Thomas Gasser
email: thomas.gasser@med.uni-tuebingen.de

Mendelian Forms of Parkinson's Disease

Over the last few years, genetic findings have changed our views on Parkinson's disease (PD), as mutations in a growing number of genes are found to cause monogenic forms of the disorder. Point mutations in the gene for -synuclein, as well as duplications and triplications of the wild-type gene cause a dominant form of PD in rare families, pointing towards mishandling of this protein as a crucial step in the molecular pathogenesis of the disorder.

Mutations in the gene for leucine-rich repeat kinase 2 (LRRK2) have recently been identified as a much more common cause for dominant PD, while mutations in the parkin gene, in DJ-1, PINK1 and ATP13A2 all cause autosomal recessive parkinsonism of early onset.

Mutations in recessive genes probably are pathogenic through loss-of-function mechanisms, suggesting that their wild-type products protect dopaminergic cells against a variety of insults.

Evidence is emerging that at least some of these genes may play a direct role in the etiology of the common sporadic form of PD. Further, it is likely that the cellular pathways identified in rare monogenic variants of the disease also shed light on the molecular pathogenesis in typical sporadic PD.

Introduction

Parkinson´s disease (PD) is one of the most common neurodegenerative disorders, second in prevalence only to Alzheimer's disease, and affects about 1 to 2 % of individuals over the age of 60. It is estimated that in the EU alone there are more than 500.000 affecteds with

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PD, and the number is steadily increasing due to demographic changes in Western countries.

It is assumed that PD is caused, in most cases, by an interplay of genetic susceptibility factors and the "environment", acting on the background of an ageing brain. In fact, ageing is probably the single most important "environmental" risk factor, that eventually tips the balance towards neurodegeneration and PD in a genetically susceptible individual.

For over one and a half centuries after its initial description by James Parkinson in 1817, the cause of the disease remained completely obscure. It is only in the past 25 years that significant advances have been made in the understanding of PD pathogenesis and most of them were made possible by the identification of genes for monogenic forms of the disease, i.e. those with a clear "Mendelian", autosomal-dominant or autosomal-recessive, pattern of inheritance.

The role of those genes and loci causing Mendelian forms of PD and of their importance for the PD population as a whole will be the subject of the present review.

Locus

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Although the notion that the etiology of PD has a significant genetic contribution had not entered the mainstream of medical thinking until recent years, there have been occasional reports on genetic clusters or even PD families already in the first half of the twentieth century. One of the earliest systematic genetic studies of PD was carried out by Henry

Mjönes in Sweden and published in 1949. His results, which depended heavily on a single extended family, which is now known to carry an -synuclein multiplication [START_REF] Fuchs | Phenotypic variation in a large Swedish pedigree due to SNCA duplication and triplication[END_REF], indicated autosomal dominant transmission with 60% penetrance [START_REF] Mjönes | Paralysis agitans. A clinical genetic study[END_REF]. These conclusions, however, were long discounted because oligosymptomatic relatives and those with an atypical clinical presentation were counted as secondary cases without clear justification. Subsequent surveys of patient and twin studies largely failed to confirm evidence of familial aggregation and the hypothesis of a genetic etiology was over-shadowed by interest in possible environmental neurotoxins. This led Roger Duvoisin to conclude, in 1987, that "the best available data do not support a role of heredity in the etiology of PD" [START_REF] Duvoisin | Genetics of Parkinson's disease[END_REF].

In the following years, however, a growing number of pedigrees with clinically more or less typical, and in some cases histologically confirmed Lewy-body PD was reported, particularly one extended family originating from Southern Italy, the "Contursi kindred" [16; 17], and three families described by Wszolek et al., [START_REF] Wszolek | Hereditary Parkinson disease: report of 3 families with dominant autosomal inheritance[END_REF] which appeared to fall completely within the spectrum of what had to be considered clinically and pathologically "typical" PD. In addition, Fluorodopa positron emission tomography (PET) studies in oligo-or asymptomatic co-twins of probands with PD showed much higher concordance rates than previously thought [START_REF] Burn | Parkinson's disease in twins studied with 18F-dopa and positron emission tomography[END_REF], so in 1992, Duvoisin concluded that "these findings favour monogenic autosomal dominant inheritance and show reason to argue against a multifactorial etiology or heteroplasmy" [START_REF] Duvoisin | Hereditary Lewy-body parkinsonism and evidence for a genetic etiology of Parkinson's disease[END_REF].

Although we know today, that this view also turned out to be somewhat simplistic, it was still of remarkable foresight. Since then, genes and mutations for a growing number of Mendelian forms of PD have been identified in rapid succession.

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Autosomal-dominant forms of Mendelian PD

The PARK1 locus (SNCA, -synuclein)

The first "PD gene" was mapped in a large family of clearly dominant inheritance, which was already mentioned above, the "Contursi-kindred", named after the village of origin of at least one branch of this pedigree, in Southern Italy. By linkage mapping, the gene was located to the long arm of chromosome 4 [START_REF] Polymeropoulos | Mapping of a gene for Parkinson's disease to chromosome 4q21-q23[END_REF] and a missense mutation (Ala53Thr) in the gene for synuclein (SNCA) was found to segregate with the disease in all but one affecteds (this one affected being a "phenocopy", i.e. an individual with PD from an apparently different cause) [START_REF] Polymeropoulos | Mutation in the -synuclein gene identified in families with Parkinson's disease[END_REF]. Interestingly, already in this first report, the mutation was also identified in two smaller families from the Western region of Greece. Haplotype data suggested that the mutations in all families originated from a common founder. The mutation was later found in several other families throughout the world, which all could be traced back to Mediterranean ancestry and had the same set of DNA-marker alleles (haplotype) on their mutation-carrying chromosomes, indicating that they all reflect a single historical mutational event and originate from a common founder [22; 23]. Only very recently, a second, independent A53T mutation was identified in a Korean Parkinson patient [START_REF] Ki | The Ala53Thr mutation in the alpha-synuclein gene in a Korean family with Parkinson disease[END_REF].

Only two additional different point mutations in the SNCA gene have subsequently been identified in families with dominantly inherited PD, A30P in a German kindred [START_REF] Krüger | Ala30Pro mutation in the gene encoding a-synuclein in Parkinson's disease[END_REF], and E46K in a family from Spain [START_REF] Zarranz | The new mutation, E46K, of alpha-synuclein causes Parkinson and Lewy body dementia[END_REF]. SNCA point mutations have so far only been found in large, multigenerational families, never in sporadic PD [START_REF] Berg | Alpha-synuclein and Parkinson's disease: implications from the screening of more than 1,900 patients[END_REF].

The phenotype of patients with SNCA A53T and E46K point mutations is that of L-doparesponsive parkinsonism with a relatively early age of onset, rapid progression and high prevalence of dementia, psychiatric and autonomic disturbances, although for all of these features the range was relatively wide, and the families included individuals indistinguishable from typical late-onset sporadic PD as well as those who would fulfil present criteria for dementia with Lewy-bodies. In a formal comparison of patients with familial PD from Greece with and without A53T mutation, only the earlier onset age (mean difference ~ 12 years) and a lower rate of tremor at onset were specific for A53T carriers [START_REF] Papapetropoulos | Clinical characteristics of the alpha-synuclein mutation (G209A)-associated Parkinson's disease in comparison with other forms of familial Parkinson's disease in Greece[END_REF]. The A30P mutation, on the other hand, seems to lead to a later age of onset and a milder phenotype, although this analysis is based on a very small number of patients [START_REF] Krüger | Ala30Pro mutation in the gene encoding a-synuclein in Parkinson's disease[END_REF].

A number of family members from these kindreds have come to autopsy and they invariably showed cell loss of dopaminergic neurons of the substantia nigra, and severe and widespread Lewy-pathology, particularly in the form of Lewy-neurites, not only in the substantia nigra and other brain-stem nuclei, but also in the limbic system, the hippocampus and throughout the neocortex. Interestingly, many inclusions were also stained with Tauantibodies [START_REF] Duda | Concurrence of alpha-synuclein and tau brain pathology in the Contursi kindred[END_REF], while classical Lewy-bodies were relatively rare.

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Therefore, the clinical and pathological picture of dominant PD with SNCA mutations clearly overlaps, but is not identical with typical sporadic late-onset PD.

A direct link between -synuclein and PD was strongly supported by the recent discovery that not only point-mutations, but also multiplications of the wild-type sequence of the synuclein gene (duplications and triplications) cause parkinsonism with or without dementia with -synuclein inclusions [4; 30] in some families. This type of mutation, also called "genomic rearrangement", has recently been found to be numerically and mechanistically important in many diseases, and has escaped detection by classic gene sequencing methods. It's role in PD has first been described in a family known as the "Spellman-Muenter kindred". Linkage mapping in this family had originally placed the gene on the short arm of chromosome 4, (4p15), hence the locus has been named PARK4. When subsequently it was found that the causative mutation was actually the triplication of a genomic region on 4q23 containing the SNCA-gene, the locus designation was nevertheless upheld as a PARK-locus involving genomic rearrangements affecting the SNCA-gene.

This finding was of major mechanistic importance because it indicates that a mere increase in SNCA gene dosage, resulting in a doubling of -synuclein protein levels, which can also be measured in the blood and in the brain in cases with triplications [START_REF] Miller | Alpha-synuclein in blood and brain from familial Parkinson disease with SNCA locus triplication[END_REF], appear to be toxic to neurons. This link between -synuclein expression levels and disease was further supported by the observation that patients with an SNCA-triplication generally have an early age of onset (mean 35 in the original triplication family) and a high prevalence of dementia, while patients described subsequently with SNCA-duplications, corresponding to a 1.5-fold increase in -synuclein protein load, have a more typical late-onset PD-phenotype [START_REF] Chartier-Harlin | Alpha-synuclein locus duplication as a cause of familial Parkinson's disease[END_REF]. This was confirmed in a single noteworthy family, the kindred described by Henry Mjönes in 1949 [START_REF] Mjönes | Paralysis agitans. A clinical genetic study[END_REF], in whom one branch was found to segregate an SNCA-duplication, corresponding to a late-onset phenotype, while in another branch the identical fragment of about 600 kb, carrying only the SNCA gene (and one other gene, MMRN1, which is not expressed in the nervous system), was found triplicated in a patient with an early disease onset of 35 years.

This "dose-dependency" of the pathogenic effect clearly supports the direct role of SNCAexpression levels as one possible mechanism underlying Parkinson´s disease. Due to the late onset, it is not surprising that SNCA-duplications can occasionally be found also in sporadic patients [START_REF] Ahn | alpha-Synuclein gene duplication is present in sporadic Parkinson disease[END_REF]. Although on a population level, dosage changes of SNCA are probably more frequent than point mutations, again overall they are a rare cause of dominant parkinsonism and are very rare in the sporadic disease [34; 35].

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In accordance with the early onset and severe rapidly progressive phenotype with the early development of dementia in the kindred described by Mjönes, neuropathology showed severe changes not only in the substantia nigra, but also in many other regions of the brain.

There was striking cortical pathology, with regions of spongiosis and gliosis accompanied by scattered glial cells with -synuclein-immunoreactive inclusions somewhat similar to glial cytoplasmic inclusions of multiple system atrophy. -Synuclein-immunoreactive round to pleomorphic inclusions were also found within neurons, mostly in lower cortical layers, consistent with what has been described as cortical Lewy-bodies in dementia with Lewybodies (DLB). This finding supports the assumption that Lewy-body diseases, including PD and DLB can be seen as part of a spectrum reflecting the same pathogenic process in variable levels of severity.

As SNCA-multiplications clearly indicate that an increase in expression of the wild-type gene can be pathogenic, there has been considerable interest in studies looking at polymorphisms in SNCA, which might influence expression levels, as a potential risk factor in the sporadic disease. Early studies looking at a complex polymorphic dinucleotide repeat polymorphism (NACP-Rep1) located about 4 kb upstream of the transcriptional start site of SNCA have initially produced controversial results, but a meta-analysis summarizing data of over 2500 individuals appeared to confirm initial findings of an association with sporadic PD [START_REF] Maraganore | Collaborative analysis of alpha-synuclein gene promoter variability and Parkinson disease[END_REF].

In a more general approach, Müller et al. first defined the haplotype structure of the entire SNCA-gene by analyzing more than 50 SNPs across the gene. They found a strong association of a haplotype comprising exons 5 and 6 and the 3'-UTR of the SNCA gene with PD, conferring a relative risk of about 1.4 in heterozygous carriers of the risk haplotype and about 2 in homozygotes [START_REF] Mueller | Multiple regions of alpha-synuclein are associated with Parkinson's disease[END_REF]. A weaker signal was also found in the promotor region, confirming earlier results [38; 39]. It is therefore likely that more than one region of the SNCA gene is involved in the regulation of -synuclein expression and consequently modulates PD risk. The association with the 3'-haplotype block with a similar relative risk has since been replicated in numerous populations from Japan [START_REF] Mizuta | Multiple candidate gene analysis identifies {alpha}-synuclein as a susceptibility gene for sporadic Parkinson's disease[END_REF], the UK [START_REF] Goris | Tau and alpha-synuclein in susceptibility to, and dementia in, Parkinson's disease[END_REF] and Germany [START_REF] Winkler | alpha-Synuclein and Parkinson disease susceptibility[END_REF].

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The mechanism, by which these variants influence the risk for PD is still unclear. The NACP-Rep1-polymorphism has been best studied in this regard. Specific binding of a nuclear protein, Poly (ADP ribose) polymerase-1 (PARP-1) to the NACP-Rep1-sequence was recently discovered [START_REF] Chiba-Falek | Regulation of alpha-synuclein expression by poly (ADP ribose) polymerase-1 (PARP-1) binding to the NACP-Rep1 polymorphic site upstream of the SNCA gene[END_REF] and allele-specific binding of this protein may be responsible, at least in part, for altered expression levels that have been found in white blood cells in vivo [START_REF] Fuchs | Genetic variability in the SNCA gene influences alpha-synuclein levels in the blood and brain[END_REF]. In addition to differential binding of transcription factors to the promotor or of enhancers or suppressors to intronic regions, it is also possible that genetic variability in the 3'-UTR modifies translation by influencing RNA-stability, conformation or transport. It could be hypothesized that this could lead to local areas of increased -synuclein concentration where oligomerization is initiated, which is then propagated in a seeding-dependent fashion. If further confirmed, pharmacologic manipulation of -synuclein expression may be a possible therapeutic strategy to prevent PD in susceptible individuals.

Although -synuclein mutations are rare, their identification was of course extremely important, as it led to the discovery that the encoded protein is the major fibrillar component of the Lewy-body [START_REF] Spillantini | Alpha-synuclein in Lewy bodies[END_REF], the proteinaceous inclusion which has, since Friedrich Lewy's original description in 1917, been considered to be the pathologic hallmark of PD in both familial and sporadic cases.

This and many other lines of evidence support a central role of -synuclein in the development of PD, and -synuclein staining has replaced staining for ubiquitin as the most sensitive tool in the detection of Lewy-bodies. The currently favored hypothesis states that the amino acid changes in the -synuclein protein associated with PD may favor the ßpleated sheet conformation of the protein over the natively unstructured form that is usually predominant. This in turn is thought to lead to an increased tendency to form aggregates [START_REF] Goedert | Filamentous nerve cell inclusions in neurodegenerative diseases[END_REF]. However, the precise relationship between the formation of aggregates and cell death is unknown. It has been hypothesized that a failure of proteasomal degradation of -synuclein and other proteins may lead to an accumulation of toxic compounds, possibly consisting of synuclein oligomers and protofibrils [START_REF] Goldberg | Is there a cause-and-effect relationship between alpha-synuclein fibrillization and Parkinson's disease?[END_REF], ultimately leading to cell death [START_REF] Mcnaught | Failure of the ubiquitin-proteasome system in Parkinson's disease[END_REF]. Another possible mechanism could be a deficient chaperone-mediated lysosomal clearance of synuclein [START_REF] Cuervo | Impaired degradation of mutant alpha-synuclein by chaperone-mediated autophagy[END_REF]. Mechanistic aspects of -synuclein pathogenicity are discussed in more detail elsewhere in this volume (ref).

-Synuclein point mutations and also gene dosage mutations are too rare to warrant mutational analysis on a routine basis except in cases when the disease clearly segregates in a dominant fashion in a large family. SNCA risk variants, on the other hand, are common in the population, but confer only a slightly increased risk to develop the disease (odds ratio of 1.5 to 2) and therefore are only of limited value for individual risk prediction. They may however be useful at some point to stratify populations, for example if preventive or diseasemodifying treatments become available that modify -synuclein expression levels.

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PARK8: Leucine-Rich Repeat Kinase 2 (LRRK2)

Another major locus for an autosomal-dominant mendelian form of PD was first mapped in a large Japanese family to the pericentromeric region of chromosome 12 and named PARK8. Affecteds in this family showed typical L-dopa responsive parkinsonism with onset in their late fifties [START_REF] Funayama | A new locus for Parkinson's disease (PARK8) maps to chromosome 12p11.2-q13.1[END_REF]. Linkage to this locus was confirmed and refined in two families of Caucasian background [START_REF] Zimprich | The PARK8 Locus in Autosomal Dominant Parkinsonism: Confirmation of Linkage and Further Delineation of the Disease-Containing Interval[END_REF], and by a positional cloning approach, different missense mutations in a large gene called leucine-rich repeat kinase 2 (LRRK2) [8; 9] were found to cosegregate with the disease in these and several other families. The observation of different missense changes in highly conserved codons linked to the disease in a number of large families is strongly supportive of their pathogenic role.

Soon it became clear that mutations in the LRRK2 gene are a much more common cause of inherited parkinsonism than those in the SNCA gene. In a number of studies across several different populations, between 5 to 15% of dominant families carried mutations in the LRRK2 gene [52; 53]. One particularly common mutation is a basepair change, a G-->A transition at position 6055, better known as the "G2019S-mutation", which is responsible for familial PD in up to 7% of cases in different Caucasian populations [54; 55; 56], but also in 1 to 2% of sporadic patients [START_REF] Gilks | A common LRRK2 mutation in idiopathic Parkinson's disease[END_REF]. Similar to the A53T SNCA mutation, haplotype analysis suggested that the mutation can be traced back to probably a few founder events [56; 58] in different populations, indicating that the mutation has been passed down for many centuries. Even higher G2019S prevalence rates of up to 40% were found in genetically more isolated populations, such as the Ashkenazi Jewish and the North African Berber Arab populations, both in sporadic and familial cases [59; 60], again explained by a genetic founder effect.

Particularly the high prevalence in sporadic cases suggests a reduced penetrance of this mutation [START_REF] Goldwurm | Evaluation of LRRK2 G2019S penetrance: relevance for genetic counseling in Parkinson disease[END_REF], which must be taken into account when counseling patients and families.

Based on the type of study (family ascertainment vs. population based) estimates of penetrance vary between 35 and 85%, but are in any case of course highly age-dependent.

Another old, population-specific mutation is commonly encountered in the Basque population in Northern Spain, involving codon 1441 (R1441G) [START_REF] Mata | LRRK2 R1441G in Spanish patients with Parkinson's disease[END_REF].

The most extensive study so far estimated the overall frequency of LRRK2 mutations in the European population to about 1.5% in sporadic and 4% in familial cases, with a geographic gradient decreasing from the Mediterranean countries (Spain, Portugal, Italy) to the North.

A large collaborative clinicogenetic study collecting clinical data on 356 LRRK2 mutation carriers showed that the average age of onset was 58 years, with a wide range from the midtwenties (rarely!) to over 90 years [START_REF] Healy | Phenotype, genotype, and worldwide genetic penetrance of LRRK2-associated Parkinson's disease: a case-control study[END_REF]. The clinical picture was that of typical asymmetric levodopa responsive parkinsonism that was indistinguishable by any single criterion from PD in non mutation-carriers. As a group however, the disease appears to be somewhat more benign in LRRK2 positive patients, which is reflected in a longer duration to reach equivalent

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Hoehn-and-Yahr stages, and a lower frequency of dementia and psychiatric complications.

Although the data are still limited it also seems that olfactory dysfunction is less frequent (~ 50% as opposed to 80 -100% in sporadic PD).

Due to its relatively high frequency, the G2019S mutation for the first time offers the possibility to look at gene-gene interactions. Dächsel et al. identified three Spanish patients simultaneously harboring heterozygous mutations in the LRRK2 and in the parkin gene. In comparison to other Spanish patients with a single LRRK2 mutation, the three doublemutation patients did not present with an earlier age-at-onset or a faster progression of disease [START_REF] Dachsel | Digenic parkinsonism: investigation of the synergistic effects of PRKN and LRRK2[END_REF]. Given the questionable pathogenicity of heterozygous parkin mutations (see below), this finding is not surprising.

The G2019S mutation also seems to be fully dominant, as homozygous mutation carriers have been identified who also do not differ from heterozygotes with respect to disease severity or age of onset [START_REF] Nichols | Genetic screening for a single common LRRK2 mutation in familial Parkinson's disease[END_REF].

The LRRK2 gene is very large and spans a genomic region of 144 Kb, with 51 exons encoding 2527 amino acids. To date, more than 20 potential mutations have been identified so far. It must be remembered, however, that only six of them can at this point be considered proven to be pathogenic (R1441C, R1441G, R1441H, Y1699C, G2019S, I2020T), because of firm evidence of cosegregation in affected families. This does not necessarily mean that other reported rare variants, that often have been found in sporadic PD patients or small families but not in several hundred controls are not, but the possibility remains that they may represent rare polymorphisms, which is of course of major importance for genetic counseling.

Early-on the possibility was investigated whether common polymorphisms in the LRRK2 gene may contribute to the risk to develop sporadic PD, as was shown for the SNCA gene (see above). Two extensive studies in European populations using tagging single nucleotide polymorphisms (tSNPs) failed to detect any associated variants or haplotypes [65; 66], while a similar study in Chinese patients detected an association signal [START_REF] Skipper | Comprehensive evaluation of common genetic variation within LRRK2 reveals evidence for association with sporadic Parkinson's disease[END_REF], indicating possible population specific effects. In fact, this assumption was clearly confirmed by the subsequent identification of at least 2 relatively common risk variants in the Asian (Chinese as well as Japanese) population: the variant G2385R was found in approximately 6 to 9% of Chinese patients with PD ([68; 69; 70], as compared to 2 to 4% in controls, conferring a relative risk of about 2. Again, haplotype analyses suggested an ancient mutation with a common founder [START_REF] Farrer | Lrrk2 G2385R is an ancestral risk factor for Parkinson's disease in Asia[END_REF]). Another risk variant R1628P of nearly similar strength was also identified in Chinese patients [72; 73], again representing a single ancient mutational event. Remarkably, both variants together explain about 10% of the risk for sporadic PD in this Asian population (population attributable risk). These studies clearly demonstrate that some mutations in the LRRK2 act as high penetrance Mendelian disease genes, whereas others operate as risk Many aspects of LRRK2 associated PD remain to be elucidated, some of which are dealt with in other contributions to this issue. For example, the mechanism by which LRRK2 mutations cause neuronal degeneration is still unknown. By sequence homology, LRRK2 can be assigned to the group of recently identified ROCO-proteins [START_REF] Bosgraaf | Roc, a Ras/GTPase domain in complex proteins[END_REF] and contains a protein kinase domain of the MAPKKK class, suggesting a role in intracellular signaling pathways.

Although the natural substrate of LRRK2 is unknown, cell culture studies suggest that at least some pathogenic mutations seem to be associated with an increase, rather than a loss, of kinase activity [75; 76; 77], and that kinase activity appears to be necessary for neurotoxicity in vitro [START_REF] Greggio | Kinase activity is required for the toxic effects of mutant LRRK2/dardarin[END_REF], raising the interesting possibility that kinase inhibition may be a potential therapeutic strategy.

Another puzzling aspect of LRRK2 related PD is its pathology. Post mortem changes in patients with LRRK2 mutations are consistent with typical Lewy-body Parkinson´s disease in most cases reported so far [START_REF] Ross | Lrrk2 and Lewy body disease[END_REF], but also include diffuse Lewy-body disease, nigral degeneration without distinctive histopathology and, rarely, even progressive supranuclear palsy-like tau aggregation. All these pathologic findings were reported in a single family with an R1441C mutation [START_REF] Zimprich | Mutations in LRRK2 Cause Autosomal-Dominant Parkinsonism with Pleomorphic Pathology[END_REF]. LRRK2 mutations may therefore be an upstream event in the cascade leading to neurodegeneration with different pathologies. On the other hand, mutation-specific effects are conceivable. The vast majority of patients with a G2019S mutation, in whom pathology has been reported, seem to conform to the typical -synuclein

Lewy-body type of PD [START_REF] Ross | Lrrk2 and Lewy body disease[END_REF], while in the original Japanese family with an I2020T mutation, nigral degeneration without prominent inclusions seems to be dominating the pathologic picture [START_REF] Hasegawa | Familial parkinsonism: Study of original Sagamihara PARK8 (I2020T) kindred with variable clinicopathologic outcomes[END_REF].

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Despite its still enigmatic role in pathogenesis, LRRK2 associated PD is of particular interest since it is the first example of a Mendelian form of PD that is common enough to provide the opportunity to study the development of the disease in a sizeable population. Longitudinal studies in presymptomatic mutation carriers may reveal pre-motor changes by clinical, biochemical or imaging methods, indicating the very early phases of the neurodegenerative process. It is in this population that studies exploring neuroprotective or preventive measures, be it specific for the LRRK2 associated pathways or more general, are most promising to yield first results.

Recessive forms of Mendelian PD

One of the interesting developments of recent years was the recognition of the relatively high proportion of patients with a monogenically inherited form of parkinsonism following autosomal-recessive transmission. So far, three of the causative genes, parkin PRKN (PARK2), PINK1 (PARK6) and DJ-1 (PARK7) have been cloned. One other recessive gene (ATP13A2, PARK9) has been identified in patients with early-onset parkinsonism, spasticity and dementia. It is likely that several other loci exist. Currently, the genes for parkin (PRKN)

and PINK1 are numerically clearly the most important causes for autosomal-recessive juvenile-onset parkinsonism (AR-JP).

Autosomal-recessive juvenile parkinsonism caused by mutations in the parkin gene

Cases of parkinsonism with early disease-onset and recessive inheritance (families with affected siblings, but usually no transmission from one generation to the next) were first recognized and described as a clinical entity in Japan [START_REF] Ishikawa | Clinical analysis of 17 patients in 12 Japanese families with autosomal-recessive type juvenile parkinsonism[END_REF]. Clinically, these patients suffered from L-dopa-responsive parkinsonism with onset in the second to fourth decade. Some of the patients showed diurnal fluctuations with symptoms becoming worse later in the day, similar to patients with dopa-responsive dystonia (DRD), which is caused by mutations in the gene for GTP-cyclohydrolase [START_REF] Ichinose | Hereditary progressive dystonia with marked diurnal fluctuation caused by mutations in the GTP cyclohydrolase I gene[END_REF], one of the genes involved in dopamine biosynthesis. In contrast to patients with DRD, however, patients with recessive parkinsonism develop often early and severe levodopa-induced motor-fluctuations and dyskinesias. In other respects, however, the disease seemed to progress slowly and development of dementia was relatively rare.

Dystonia at onset of the disease was found to be a common feature. The disorder was called autosomal-recessive juvenile parkinsonism (AR-JP).

The first genetic locus for AR-JP was mapped by linkage analysis in a set of 13 Japanese families to the long arm of chromosome 6 [START_REF] Matsumine | Localization of a gene for an autosomal recessive form of juvenile Parkinsonism to chromosome 6q25.2-27[END_REF]. This region was analyzed as a candidate locus bearing the gene for mitochondrial superoxide dismutase 2 (Mn-SOD2), based on

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hypotheses of oxidative stress and mitochondrial dysfunction being involved in the pathogenesis of PD. The locus was later confirmed and refined in different ethnic groups [84; 85], excluding SOD2 from the refined region.

Positional cloning eventually identified several homozygous deletions in consanguineous families with AR-JP in a large gene in this region that was named parkin (PRKN) [2; 86].

PRKN mutations turned out to be a common cause of parkinsonism with early onset, and novel mutations were reported in rapid succession [87; 88; 89]. All types of mutations, including missense mutations leading to amino acid exchanges, nonsense mutations resulting in premature termination of translation and exonic rearrangements (deletions, duplications and triplications) were identified [START_REF] Lucking | Pseudo-dominant inheritance and exon 2 triplication in a family with parkin gene mutations[END_REF].

A number of studies have delineated the clinical spectrum associated with parkin mutations.

By and large, the clinical picture is frequently indistinguishable from that of the sporadic disease, with the notable exception of course of the earlier age-at-onset. The vast majority of patients manifest the disease before the age of 40, very few patients with disease-onset in their fifties have been described [START_REF] Klein | Parkin deletions in a family with adult-onset, tremor-dominant parkinsonism: expanding the phenotype[END_REF].

For example, a large collection of European sib pairs collected by the European Consortium on Genetic Susceptibility in PD (GSPD) allowed the characterization of the clinical spectrum of parkin-associated parkinsonism [START_REF] Lücking | Association between Early-Onset Parkinson's Disease and Mutations in the Parkin Gene[END_REF]. Mean age at onset was 32 years, progression of the disease was slow but L-dopa associated fluctuations and dyskinesias occurred frequently and early. Brisk reflexes of the lower limbs were present in 44%.

Comparing young-onset carriers and non-carriers of parkin mutations, those with a mutation tended to have earlier and more symmetrical onset, slower progression of the disease and greater response to L-dopa despite lower doses. Lower limb dystonia at disease-onset occurred in about a third of the patients, but this feature did not appear to be specific for parkin-related disease, and was more correlated with the age at onset than with genetic status [START_REF] Lohmann | How much phenotypic variation can be attributed to parkin genotype?[END_REF]. Nevertheless, dystonia at disease-onset can be prominent enough to render distinction from dopa-responsive dystonia difficult [START_REF] Tassin | Levodopa-responsive dystonia: GTP cyclohydrolase I or parkin mutations?[END_REF].

In the initial European study by Lücking and co-workers, there was no discernable difference in the clinical phenotype between patients with missense mutations, truncating point mutations or deletions, suggesting that a complete loss of Parkin function is associated with all of these mutations and with the full phenotype of early-onset parkinsonism. More recent phenotype-genotype studies by the same group in an extended sample however suggested that the type of mutation may in fact influence the clinical phenotype to a certain degree:

patients with at least one missense mutation showed a faster progression of the disease with a higher UPDRS (United Parkinson´s Disease Rating Scale) motor score than carriers of truncating mutations. Missense mutations in functional domains of the parkin gene resulted

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in earlier onset [START_REF] Lohmann | How much phenotypic variation can be attributed to parkin genotype?[END_REF]. A very similar clinical picture was described in Japanese patients, but age of onset was still lower (average about 26 years) [START_REF] Yamamura | Autosomal recessive early-onset parkinsonism with diurnal fluctuation: clinicopathologic characteristics and molecular genetic identification[END_REF] .

Although the majority of patients with parkin mutations in most series conform to this relatively well defined clinical picture, a minority will present with unusual features, including cervical dystonia, autonomic dysfunction and peripheral neuropathy, exercise-induced dystonia [START_REF] Khan | Parkin disease: a phenotypic study of a large case series[END_REF], or camptocormia [START_REF] Inzelberg | Camptocormia, axial dystonia, and parkinsonism: Phenotypic heterogeneity of a parkin mutation[END_REF]. Further clinical features that can be seen in parkin disease: focal dystonia; early postural instability; severe and early freezing; festination or retropulsion; concurrent autonomic failure; dramatic response to anticholinergics; early or atypical L-dopa-induced dyskinesias; exquisite sensitivity to small doses of L-dopa; and recurrent psychosis, even taking L-dopa alone [START_REF] Khan | Parkin disease: a phenotypic study of a large case series[END_REF]. Psychiatric and behavioural disorders prior to the onset of parkinsonism have also been recognized in patients with parkin mutations [94; 95]. Exceptionally, patients with parkin mutations may even present with a clinical picture of levodopa unresponsive parkinsonism complicated by cerebellar and pyramidal tract dysfunction [START_REF] Kuroda | Homozygous deletion mutation of the parkin gene in patients with atypical parkinsonism[END_REF].

Functional neuroimaging in parkin-linked parkinsonism consistently demonstrated a severely reduced uptake of dopamine tracer in both hemispheres in the putamen und caudate nucleus [START_REF] Portman | The nigrostriatal dopaminergic system in familial early onset parkinsonism with parkin mutations[END_REF], which differs at least quantitatively from the initially unilateral reduction in dopa uptake of sporadic PD patients [START_REF] Leenders | Parkinson's disease: clinical signs and symptoms, neural mechanisms, positron emission tomography, and therapeutic interventions[END_REF]. In terms of the rostro-caudal gradient of tracer-uptake in the putamen (the decrease being more severe in the posterior portion), the pattern of uptake in YOPD patients is similar to that of patients with idiopathic Parkinson´s disease and does not depend on the presence or absence of mutations of the parkin gene [START_REF] Thobois | Young-onset Parkinson disease with and without parkin gene mutations: a fluorodopa F 18 positron emission tomography study[END_REF].

Khan et al. have used [ 18 F

]dopa PET serially to study members of a family with young-onset parkinsonism who are compound heterozygous for mutations in the parkin gene, to assess disease progression. The parkin patients showed a significantly slower loss of putamen [ 18 F]dopa uptake compared with a group of idiopathic Parkinson's disease (IPD) patients who had baseline putamen [ 18 F]dopa uptake and disease severity similar to the parkin group. One study also provided evidence for an alteration of postsynaptic mechanisms, indicated by a reduced binding of the D2-receptor ligand 11C-raclopride [START_REF] Hilker | Positron emission tomographic analysis of the nigrostriatal dopaminergic system in familial parkinsonism associated with mutations in the parkin gene[END_REF]. This finding is remarkable, as it is usually associated with an impaired response to dopaminergic medication, which is clearly not typical of parkin-disease.

Numerically, parkin mutations are the major cause of early-onset parkinsonism in cases with recessive inheritance. In a group of 73 sibling pairs with PD, at least one of them having an age at onset before 45 years, 50% were due to parkin mutations [START_REF] Lücking | Association between Early-Onset Parkinson's Disease and Mutations in the Parkin Gene[END_REF]. The prevalence of Parkin-associated parkinsonism in a population of sporadic patients strongly depends on age at onset. Taking into account a number of different studies, PRKN mutations probably

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account for about 5 to 10% of early-onset cases (onset before 45 years) without family history. Within this group, the proportion of mutation-carriers increases significantly with decreasing age at onset (82 % in those with an onset before age 20 years) [START_REF] Periquet | Parkin mutations are frequent in patients with isolated early-onset parkinsonism[END_REF]. A similar distribution has been found in Japanese patients [START_REF] Ujike | Prevalence of homozygous deletions of the parkin gene in a cohort of patients with sporadic and familial Parkinson's disease[END_REF].

AR-JP caused by PRKN mutations is an autosomal-recessive disorder with high penetrance.

Homozygous or compound heterozygous disease gene carriers will develop the disease with an age dependent penetrance. Apparently dominant inheritance of parkin-related disease may be found when multiple disease alleles are segregating within a single family (which may occur particularly in the context of consanguinity) [START_REF] Maruyama | Novel mutations, pseudo-dominant inheritance, and possible familial affects in patients with autosomal recessive juvenile parkinsonism[END_REF].

The question whether (specific?) heterozygous parkin mutations may cause or increase the susceptibility to late-onset typical PD is still a matter of debate. It is well established that heterozygous carriers of parkin mutations have reduced [ 18 F]-Fluorodopa-uptake by PET [101; 105], even in the absence of clinical symptoms, which has been interpreted as a "first hit" by the genetic abnormality to the dopaminergic system. Some larger multicase-pedigrees have been ascertained on the basis of typical early-onset parkin-related cases of parkinsonism in patients with two parkin mutations. Heterozygous carriers of parkin mutations in these families are obviously common and have been found to exhibit either minor signs of basal ganglia dysfunction [START_REF] Pramstaller | Phenotypic variability in a large kindred (Family LA) with deletions in the parkin gene[END_REF] or present with late-onset parkinsonism [91; 107]. In at least two of these cases, typical Lewy-bodies have been described at autopsy, suggesting that a hetero-or hemizygous parkin mutation may be a risk factor for typical LBpositive PD. However, PD is a common condition, and an occasional occurrence of PD due to different causes in a single family is a well documented possibility. Therefore, the causal relationship between heterozygous parkin mutations and disease status is still not completely clear. Studies screening late-onset PD cases and healthy, age-matched controls for parkin mutations found similar frequencies of genetic variants, on the order of about 3%, suggesting that heterozygous parkin mutations generally do not confer a significantly increased risk for late-onset PD [108; 109]. On the other hand, "healthy" controls in these studies have usually not been examined carefully for minor signs of parkinsonism.

Even ten years after the identification of the parkin gene, there is still relatively little information available on the neuropathology of molecularly confirmed cases of parkin-related AR-JP. Severe and selective degeneration of dopaminergic neurons and gliosis in the substantia nigra pars compacta, and, to a somewhat lesser degree, of the locus coeruleus, has been described [110; 111; 112]. Lewy-bodies or other -synuclein-containing Lewypathology have not been described in several studies, suggesting that the disease process in parkin-related cases may differ in some important ways from that in typical idiopathic PD.

Lewy-bodies are thought to represent a mechanism whereby cells sequester damaged and

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toxic proteins in an inactive form. The ubiquitine-proteasome system (UPS) is involved in the detoxification of these proteins, so complete loss of Parkin, which is an E3-ubiquitin ligase and therefore a component of the UPS, may be sufficient to prevent LB formation. On the other hand, however, several cases have also been described with parkinsonism, parkin mutations and positive Lewy-bodies at autopsy [107; 113; 114]. The significance of this finding is still unclear but it may indicate that some specific parkin mutations retain sufficient activity to allow LB-formation. This has been supported by experimental studies indicating that mutations in the RING-domain retain ubiquitine ligase activity [START_REF] Farrer | Lewy bodies and parkinsonism in families with parkin mutations[END_REF], and that certain point mutations may confer a toxic gain of function, leading to Parkin protein aggregation and aggresome formation [START_REF] Gu | The C289G and C418R missense mutations cause rapid sequestration of human Parkin into insoluble aggregates[END_REF].

To further address the possible role of Parkin in sporadic PD, several association studies with coding and non-coding polymorphisms within the parkin gene have been performed.

Overall, the results of these studies are still inconclusive. Some positive results have been reported [116; 117; 118] but not fully corroborated by other studies [119; 120; 121; 122]. The general feeling is that common variants in the parkin gene are probably not of major relevance in sporadic late-onset PD.

Autosomal-recessive juvenile parkinsonism caused by mutations in the PINK1-gene

In a large Italian family with autosomal recessive early-onset parkinsonism, Valente et al.

(2001) identified a novel locus, PARK6, in a 12.5-cM region of 1p36-p35. The large Sicilian family, which the authors designated the Marsala kindred, had 4 definitely affected members.

The phenotype was characterized by early-onset (range 32 to 48 years) parkinsonism, with slow progression and sustained response to levodopa. In this and two other consanguineous PARK6-linked families, Valente et al. identified two different mutations in the gene for PINK1 [START_REF] Valente | Hereditary early-onset Parkinson's disease caused by mutations in PINK1[END_REF]. Cell culture studies indicated that PINK1 is mitochondrially located and may exert a protective effect on the cell that is abrogated by the mutations, resulting in increased susceptibility to cellular stress.

A number of studies confirmed the presence of PINK1 mutations in patients with early-onset PD [123; 124; 125; 126; 127]. Most mutations were missense mutations in conserved regions, but also whole gene deletions have been described [START_REF] Marongiu | Whole gene deletion and splicing mutations expand the PINK1 genotypic spectrum[END_REF].

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In a review of 21 patients with PINK1-related parkinsonism reported in the literature, Albanese et al. found that most patients (95 to 100%) had slow disease progression, good response to levodopa, bradykinesia, and rigidity. Also common were levodopa-induced dyskinesias, resting tremor, and asymmetry at onset, postural instability (63%), on/off phenomenon (60%), and gait impairment (55%). As in parkin disease, except of the earlier average age of onset, no single feature can separate PINK1-disease from idiopathic PD [START_REF] Albanese | The PINK1 phenotype can be indistinguishable from idiopathic Parkinson disease[END_REF]. There are some indications that PINK1-mutated patients have a higher prevalence of psychiatric disturbances, particularly anxiety and depression, which is only relatively rarely observed in parkin cases [START_REF] Ephraty | Neuropsychiatric and cognitive features in autosomal-recessive early parkinsonism due to PINK1 mutations[END_REF].

Exceptionally, a compound heterozygous PINK1 mutation can be found also in sporadic patients with a phenotype indistinguishable from idiopathic Parkinson's disease (PD), with onset in the late seventh decade, rapid progression and good response to levodopa that waned with time, underlining the existing difficulties in discriminating between mendelian parkinsonism and idiopathic PD [START_REF] Gelmetti | Late onset sporadic Parkinson's disease caused by PINK1 mutations: Clinical and functional study[END_REF].

The PINK1 protein has been located, at least in good part, to the mitochondrial compartment, where it may act as a protein kinase [START_REF] Valente | PINK1 mutations are associated with sporadic early-onset parkinsonism[END_REF]. In drosophila, a deletion of the PINK1 gene leads to a marked mitochondrial phenotype. Interestingly, this phenotype can be largely rescued by the overexpression of the parkin gene, suggesting that both proteins act in a common pathway, which is related to mitochondrial function and integrity [START_REF] Yang | Mitochondrial pathology and muscle and dopaminergic neuron degeneration caused by inactivation of Drosophila Pink1 is rescued by Parkin[END_REF].

No pathology has yet been reported in patients with PINK1 mutations.

Autosomal-recessive juvenile parkinsonism caused by mutations in the DJ-1-gene

The third locus for AR-JP, PARK7, was mapped also to chromosome 1p36, in a Dutch family [START_REF] Van Duijn | Park7, a novel locus for autosomal recessive early-onset parkinsonism, on chromosome 1p36[END_REF], and the gene was identified as the oncogene DJ-1 [START_REF] Bonifati | Mutations in the DJ-1 Gene Associated with Autosomal Recessive Early-Onset Parkinsonism[END_REF]. Again, the phenotype closely resembles that found in patients with parkin and PINK1 mutations, but this statement is based on a rather small number of identified patients [134; 135]. On the other hand, one recessive family has been described with early-onset parkinsonism, dementia and ALS, which carries two homozygous mutations in the DJ-1 gene, suggesting that the clinical phenotype associated with mutations in this gene, although rare, may be rather wide [START_REF] Annesi | DJ-1 mutations and parkinsonism-dementia-amyotrophic lateral sclerosis complex[END_REF].

The associated pathology is still unknown, as no autopsies have been reported.

The function of the DJ-1 protein is not entirely clear. The favoured hypothesis is that it acts as a sensor for oxidative stress, providing neuroprotection in situations of increased demand [START_REF] Canet-Aviles | The Parkinson's disease protein DJ-1 is neuroprotective due to cysteine-sulfinic acid-driven mitochondrial localization[END_REF].

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Relevance of recessive PD-genes to the late-onset sporadic disease

The question of the relevance of mutations in the recessive PD-genes for the development of sporadic PD and the relationship of the pathogenic pathways is still controversial. Differences in pathology of AR-JP and sporadic PD may point to different pathogenic cascades, but they may also be related to differences in age and disease dynamics between these groups.

Current evidence, also presented elsewhere in this volume, suggests that the recessive PDgenes exert a neuroprotective function against a variety of insults, with a common theme emerging with relation to mitochondrial function and integrity [138; 139; 140]. Whether enhancing this protective function can be a future therapeutic or disease-modifying strategy in PD remains to be elucidated. In any case, recessive forms of PD certainly represent a valuable "model" to study the effects of the loss of specifically defined protective mechanisms on the survival of dopaminergic neurons.

OTHER GENES AND LOCI

Several other loci have been mapped in families with PD and apparently mendelian inheritance, but either the genes have not yet been identified, or their role is still somewhat controversial. A dominant locus has been described on chromosome 2p13 (PARK3) [START_REF] Gasser | A susceptibility locus for Parkinson's disease maps to chromosome 2p13[END_REF].

Clinical features relatively closely resemble those of sporadic PD. So far, however, the gene has not been identified in these families. Interestingly, several independent recent reports implicate the PARK3-locus as a disease modifying locus influencing age at onset in two sib pair cohorts with PD [141; 142] and a European sib pair study also identified a linkage peak in this region [START_REF] Martinez | Genomewide scan linkage analysis for Parkinson's disease: the European genetic study of Parkinson's disease[END_REF]. A further study refined this association to a region containing the sepiapterine reductase gene [START_REF] Karamohamed | A haplotype at the PARK3 locus influences onset age for Parkinson's disease: the GenePD study[END_REF], which was confirmed in another study [START_REF] Sharma | The sepiapterin reductase gene region reveals association in the PARK3 locus: analysis of familial and sporadic Parkinson's disease in European populations[END_REF]. Sepiapterine reductase is involved in dopamine synthesis. This finding may indicate that the SPR gene is modifying age of onset of PD, but the potentially causative variants have not been identified.

A heterozygous missense mutation in the gene for ubiquitin carboxy-terminal hydrolase L1 gene (UCHL1, PARK5), which is located on chromosome 4p has been identified in affecteds in a single family of German ancestry [START_REF] Leroy | The ubiquitin pathway in Parkinson's disease [letter[END_REF]. To date, no other bona fide pathogenic mutations of this gene have been identified, and the segregation of the mutation in the original family does not sufficiently support the contention of UCHL1 being a dominant disease gene. Nevertheless, a common polymorphism in the UCHL1 gene, the S18Y variant, has been suggested to be associated with sporadic PD in several studies, including a large meta-analysis [START_REF] Maraganore | UCHL1 is a Parkinson's disease susceptibility gene[END_REF]. Still, whether UCHL1 is really a PD-gene is not clear yet.

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Another locus has been mapped on chromosome 1 in an Icelandic population (PARK10) [START_REF] Hicks | A susceptibility gene for late-onset idiopathic Parkinson's disease[END_REF], but the gene has not yet been identified and it is unclear, whether the locus plays a role in other populations also.

Finally, a locus on chromosome 2q has been mapped in a mixed Caucasian population from the US [START_REF] Pankratz | Significant linkage of Parkinson disease to chromosome 2q36-37[END_REF], but the locus has not been confirmed in another cohort of European families [START_REF] Prestel | PARK11 is not linked with Parkinson's disease in European families[END_REF]. Nevertheless, recent report provided evidence for mutations in a gene in this region, GIGYF2, or GBR10 interacting GYF protein-2, which encodes a component of the insulin signaling pathway [START_REF] Lautier | Mutations in the GIGYF2 (TNRC15) Gene at the PARK11 Locus in Familial Parkinson Disease[END_REF]. Four different sequence changes have been identified in conserved regions of the gene, mostly in relatively small families with PD. In some cases, unaffected family members were identified to also carry the mutation, and none of the families were large enough to unequivocally demonstrate cosegregation of mutation and disease. A recent study failed to confirm a strong association of mutations in the GIGYF2 gene in two other populations with PD [START_REF] Bras | Lack of replication of association between GIGYF2 variants and Parkinson disease[END_REF] so that their pathogenic role is still uncertain.

Mutations in several other genes have been reported in PD patients without linkage evidence from large families. Therefore, the genetic proof of pathogenicity is lacking.

Although functional studies in cell culture support a possible pathogenic effect of these mutations, as is the case for the mitochondrial protease OMI/HtrA2 [START_REF] Strauss | Loss of function mutations in the gene encoding Omi/HtrA2 in Parkinson's disease[END_REF]. In still other cases, the possible role of a gene in PD is suggested by the clinical observation of association, as found in the case of Gaucher's disease. Relatives of patients with this well characterized recessive neurometabolic disease, caused by mutations in the glucocerebrosidase gene (GBA) have a conspicuously high prevalence of PD. Screening of PD patients for GBA mutations then found a higher number of heterozygous mutations carriers as compared to healthy controls [151]. This has now been confirmed in several other populations with very similar odds ratios [152; 153; 154]. Mutations have been found in about 2 to 4% of Caucasian PD patients, and less than one percent of controls. However, those variants are likely to act as risk factors rather than as high penetrance Mendelian disease genes.

Conclusion

The identification and characterization of Mendelian forms of PD has sparked an extremely fruitful line of research, delineating molecular pathways that are involved in dopaminergic cell dysfunction and death. It is becoming increasingly clear that those pathways are not only relevant for these rare Mendelian variants of PD, but also to the PD population as a whole.

A

  C C E P T E D M A N U S C R I P TACCEPTED MANUSCRIPTalleles. Molecular studies will have to show whether the mechanism of action in these instances is the same, just of different strengths, or different. As in the case of SNCA risk variants, determination of risk alleles will only have limited clinical usefulness for individual risk prediction, as obviously other factors are required to lead to the manifestation of the disease, at least during a usual lifespan. Once the molecular mechanisms of the risk modification are deciphered, however, the path is open to stratify patient cohorts for specific LRRK2 directed disease-modifying treatments. If only the 10% disease risk attributable to these two variants could be significantly reduced, this would be a tangible success on the population level.

A

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

Table 1 :

 1 PD genes

		Inheri-	Gene	Comments	Ref.
		tance
	PARK1	AD	-Synuclein First PD gene identified. Protein is major	[1]
				component of Lewy-body
	PARK2	AR	Parkin	Most common form of autosomal-recessive	[2]
				juvenile parkinsonism
	PARK3	AD	SPR?	Gene not known with certainty	[3]
	PARK4	AD	-Synuclein Duplications and triplications of the SNCA	[4]
				gene
	PARK5	AD	UCH L1	Role uncertain	[5]
	PARK6	AR	PINK1	Second most common form of AR-JP	[6]
	PARK7	AR	DJ-1	Rare	[7]
	PARK8	AD	LRRK2	Most common form of dominant PD	[8; 9]
	PARK9	AR	ATP13A2	Complex phenotype with parkinsonism,	[10]
				spasticity and dementia
	PARK11	AD	GIGYF2 ?	Role of gene not yet certain	[11]
	PARK13	AD ?	OMI/HtrA2	No cosegregation shown to support	[12]
				pathogenicity


Roger Nakad May
email: nakad@iecn.u-nancy.fr

Lower Bounds for the Eigenvalues of the Dirac Operator on Spin c Manifolds

Keywords: Spin c structures, Dirac operator, eigenvalues, Energy-Momentum tensor, perturbed Yamabe operator, conformal geometry

In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on Spin c manifolds without boundary. The limiting case is then studied and an example is given.

Introduction

On a compact Riemannian spin manifold (M n , g) of dimension n 2, Th. Friedrich [START_REF] Friedrich | Der erste Eigenwert des Dirac-operators einer kompakten Riemannschen Mannigfaltigkeit nichtnagativer Skalarkrümmung[END_REF] showed that any eigenvalue λ of the Dirac operator satisfies

λ 2 λ 2 1 := n 4(n -1) inf M S g , (1)
where S g denotes the scalar curvature of M . The limiting case of (1) is characterized by the existence of a special spinor called real Killing spinor. This is a section ψ of the spinor bundle satisfying for every X ∈ Γ(T M ),

∇ X ψ = - λ 1 n X • ψ, 1
where X • ψ denotes the Clifford multiplication and ∇ is the spinorial Levi-Civita connection [START_REF] Lawson | Spin Geometry[END_REF]. On the complement set of zeroes of any spinor field φ, we define φ the field of symmetric endomorphisms associated with the field of quadratic forms, denoted by T φ , called the Energy-Momentum tensor which is given, for any vector field X, by

T φ (X) = g( φ (X), X) = Re < X • ∇ X φ, φ |φ| 2 > .
The associated symmetric bilinear form is then given for every X, Y ∈ Γ(T M ) by

g( φ (X), Y ) = 1 2 Re < X • ∇ Y φ + Y • ∇ X φ, φ |φ| 2 > .
Note that if the spinor field φ is an eigenspinor, C. Bär showed that the zero set is contained in a countable union of (n -2)-dimensional submanifolds and has locally finite (n -2)-dimensional Hausdroff density [START_REF] Bär | Zero sets of solutions to semilinear elliptic systems of first order[END_REF]. In 1995, O. Hijazi [START_REF] Hijazi | Lower bounds for the eigenvalues of the Dirac operator[END_REF] modified the connection ∇ in the direction of the endomorphism ψ where ψ is an eigenspinor associated with an eigenvalue λ of the Dirac operator and established that

λ 2 inf M ( 1 4 S g + | ψ | 2 ). (2)
The limiting case of (2) is characterized by the existence of a spinor field ψ satisfying for all X ∈ Γ(T M ),

∇ X ψ = -ψ (X) • ψ. (3)
The trace of ψ being equal to λ, Inequality (2) improves Inequality [START_REF] Aubin | Nonlinear Analysis on Manifolds, Monge-Ampre Equations[END_REF] since by the Cauchy-Schwarz inequality,

| ψ | 2 (tr( ψ )) 2 n
, where tr denotes the trace of ψ . N. Ginoux and G. Habib showed in [START_REF] Ginoux | A spectral estimate for the Dirac operator on Riemannian flows[END_REF] that the Heisenberg manifold is a limiting manifold for (2) but equality in (1) cannot occur.

Using the conformal covariance of the Dirac operator, O. Hijazi [START_REF] Hijazi | A conformal Lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors[END_REF] showed that, on a compact Riemannian spin manifold (M n , g) of dimension n 3, any eigenvalue of the Dirac operator satisfies

λ 2 n 4(n -1) µ 1 , (4)
where µ 1 is the first eigenvalue of the Yamabe operator given by

L := 4 n -1 n -2 g +S g ,
g is the Laplacian acting on functions. In dimension 2, C. Bär [START_REF] Bär | Lower eigenvalue estimates for Dirac operators[END_REF] proved that any eigenvalue of the Dirac operator on M satisfies

λ 2 2πχ(M ) Area(M, g) , (5)
where χ(M ) is the Euler-Poincaré characteristic of M . The limiting case of ( 4) and ( 5) is also characterized by the existence of a real Killing spinor. In terms of the Energy-Momentum tensor, O. Hijazi [START_REF] Hijazi | Lower bounds for the eigenvalues of the Dirac operator[END_REF] proved that, on such manifolds any eigenvalue of the Dirac operator satisfies the following

λ 2      1 4 µ 1 + inf M | ψ | 2 if n 3, πχ(M ) Area(M,g) + inf M | ψ | 2 if n = 2. ( 6
)
Again, the trace of ψ being equal to λ, Inequality (6) improves Inequalities (4) and ( 5). The limiting case of ( 6) is characterized by the existence of a spinor field ϕ satisfying for all X ∈ Γ(T M ),

∇ X ϕ = -ϕ (X) • ϕ, (7)
where ϕ = e -n-1 2 u ψ, the spinor field ψ is an eigenspinor associated with the first eigenvalue of the Dirac operator and ψ is the image of ψ under the isometry between the spinor bundles of (M n , g) and (M n , g = e 2u g). Suppose that on a spin manifold M , there exists a spinor field φ such that for all X ∈ Γ(T M ),

∇ X φ = -E(X) • φ, (8)
where E is a symmetric 2-tensor defined on T M . It is easy to see that E must be equal to φ . If the dimension of M is equal to 2, Th. Friedrich [START_REF] Friedrich | On the spinor representation of surfaces in Euclidean 3spaces[END_REF] proved that the existence of a pair (φ, E) satisfying ( 8) is equivalent to the existence of a local immersion of M into the euclidean space R 3 with Weingarten tensor equal to E. In [START_REF] Morel | Tenseur d'impulsion-énergie et géométrie spinorielle extrinsèque[END_REF], B. Morel showed that if M n is a hypersurface of a manifold N carrying a parallel spinor, then the Energy-Momentum tensor (associated with the restriction of the parallel spinor) appears, up to a constant, as the second fundamental form of the hypersurface. G. Habib [START_REF] Habib | Energy-Momentum tensor on foliations[END_REF] studied Equation (8) for an endomorphism E not necessarily symmetric. He showed that the symmetric part of E is φ and the skew-symmetric part of E is q φ defined on the complement set of zeroes of φ by

g(q φ (X), Y ) = 1 2 Re < Y • ∇ X φ -X • ∇ Y φ, φ |φ| 2 >,
for all X, Y ∈ Γ(T M ). Then he modifies the connection in the direction of ψ + q ψ where ψ is an eigenspinor associated with an eigenvalue λ and gets that

λ 2 inf M ( 1 4 S g + | ψ | 2 + |q ψ | 2 ). ( 9
)
The Heisenberg group and the solvable group are examples of limiting manifolds [START_REF] Habib | Energy-Momentum tensor on foliations[END_REF]. For a better understanding of the tensor q φ , he studied Riemannian flows and proved that if the normal bundle carries a parallel spinor, the tensor q φ plays the role of the O'Neill tensor of the flow. Here we prove the corresponding inequalities for Spin c manifolds:

Theorem 1.1 Let (M n , g) be a compact Riemannian Spin c manifold of dimension n 2, and denote by iΩ the curvature form of the connection A on the S 1 -principal fibre bundle (S 1 M, π, M ). Then any eigenvalue of the Dirac operator to which is attached an eigenspinor ψ satisfies

λ 2 inf M 1 4 S g - c n 4 |Ω| g + | ψ | 2 + |q ψ | 2 , ( 10
)
where c n = 2[ n 2 ]
1 2 and |Ω| g is the norm of Ω with respect to g.

In this paper, we only consider the deformation of the connection in the direction of the symmetric endomorphism φ and hence under the same conditions as Theorem 1.1, one gets

λ 2 inf M 1 4 S g - c n 4 |Ω| g + | ψ | 2 . (11)
In 1999, A. Moroianu and M. Herzlich [START_REF] Herzlich | Generalized Killing spinors and conformal eigenvalue estimates for Spin c manifold[END_REF] proved that on Spin c manifolds of dimension n 3, any eigenvalue of the Dirac operator satisfies

λ 2 λ 2 1 := n 4(n -1) µ 1 , (12)
where µ 1 is the first eigenvalue of the perturbed Yamabe operator defined by

L Ω = L -c n |Ω| g .
The limiting case of ( 12) is characterized by the existence of a real Killing spinor ψ satisfying Ω • ψ = i cn 2 |Ω| g ψ. In terms of the Energy-Momentum tensor we prove: Theorem 1.2 Under the same conditions as Theorem 1.1, any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies

λ 2      1 4 µ 1 + inf M | ψ | 2 if n 3, πχ(M ) Area(M,g) -1 2 R M |Ω|gvg Area(M,g) + inf M | ψ | 2 if n = 2, ( 13
)
where µ 1 is the first eigenvalue of the perturbed Yamabe operator.

Using the Cauchy-Schwarz inequality in dimension n 3, we have that Inequality [START_REF] Hebey | Introduction à l'analyse non linéaire sur les variétés[END_REF] implies Inequality [START_REF] Habib | Energy-Momentum tensor on foliations[END_REF]. As a corollary of Theorem 1.2, we compare the lower bound to a conformal invariant (the Yamabe number) and to a topological invariant, in case of 4-dimensional manifolds whose associated line bundle has self dual curvature (see Corollary (4.1) and Corollary (4.2)). Finally, we study the limiting case of ( 11) and ( 13), and we give an example.

Even though the number inf M | ψ | 2 is not a nice geometric invariant, it appears naturally in some situations. For example, on hypersurfaces of certain limiting Spin c manifolds it is easy to see, with the help of the Spin c Gauss formula, that it is precisely the second fundamental form. Also, when deforming the Riemannian metric in the direction of the Energy-Momentum tensor, the eigenvalues of the Dirac operator on a Spin c manifold are then critical (see [START_REF] Nakad | The Energy-Momentum tensor on Spin c manifolds[END_REF]). The author would like to thank Oussama Hijazi for his support and encouragements.

Spin c geometry and the Dirac operator

In this section, we briefly introduce basic notions concerning Spin c manifolds and the Dirac operator. Details can be found in [START_REF] Friedrich | Dirac operator's in Riemannian Geometry[END_REF], [START_REF] Lawson | Spin Geometry[END_REF] and [START_REF] Moroianu | Parallel and Killing spinors on Spin c manifolds[END_REF].

Let (M n , g) be a compact connected oriented Riemannian manifold of dimension n 2 without boundary. Furthermore, let SOM be the SO n -principal bundle over M of positively oriented orthonormal frames. A Spin c structure of M is a Spin c n -principal bundle (Spin c M, π, M ) and a S 1 -principal bundle (S 1 M, π, M ) together with a double covering given by θ : Let Σ c M := Spin c M × ρn Σ n be the associated spinor bundle where

Spin c M -→ SOM × M S 1 M such that θ(ua) = θ(u)ξ(a),
Σ n = C 2 [ n 2 ]
and ρ n : Spin c n -→ End(Σ n ) the complex spinor representation. A section of Σ c M will be called a spinor and the set of all spinors will be denoted by Γ(Σ c M ). The spinor bundle Σ c M is equipped with a natural Hermitian scalar product, denoted by < ., . > and satisfies

< X • ψ, ϕ >= -< ψ, X • ϕ > for every X ∈ Γ(T M ) and ψ, ϕ ∈ Γ(Σ c M ),
where X • ψ denotes the Clifford multiplication of X and ψ. With this Hermitian scalar product we define an L 2 -scalar product

(ψ, φ) = M < ψ, φ > v g ,
for any spinors ψ and φ. Additionally, given a connection 1-form A on S 1 M , A : T (S 1 M ) -→ iR and the connection 1-form ω M on SOM for the Levi-Civita connection ∇ M , induce a connection on the principal bundle SOM × M S 1 M , and hence a covariant derivative ∇ on Γ(Σ c M ) [START_REF] Friedrich | Dirac operator's in Riemannian Geometry[END_REF], given by

∇ e i ψ = b, e i (σ) + 1 4 n j=1 e j • ∇ M e i e j • σ + 1 2 A(s * (e i ))σ , (14)
where ψ = [b, σ] is a locally defined spinor field, (e 1 , . . . , e n ) is a local oriented orthonormal tangent frame and s :

U -→ S 1 M is a local section of S 1 M .
The curvature of A is an imaginary valued 2-form denoted by F A = dA, i.e., F A = iΩ, where Ω is a real valued 2-form on S 1 M . We know that Ω can be viewed as a real valued 2-form on M [START_REF] Friedrich | Dirac operator's in Riemannian Geometry[END_REF]. In this case iΩ is the curvature form of the associated line bundle L. It's the complex line bundle associated with the S 1 -principal bundle via the standard representation of the unit circle. The spinorial curvature R associated with the connection ∇, is given by

R X,Y = 1 4 n i,j=1 g R X,Y e i , e j e i • e j • + i 2 Ω(X, Y ).
In the Spin c case, the Ricci identity translates to

j e j • R e j ,X ψ = 1 2 Ric(X) • ψ - i 2 (X Ω) • ψ, (15)
where denotes the interior product. For every spinor ψ, the Dirac operator is locally defined by

Dψ = n i=1 e i • ∇ e i ψ.
It is an elliptic, self-adjoint operator with respect to the L 2 -scalar product and verifies the Schrödinger-Lichnerowicz formula

D 2 = ∇ * ∇ + 1 4 S g Id Γ(Σ c M ) + i 2 Ω•,
where Ω• is the extension of the Clifford multiplication to differential forms given by (e * i ∧ e * j ) • ψ = e i • e j • ψ.

Eigenvalue estimates on Spin c manifolds

In this section, we prove the lower bound [START_REF] Ginoux | A spectral estimate for the Dirac operator on Riemannian flows[END_REF]. This proof is based on the following Lemma given by A. Moroianu and M. Herzlich in [START_REF] Herzlich | Generalized Killing spinors and conformal eigenvalue estimates for Spin c manifold[END_REF]:

Lemma 3.1 [START_REF] Herzlich | Generalized Killing spinors and conformal eigenvalue estimates for Spin c manifold[END_REF]. Let (M n , g) be a Spin c manifold. For any spinor ψ ∈ Γ(Σ c M ) and a real 2-form Ω, we have

< iΩ • ψ, ψ > - c n 2 |Ω| g |ψ| 2 , ( 16
)
where |Ω| g is the norm of Ω, with respect to g given by |Ω| 2 g = i<j (Ω ij ) 2 , in any orthonormal local frame. Moreover, if equality holds in [START_REF] Hijazi | Première valeur propre de l'opérateur de Dirac et nombre de Yamabe[END_REF], then

Ω • ψ = i c n 2 |Ω| g ψ. (17)
Proof of Theorem 1.1: Let E (resp. Q) be a symmetric (resp. skewsymmetric) 2-tensor defined on T M . For any spinor field φ, the modified connection

∇ X φ := ∇ X φ + E(X) • φ + Q(X) • φ, satisfies | ∇φ| 2 = |∇φ| 2 -|E| 2 |φ| 2 -|Q| 2 |φ| 2 .
After integration on M , the Schrödinger-Lichnerowicz formula gives

M | ∇φ| 2 v g = M |Dφ| 2 v g - M 1 4 S g |φ| 2 v g - M (|E| 2 + |Q| 2 )|φ| 2 v g - M < i 2 Ω • φ, φ > v g .
Let ψ be an eigenspinor corresponding to the eigenvalue λ of D. For E = ψ , Q = q ψ and by Lemma 3.1, it follows

λ 2 M |ψ| 2 v g 1 4 M S g |ψ| 2 v g + M (| ψ | 2 + |q ψ | 2 )|ψ| 2 v g + M < i 2 Ω • ψ, ψ > v g M 1 4 S g - c n 4 |Ω| g + | ψ | 2 + |q ψ | 2 |ψ| 2 v g .
Finally,

λ 2 inf M 1 4 S g - c n 4 |Ω| g + | ψ | 2 + |q ψ | 2 .

Conformal geometry and eigenvalue estimates

Before proving Theorem 1.2, we give some basic facts on conformal Spin c geometry. The conformal class of g is the set of metrics g = e 2u g, for a real function u on M . At a given point x of M , we consider a g-orthonormal basis {e 1 , . . . , e n } of T x M . The corresponding g -orthonormal basis is denoted by {e 1 = e -u e 1 , . . . , e n = e -u e n } . This correspondence extends to the Spin c level to give an isometry between the corresponding spinor bundles. We put a " " above every object which is naturally associated with the metric g, except for the scalar curvature where S g (resp. S u or S h ) denotes the scalar curvature associated with the metric g (resp. g = e 2u g = h 4 n-2 g). Then, for any spinor fields ψ and ϕ, one has < ψ, ϕ >=< ψ, ϕ > , where < ., . > denotes the natural Hermitian scalar products on Γ(Σ c M ), and on Γ(Σ c M ). The corresponding Dirac operators satisfy

D ( e -(n-1) 2 u ψ ) = e -(n+1) 2 u Dψ.
The norm of any real 2-form Ω with respect to g and g are related by

|Ω| g = e -2u |Ω| g .
O. Hijazi [START_REF] Hijazi | Lower bounds for the eigenvalues of the Dirac operator[END_REF] showed that on a spin manifold the Energy-Momentum tensor verifies

| ϕ | 2 = e -2u | ϕ | 2 = e -2u | ψ | 2 ,
where ϕ = e -(n-1) 2 u ψ. We extend the result to a Spin c manifold and get the same relation.

n-2 -c n |Ω| g . For any positive function H, we write f H = h, where f is a positive function, and refering to [START_REF] Hijazi | Première valeur propre de l'opérateur de Dirac et nombre de Yamabe[END_REF] we get

µ 1 = (H -1 LH)f 2 H 2 v g -c n M |Ω| g f 2 H 2 v g + M H 2 |df | 2 v g . Finally, µ 1 inf M (H -1 L Ω H) = inf M (S v e 2v -c n |Ω| g ),
where

e 2v = H 4 n-2 , then µ 1 = sup u inf M (S u e 2u -c n |Ω| g ).
For n = 2 and for every u we have S u e 2u = S g + 2 g u. The Stokes and Gauβ-Bonnet theorems yield inf

M (S u e 2u -2|Ω| g ) M S u e 2u -2|Ω| g v g Area(M, g) = 4πχ(M ) -2 M |Ω| g v g Area(M, g) .
Let u 0 be a solution of the following equation [START_REF] Aubin | Nonlinear Analysis on Manifolds, Monge-Ampre Equations[END_REF] 2

g u = M (S g -2|Ω| g )v g Area(M, g) -S g + 2|Ω| g , (21)
hence,

S u 0 e 2u 0 -2|Ω| g = 2 g u 0 + S g -2|Ω| g = 4πχ(M ) -2 M |Ω| g v g Area(M, g) .
Proof of Theorem 1.2: Combining Lemma 4.2 and Lemma 4.1, Theorem 1.2 follows.

Remark 4.1 Inequality (11) improves Inequality [START_REF] Habib | Energy-Momentum tensor on foliations[END_REF], which itself implies the Friedrich Spin c inequality given by

λ 2 n 4(n -1) inf M (S g -c n |Ω| g ). ( 22
)
Equality holds in [START_REF] Morel | Tenseur d'impulsion-énergie et géométrie spinorielle extrinsèque[END_REF] if and only if equality holds in [START_REF] Habib | Energy-Momentum tensor on foliations[END_REF], i.e., if and only if the eigenspinor ψ associated with the first eigenvalue of D is a real Killing spinor and

Ω • ψ = i cn 2 |Ω| g ψ.
Corollary 4.1 Any eigenvalue of the Dirac operator on a compact Riemannian Spin c manifold of dimension n 3, satisfies

λ 2 1 4 vol(M, g) -2 n Y (M, [g]) -c n Ω n 2 + inf M | ψ | 2 ,
where Y (M, [g]) is the Yamabe number given by

Y (M, [g]) = inf η =0 M 4 n-1 n-2 |dη| 2 + S g η 2 M |η| 2n n-2 n-2 n
.

Proof : Using the Hölder inequality, it follows

µ 1 = inf η =0 M 4 n-1 n-2 |dη| 2 + (S g -c n |Ω| g )η 2 M η 2 inf η =0 M 4 n-1 n-2 |dη| 2 + (S g -c n |Ω|)η 2 M |η| 2n n-2 n-2 n vol(M, g) 2 n
.

Using the Hölder inequality again, we deduce

µ 1 vol(M, g) 2 n inf η =0 M 4 n-1 n-2 |dη| 2 + Sη 2 M |η| 2n n-2 n-2 n -c n M |Ω| n 2 2 n = Y (M, [g])-c n Ω n 2 .
Finally, replacing in (13), we get the result.

Corollary 4.2 On a compact 4-dimensional Spin c manifold with self-dual curvature form iΩ, any eigenvalue of the Dirac operator satisfies

λ 2 1 4 vol(M, g) -1 2 Y (M, [g]) -4π √ 2 c 1 (L) 2 + inf M | ψ | 2 ,
where c 1 (L) is the Chern number of the line bundle L associated with the Spin c structure.

Proof: It follows directly from Corollary 4.1 and the fact that if n = 4 and Ω self-dual, then M |Ω| 2 g v g = 4π 2 c 1 (L) 2 (see [START_REF] Friedrich | Dirac operator's in Riemannian Geometry[END_REF]).

Equality case

In this section, we study the limiting case of ( 11) and ( 13). An example is then given.

Proposition 5.1 Under the same conditions as Theorem 1.1,

Equality in (11) holds ⇐⇒ ∇ X ψ = -ψ (X) • ψ, Ω • ψ = i cn 2 |Ω| g ψ, (23)
for any X ∈ Γ(T M ) and where ψ is an eigenspinor associated with the first eigenvalue of the Dirac operator.

Proof: If equality in [START_REF] Gursky | Yamabe invariants and Spin c structure[END_REF] is achieved, the two conditions follow directly. Now, suppose that ∇ X ψ = -ψ (X)•ψ and Ω•ψ = i cn 2 |Ω| g ψ. The condition ∇ X ψ = -ψ (X)•ψ implies that |ψ| 2 is constant. Denoting by R The curvature tensor on the Spin c bundle associated with the connection ∇, one easily gets the following relation

R X,Y ψ + d ψ (X, Y ) • ψ + [ ψ (X), ψ (Y )] • ψ = 0,
where d ψ is a 2-form with values in Γ(T M ) given by

d ψ (X, Y ) = (∇ X ψ )Y -(∇ Y ψ )X.
Taking Y = e j and performing its Clifford multiplication by e j yields by the Ricci identity (15) on a Spin c manifold

- 1 2 Ric(X) • ψ + i 2 (X Ω) • ψ + j e j • d ψ (X, e j ) • ψ + j e j • [ ψ (X), ψ (e j )] • ψ = 0. ( 24
)
We then decompose the last two terms in [START_REF] Montiel | Using spinors to study submanifolds[END_REF] using that X •α = X ∧α-X α for any form α, it follows

j e j • d ψ (X, e j ) • ψ = j [e j ∧ d ψ (X, e j )] • ψ -[X(tr ψ ) + div ψ (X)]ψ. j e j • [ ψ (X), ψ (e j )] • ψ = 2 (tr ψ ) ψ (X) • ψ -2 j g(X, ψ (e j )) ψ (e j ) • ψ.
Taking the scalar product of (24) with ψ, and after seperating real and imaginary parts, yields for every vector field X the relation

X(tr ψ ) + div ψ (X) |ψ| 2 = i 2 < (X Ω) • ψ, ψ > . (25)
But since Equality (17) holds we compute

< (X Ω) • ψ, ψ > = < (X ∧ Ω) • ψ, ψ > -< X • Ω • ψ, ψ > = < (X ∧ Ω) • ψ, ψ > -i n 2 1 2 |Ω| g < X • ψ, ψ > .
After separating real and imaginary parts, < (X Ω) • ψ, ψ > must vanish. Using this and n j=1 e j • (e j Ω) = 2Ω, Clifford multiplication of ( 24) with e k , and for X = e k , gives

- 1 2 S g ψ -iΩ • ψ = k,j e j • (e k ∧ d ψ (e j , e k )) • ψ -2(tr ψ ) 2 ψ + 2| ψ | 2 ψ.
An easy computation implies that k,j e j • (e k ∧ d ψ (e j , e k )) • ψ = 0, hence

- 1 2 S g + n 2 1 2 |Ω| g = -2(tr ψ ) 2 + 2| ψ | 2 , (26)
which implies Equality in [START_REF] Gursky | Yamabe invariants and Spin c structure[END_REF].

Proposition 5.2 On a compact Riemannian Spin c manifold (M n , g) of dimension n 3, assume that the first eigenvalue λ 1 of the Dirac operator to which is attached an eigenspinor ψ satisfies the equality case in [START_REF] Hebey | Introduction à l'analyse non linéaire sur les variétés[END_REF]. Then, | ψ | is constant and if h > 0 denotes an eigenfunction of the Yamabe operator corresponding to µ 1 , then for any vector field X g(X, ψ (dh) -

λ 1 dh) = g(λ 1 X -ψ (X), dh) = 0. (27)
Proof: If n 3 and equality holds in [START_REF] Hebey | Introduction à l'analyse non linéaire sur les variétés[END_REF], we consider the positive function v > 0 defined by e 2v = h 4 n-2 where h is an eigenfunction of the Yamabe operator corresponding to µ 1 . Inequality [START_REF] Kim | The Einstein-Dirac equation on Riemannian spin manifolds[END_REF]

with u = v gives | ψ | is constant, ∇ X ϕ = -ϕ (X) • ϕ and Ω • ϕ = i cn
2 |Ω| g ϕ. By Proposition 5.1, Equality ( 26) and ( 25) can be considered for the conformal metric g = e 2v g = h 4 n-2 g to get

(tr ϕ ) 2 := f 2 = 1 4 S v - c n 4 |Ω| g + | ϕ | 2 , grad f = -div ϕ .
It is straightforward to see that these two equalities give (27).

Example: If the lower bound ( 22) is achieved, automatically equality holds in [START_REF] Gursky | Yamabe invariants and Spin c structure[END_REF]. Here we will give an example where equality holds in [START_REF] Gursky | Yamabe invariants and Spin c structure[END_REF] but not in [START_REF] Morel | Tenseur d'impulsion-énergie et géométrie spinorielle extrinsèque[END_REF]. Let (M 3 , g) = (S 3 , can) be endowed with its unique spin structure and consider a real Killing spinor ψ with Killing constant 1 2 . As the norm of ψ is constant, we may suppose that |ψ| = 1. Let ξ be the Killing vector field on M defined by ig(ξ, X) =< X • ψ, ψ > .

In [START_REF] Herzlich | Generalized Killing spinors and conformal eigenvalue estimates for Spin c manifold[END_REF], it is shown that: Let h be a real constant such that h > 1. We define the metric g h on M , by:

1. idξ(X, Y ) = -< X ∧ Y • ψ, ψ > for any X, Y ∈ Γ(T M ).
g h (ξ, X) = g(ξ, X) pour tout X ∈ Γ(T M ), g h (X, Y ) = h -2 g(X, Y ) pour X, Y ⊥ ξ.
Using the following isomorphism

(T M, g) -→ (T M, g h ) Z -→ Z h = Z si Z = ξ, hZ si Z ⊥ ξ, if u = {ξ, e 1 , e 2 } is a positive local g-orthonormal frame defined in a neigh- borhood U of x, then u h = {ξ h = ξ, e h 1 = he 1 , e h 2 = he 2 } is a positive local g h -orthonormal frame defined in a neighborhood U of x.
There exists an isomorphism of vector bundles (see [START_REF] Herzlich | Generalized Killing spinors and conformal eigenvalue estimates for Spin c manifold[END_REF]) given by:

Σ g M -→ Σ g h M ψ = [ũ, φ] -→ ψ h = [ũ h φ], satisfying, < ψ 1 , ψ 2 > ΣgM = < ψ h 1 , ψ h 2 > Σ g h M and (X•ψ) h = X h •ψ h for any X ∈ Γ(T M ).
The covariant derivative of the spinor ψ h = [ũ h , φ] is given by (see [START_REF] Herzlich | Generalized Killing spinors and conformal eigenvalue estimates for Spin c manifold[END_REF]):

∇ h X h ψ h = h 2 2 X h • ψ h + i((1 -h 2 )ξ)(X h )ψ h .
Let α = (1 -h 2 )ξ be a 1-form on M . We may view iα as a connection 1-form on the trivial S 1 bundle. Let L = M × C be the induced trivial line bundle over M . We denote by σ the global section of L and by ∇ 0 the covariant derivative on L induced by the above connection. It satisfies ∇ 0 X σ = iα(X)σ, for any X ∈ Γ(T M ).

On the twisted bundle Σ g h M ⊗ L, we consider the connection ∇ = ∇ h ⊗ ∇ 0 and we calculate

∇ e h 1 (ψ h ⊗ σ) = h 2 2 e h 1 • (ψ h ⊗ σ), ∇ e h 2 (ψ h ⊗ σ) = h 2 2 e h 2 • (ψ h ⊗ σ), ∇ ξ (ψ h ⊗ σ) = ( -3h 2 2 + 2)ξ • (ψ h ⊗ σ).
The spinor ψ h ⊗ σ is a section of Σ g h M ⊗ L, which is, of course, the spinor bundle associated to the Spin c structure with auxiliary line bundle L 2 . It is easy to see that ψ h ⊗ σ is an eigenspinor associated with the eigenvalue

h 2
2 -2, and it is clear that ψ h ⊗ σ is not a real Killing spinor since h = 1, so (M, g h ) not is a limiting manifold for the Friedrich Spin c inequality. But it is a limiting manifold for the lower bound [START_REF] Gursky | Yamabe invariants and Spin c structure[END_REF], in fact we will prove that (23) holds.

The complex 2-form idα is the curvature form associated with the connection ∇ 0 on L. We have:

dα • (ψ h ⊗ σ) = (1 -h 2 )dξ • (ψ h ⊗ σ) = i(h 2 -1)h 2 ψ h ⊗ σ.
The norm of dα with respect to the metric g h is given by

|dα| 2 g h = (1 -h 2 ) 2 |dξ| 2 g h = (1 -h 2 ) 2 (dξ(e h 1 , e h 2 )) 2 = h 4 (1 -h 2 ) 2 .
Since h > 1, |dα| g h = h 2 (h 2 -1), then the second equation of ( 23) is verified. Futhermore, it is easy to check that T ψ h ⊗σ (e h 1 ) = T ψ h ⊗σ (e h 2 ) = g h ( ψ h ⊗σ (e h 1 ), e h 1 ) = g h ( ψ h ⊗σ (e h 2 ), e h 2 ) = -h 2 2 , g h ( ψ h ⊗σ (e h 1 ), ξ) = g h ( ψ h ⊗σ (e h 2 ), ξ) = g h ( ψ h ⊗σ (e h 1 ), e h 2 ) = 0,

T ψ h ⊗σ (ξ) = g h ( ψ h ⊗σ (ξ), ξ) = 3h 2 2 -2.
Finally, it is straightforward to verify that the first equation of ( 23) holds:

-ψ h ⊗σ (e h 1 ) • (ψ h ⊗ σ) = h 2 2 e h 1 • (ψ h ⊗ σ) = ∇ e h 1 (ψ h ⊗ σ), -ψ h ⊗σ (e h 2 ) • (ψ h ⊗ σ) = h 2 2 e h 2 • (ψ h ⊗ σ) = ∇ e h 2 (ψ h ⊗ σ), -ψ h ⊗σ (ξ) • (ψ h ⊗ σ) = ( -3h 2 2 + 2)ξ • (ψ h ⊗ σ) = ∇ ξ (ψ h ⊗ σ).

  for every u ∈ Spin c M and a ∈ Spin c n , where ξ is the 2-fold covering of Spin c n over SO n × S 1 . A Riemannian manifold that admits a Spin c structure is called a Riemannian Spin c manifold.

2. d|ξ| 2 =

 2 -2dξ(ξ, .) = -2g(∇ ξ ξ, .) -2∇ ξ ξ = 0. 3. ξ • ψ = iψ and |ξ| = 1. 4. ξ•ψ = -e 1 •e 2 •ψ, where {ξ/|ξ|, e 1 , e 2 } is an oriented local orthonormal frame.

Lemma 4.1 Under the same conditions as Theorem 1.1, any eigenvalue λ of the Dirac operator to which is attached an eigenspinor ψ satisfies

Proof: For any spinor field φ and for any symmetric 2-tensor E defined on T M , the modified connection introduced in [START_REF] Hijazi | Lower bounds for the eigenvalues of the Dirac operator[END_REF]:

Using the Schrödinger-Lichnerowicz formula on M , applied to the spinor field φ with respect to the metric g, yields

For the spinor ϕ = e -(n-1) 2 u ψ with Dψ = λψ, one gets D ϕ = λe -u ϕ, and hence by Lemma 3.1 and for

Lemma 4.2 Let (M n , g) be a compact Riemannian Spin c manifold of dimension n 2 and S g (resp. S u or S h ) the scalar curvature associated with the metric g (resp. g = e 2u g = h 4 n-2 g). The 2-form iΩ denotes the curvature form on the S 1 -principal bundle associated with the Spin c structure. We have

where µ 1 is the first eigenvalue of the perturbed Yamabe operator L Ω .

Proof: For n 3, let h > 0 be an eigenfunction of L Ω associated with the eigenvalue µ 1 such that M h 2 v g = 1. For a conformal metric g = e 2u g = h