry[END_REF].

Self-duality can be imposed alternatively on the Weyl tensor, and leads thus to solutions with cosmological constant like Fubini-Study or Pedersen[START_REF] Pedersen | Einstein metrics, spinning top motions and monopoles[END_REF]. Note also that anti-self-solutions are obtained by parity or time reversal.

The three-dimensional group G3 acts simply transitively on the leaves, endowed thus with the structure of a group manifold. Hence we exclude H 3 , H 2 × S 1 or S 2 × S 1 , which are the alternatives to the nine Bianchi classes.

To see this is a valid choice, note that under a time re-parameterization t → t = f (t) we could pick f = N/θ. Assuming the metric is invertible, θ = 0, this transformation is well defined.

However in Ref.[START_REF] Lorenz | Gravitational instanton solutions for Bianchi types I-IX[END_REF][START_REF] Lorenz-Petzold | Gravitational instanton solutions[END_REF] the metrics were assumed to be a priori diagonal, a technical assumption just introduced to facilitate the integration of the flow equations. Here we justify that they furnish the most general ones in case of rank 0 cases, and when necessary shall discuss the integration of the relevant non-diagonal metrics.

We use the subscript 0 to indicate the value of the corresponding function at the instant considered : a(t 0 ) = a 0 , etc.

The metric can be chosen diagonal; the off-diagonal entries are consistently set to zero in this class.

Acknowledgements

The authors thank P. Bieliavsky for stimulating discussions and the referees for having indicated us the paper [22]. François Bourliot, J. Estes and M. Petropoulos would like to thank the Service de Physique Théorique de l'Université Libre de Bruxelles as well as the Service de Mécanique et Gravitation de l'Université de Mons-Hainaut for kind hospitality. Philippe Spindel thanks the IHES where the present collaboration was initiated and the CPHT of Ecole Polytechnique, and acknowledges financial support from IISN-Belgium (convention 4.4511.06). John Estes acknowledges financial support from the Groupement d'Intérêt Scientifique P2I. This research was partially supported by the French Agence Nationale pour la Recherche, contract 09-BLAN-NT09-573739 and by the CNRS-FNRS-CGRI-2009.

Appendix: G3 self-dual metrics

We summarize here all known (real) self-dual G3-homogeneous gravitational instantons, following the classification pattern we have developed in terms of homomorphisms g 3 → so(3). These can be of rank 0, 1 or 3. Whenever rank-0 and rank-1 solutions coexist (i.e. for Bianchi I, II, VI 0 and VII 0 ), they are both captured by a single expression with a two-valued parameter = 0, 1. Bianchi IX is the only case which possess rank-0 and rank-3 solutions. Bianchi III is the