MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS

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1 MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT Abstract. We rove that the mass endomorhism associated to the Dirac oerator on a Riemannian manifold is non-zero for generic Riemannian metrics. The roof involves a study of the mass endomorhism under surgery, its behavior near metrics with harmonic sinors, and analytic erturbation arguments.. Introduction Let (M, g) be a comact Riemannian sin manifold. We always assume that a sin manifold comes equied with a choice of orientation and sin structure. Assume that the metric g is flat in a neighborhood of a oint M and has no harmonic sinors. Then the Green s function G g at for the Dirac oerator D g exists. The constant term in the exansion of G g at is an endomorhism of Σ M called the mass endomorhism. The terminology is motivated by the analogy to the ADM mass which is the constant term in the Green s function of the Yamabe oerator. The non-nullity of the mass endomorhism has many interesting consequences. In articular, combining the results resented here with inequalities in [7] and [4], one obtains a solution of the Yamabe roblem. Finding examles for which the mass endomorhism does not vanish is then a natural roblem. In [2], see also [3], it is roven that for a generic metric on a manifold of dimension 3, the mass endomorhism does not vanish in a given oint. The aim of this aer is to extend this result to all dimensions at least 3, see Theorem Definitions and main result The goal of this section is to give a recise statement of the main results. At first, the mass endomorhism is defined. Then, in Subsection 2.2, we define suitable sets of metrics to work with. Further, in Subsection 2.3, we exlain some well known facts on the α-genus. Finally, in Subsection 2.4 we state Theorem 2.4, which is the main result of this article. 2.. Mass endomorhism. In this section we will recall the mass endomorhism introduced in [7]. Let (M, g) be a comact Riemannian sin manifold of dimension n 2 and let M. Assume that the metric g is flat in a neighborhood of and that the Dirac oerator D g is invertible. The Green s function G g (, ) = G g ( ) of D g at is defined by D g G g = δ Id ΣM, Date: November 9, 200. Key words and hrases. Dirac oerator, mass endomorhism, surgery MSC C27 (rimary), 57R65, 58J05, 58J60 (secondary).

2 2 BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT where δ is the Dirac distribution at and G g is viewed as a linear ma which associates to each sinor in Σ M a smooth sinor field on M \ {}. The distributional equation satisfied by G g should be interreted as G g (x)ψ 0, D g ϕ(x) dv g (x) = ψ 0, ϕ() M for any ψ 0 Σ M and any smooth sinor field ϕ. Let ξ denote the flat metric on R n, it then holds that G ξ ψ = ω n x n x ψ. at = 0, where ω n is defined as the volume of S n. The following Proosition is roved in [7]. Proosition 2.. Let (M, g) be a comact Riemannian sin manifold of dimension n 2. Assume that g is flat on a neighborhood U of a oint M. Then, for ψ 0 Σ M we have G g (x)ψ 0 = ω n x n x ψ 0 + v g (x)ψ 0, where the sinor field v g (x)ψ 0 satisfies D g (v g (x)ψ 0 ) = 0 in a neighborhood of. This allows us to define the mass endomorhism. Definition 2.2. The mass endomorhism α g : Σ M Σ M for a oint U M is defined by α g (ψ 0 ) := v g ()ψ 0. In articular, we have α g (ψ 0 ) = lim x 0 ( G g (x)ψ 0 + ) ω n x n x ψ 0. The mass endomorhism is thus (u to a constant) defined as the zero order term in the asymtotic exansion of the Green s function in normal coordinates around Metrics flat around a oint. Let M be a connected sin manifold, U where U is an oen subset of M. A Riemannian metric on U will be called extendible if it ossesses a smooth extension to a (not necessarily flat) Riemannian metric on M. Fix a flat extendible metric g flat on U. The set of all smooth extensions of g flat is denoted by R U,gflat (M) := {g g is a metric on M such that g U = g flat }. Inside this set of metrics we study those with invertible Dirac oerator R inv (M) := {g R U,gflat (M) D g is invertible}. The main subject of the article is the set R 0, (M) := {g R inv (M) the mass endomorhism at is not 0}. Note that R inv (M) can be emty (see Subsection 2.3). We say that a subset A R U,gflat (M) is generic in R U,gflat (M) if it is oen in the C -toology and dense in the C -toology in R U,gflat (M).

3 MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS The α-genus. The α-genus is a ring homomorhism α : Ω sin (t) KO (t) where Ω sin (t) is the sin bordism ring and KO (t) is the ring of coefficients for KO-theory. In articular, the well-definedness of the ma means that the α-genus α(m) of a sin manifold M deends only on its sin bordism class, and the homomorhism roerty means that it is additive with resect to the disjoint union and multilicative with resect to the roduct of sin manifolds. We recall that if the dimension of M is n then α(m) KO n (t) and as grous we have Z if n 0 mod 4; KO n (t) = Z/2Z if n, 2 mod 8; 0 otherwise. Let (M, g) be a comact Riemannian sin manifold. The Atiyah-Singer index theorem states that the Clifford index of D g coincides with α(m), see [6]. This imlies that a manifold M with α(m) 0 cannot have a metric with invertible Dirac oerator. If M is not connected, one can aly the argument in each connected comonent. Thus there are many non-connected examles M, with α(m) = 0, but admitting no metric with invertible Dirac oerator. However, the converse holds true under the additional assumtion that M is connected, see [5]. The roof of the converse relies on a surgery construction reserving invertibility of the Dirac oerator together with Stolz s examles of manifolds with ositive scalar curvature in every sin bordism class [20]. Secial cases were roved reviously in [8] and [8]. For our uroses, it is more convenient to use a slightly stronger version, resented in [4]: Theorem 2.3. Let M be a connected comact sin manifold and let M. Let U be an oen subset of M, U M, and let g flat be a flat extendible metric on U. Then R inv (M) if and only if α(m) = 0. Using real analyticity one obtains that R inv (M) is oen and dense in R U,gflat (M) Main result. The main result of this aer is the following: If α(m) = 0, so that the mass endomorhism is defined for metrics in the non-emty set R inv (M), then a generic metric has a non-zero mass endomorhism. Theorem 2.4. Let M be a comact connected n-dimensional sin manifold with n 3 and with vanishing α-genus. Let M and assume that g flat is an extendible metric which is flat around. Then there exists a neighborhood U of for which R 0, (M) is generic in R U,gflat (M). Theorem 2.4 will follow from Theorems 4. and 7. below The relation to the ADM mass. Let (M, g) be a comact sin manifold of dimension n 3. Assume that g is flat in a neighborhood U of a oint M. The conformal Lalacian is then defined by L g 4(n ) := n 2 g + scal g, where g is the non-negative Lalacian and where scal g is the scalar curvature of the metric g. As for the Dirac oerator D g, we say that a function H g L (M) C (M \ {}) is the Green s function for L g if L g H g = δ

4 4 BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT in the sense of distributions. Assume that the metric g is conformal to a metric with ositive scalar curvature. Then it is well known (see for instance [7]) that the Green s function H g of L g exists, is ositive everywhere and has the following exansion at : H g (x) = 4(n )ω n d g (x, ) n 2 + Ag + o(x), where A g R and o(x) is a smooth function with o() = 0. Set M = M \ {} and g = H 4 n 2 g. Schoen [9] observed that the comlete non-comact manifold ( M, g) is asymtotically flat and its ADM mass is a n A g, where a n > 0 deends only on n. We recall that an asymtotically flat manifold, if interreted as a time symmetric sacelike hyersurface of a Lorentzian manifold, is obtained by considering an isolated system at a fixed time in general relativity. The ADM mass gives the total energy of this system. With this remark, the number A g is often called the mass of the comact manifold (M, g). By analogy, the oerator α g (), which is by construction the sin analog of A g, is called the mass endomorhism of (M, g) at. We will also see in Subsection 2.6 that the mass endomorhism lays the same role as the number A g in a Dirac oerator version of the Yamabe roblem Conclusions of non-zero mass. In this Subsection we will summarize why we are interested in metrics with non-zero mass endomorhism. Let (M, g) be a comact Riemannian sin manifold of dimension n 2. For a metric g in the conformal class [g] of g, let λ ( g) be the eigenvalue of the Dirac oerator D g with the smallest absolute value (it may be either ositive or negative). We define λ + min (M, [g]) = inf λ( g) Vol g (M) /n. g [g] For this conformal invariant λ + min (M, [g]) it was roven in [, 2] and [6] that The strict inequality has several alications, see [3, 6, 7]: 0 < λ + min (M, [g]) λ+ min (Sn ) = n 2 ω/n n. λ + min (M, [g]) < n 2 ω/n n () Inequality () imlies that the invariant λ + min (M, [g]) is attained by a generalized metric, that is, a metric of the form f 2/(n ) g where f C 2 (M) can have some zeros; Inequality () gives a solution of a conformally invariant artial differential equation which can be read as a nonlinear eigenvalue equation for the Dirac oerator, a tye of Yamabe roblem for the Dirac oerator; using Hijazi s inequality [4] one obtains a solution of the standard Yamabe roblem which consists of finding a metric with constant scalar curvature in the conformal class of g in the case of n 3. The first two alications can for several reasons be interreted as a sin analog of the Yamabe roblem, see []. The third alication says that a non-zero mass endomorhism can be used in the Yamabe roblem instead of the ositivity of the mass A g defined in Subsection 2.5.

5 MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS 5 Now, let us reconnect to the subject of this aer. In [7], we rove that a non-zero mass endomorhism imlies Inequality (). In articular we see with Theorem 2.4 that Inequality () holds for generic metric in R U,gflat (M). As a consequence, for generic metrics in R U,gflat (M), we have all the alications stated above. This can be comared to the Yamabe roblem: Schoen roved that the ositivity of the number A g, that is the mass of (M, g) defined in Subsection 2.5, imlies a solution of the standard Yamabe roblem. The ositive mass theorem imlies that A g 0. Hence, we get a solution of the Yamabe roblem as soon as A g 0. In articular, the mass endomorhism lays the same role in the Yamabe roblem for the Dirac oerator as the ADM mass in the classical Yamabe roblem Overview of the aer. We here give a short overview of the aer. In Section 3 we introduce notation and collect basic facts concerning sinors and Dirac oerators. In Section 4 we exlain how to find one metric with non-zero mass endomorhism on a given manifold, this uses the results of the following two sections. In Section 5 we show that under certain assumtions the mass endomorhism tends to infinity when the Riemannian metric varies and aroaches a metric with harmonic sinors. In Section 6 we show that the roerty of non-zero mass endomorhism can be reserved under surgery on the underlying manifold. Finally, in Section 7 we use analytic erturbation techniques to show that the existence of one metric with non-zero mass endomorhism imlies that a generic metric has this roerty. 3. Notations and reliminaries 3.. Notation and some basic facts. In this article we use the following notations for balls and sheres: B k (R) := {x R k x < R}, B k := B k (), S k (R) := {x R k x = R}, S k := S k (). As background for basic facts on sinors and Dirac oerators we refer to [6] and []. For the convenience of the reader we summarize here a few definitions and facts. On a comact Riemannian sin manifold (M, g) one defines the Dirac oerator D g acting on sections of the sinor bundle. The Dirac oerator is essentially self-adjoint and extends to a self-adjoint oerator H L 2 where H is the sace of L 2 -sinors whose first derivative is L 2 as well, and L 2 is the sace of square integrable sinors. A smooth sinor is called harmonic, if it is in the kernel of the Dirac oerator D g. Any L 2 -sinor satisfying D g ϕ = 0 in the weak sense, is already smooth, thus it is a harmonic sinor. If the kernel of D g is trivial, then the Dirac oerator is invertible with a bounded inverse L 2 H. The inverse has an integral kernel called the Green s function of D g. The Green s function of D g has already been used in Subsection 2. to define the mass endomorhism Comaring sinors for different metrics. Let g and h be Riemannian metrics on the sin manifold M. The goal of this section is to recall how sinors on (M, g) are identified with sinors on (M, h) using the method of Bourguignon and Gauduchon [0], see also [5]. Given the metrics g and h there exists a unique bundle endomorhism a g h of T M which satisfies g(a g hx, Y ) = h(x, Y ) for all X, Y T M. It is g-self-adjoint and ositive definite. Define b g h := (ag h ) /2, where (a g h )/2 is the unique ositive ointwise square root of a g h. The ma bg h mas g-orthonormal frames to h-orthonormal frames and defines an SO(n)-equivariant bundle morhism b g h : SO(M, g) SO(M, h)

6 6 BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT of the rincial bundles of orthonormal frames. The ma b g h lifts to a Sin(n)- equivariant bundle morhism β g h : Sin(M, g) Sin(M, h) of the corresonding sin structures. From this we obtain a homomorhism of vector bundles β g h : Σg M Σ h M (2) which is a fiberwise isometry with resect to the inner roducts on Σ g M and Σ h M. We let the Dirac oerator D h act on sections of Σ g M by defining D h g := (βg h ) D h β g h. In [0, Thm. 20] an exression for Dg h is comuted in terms of a local g-orthonormal frame {e i } n i=. The result is n Dg h ϕ = e i g b g (ei)ϕ + n e i ((b g h h 2 ) h b g h (ei)bg h g b g h (ei)) ϕ, (3) i= i= where for any vector field X the oerator (b g h ) h X bg h g X is g-antisymmetric and therefore considered as an element of the Clifford algebra. It follows that D h g ϕ = D g ϕ + A h g ( g ϕ) + B h g (ϕ), (4) where A h g and B h g are ointwise vector bundle mas whose ointwise norms are bounded by C h g g and C( h g g + g (h g) g ) resectively. 4. Finding one metric with non-vanishing mass endomorhism The goal of this section is to rove the following Theorem. Theorem 4.. Let M be a comact connected sin manifold of dimension n 3 and let M. Assume that α(m) = 0. Then there exists a neighborhood U of and a flat metric g flat on U such that R 0, (M) is non-emty. Proof. We start by roving the theorem when the manifold is a torus. Consider the torus T n equied with the Lie grou sin structure for which the standard flat metric g 0 has a sace of arallel sinors of maximal dimension. Choose T n and let U be a small oen neighborhood of. Further, let g flat be the restriction of g 0 to U. Since n 3 we have that α(t n ) = 0 so by [5] there is a metric g on T n with invertible Dirac oerator. The construction of g is done through a sequence of surgeries which starts with the disjoint union of T n and some other manifolds, and ends with the torus T n. These surgeries can be arranged so that they do not change the oen set U in the initial T n, so the resulting metric satisfies g = g 0 on U, or g R inv (T n ). Define the family of metrics g t := tg + ( t)g 0. Since the eigenvalues of D gt deend analytically on t it follows that D gt is invertible excet for isolated values of t, it follows that g t R inv (T n ) excet for isolated values of t. Choose a sequence t k 0 for which g tk R inv (T n ), we can then aly Theorem 5. below to the sequence g tk converging to g 0 and conclude that g tk R 0, (T n ) for k large enough. In articular R 0, (T n ) is not emty, and we choose a metric h 0 from this set.

7 MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS 7 Now let M be a manifold of dimension n as in the theorem. Since α(m) = 0 we know that there is a metric g on M with invertible Dirac oerator. We consider the disjoint union M 0 = T n ( T n ) M. Here T n denotes T n with the oosite orientation, so that T n ( T n ) is a sin boundary and M 0 is sin bordant to M. Since M is connected it follows that M can be obtained from M 0 by a sequence of surgeries of codimension 2 and higher, see [5, Proosition 4.3]. Again, these surgeries can be arranged to miss the oen set U in the first T n. We equi M 0 with the Riemannian metric h 0 h 0 g R 0, (T n ( T n ) M) and when we use Theorem 6. below for the sequence of surgeries we end u with a metric g R 0, (M). Finally, the oint M we end u with after the sequence of surgeries might of course not be equal to the oint in the assumtions of the theorem. If we set this right by a diffeomorhism we have roved that R 0, (M) is non-emty. Note that this roof does not work in dimension 2. Indeed, we strongly use that the α-genus of the torus T n vanishes. This fact is only true in dimension n 3. If the flat torus T 2 is equied with the Lie grou sin structure with two arallel sinors, then α(t 2 ) =. By the way, it is roven in [7] that the mass endomorhism always vanishes in dimension Mass endomorhism of metrics close to a metric with harmonic sinors Finding examles of metrics with non-zero mass endomorhism seems to be a difficult issue. The only exlicit examles we have until now are the rojective saces RP n, n 3 mod 4, equied with its standard metric, see [7]. The goal of this section is to show that metrics g R inv (M) sufficiently close to a metric h R,U,gflat \ R inv (M) will under some additional assumtions rovide such examles. This is the object of Theorem 5. below, which in our mind has an interest indeendently of the alication to Theorem 2.4. Theorem 5.. Let U be a neighborhood of M. Assume that h R U,gflat (M) has ker D h {0}. Further assume that the evaluation ma of harmonic sinors at, ker D h ψ ψ() Σ h M, is injective. Set m := dim ker D h Let g k R inv (M), k =, 2,..., be a family of metrics on M converging to h in the C -toology. Then the mass endomorhism α g k at has at least m eigenvalues tending to as k. In articular, g k R 0, (M) for large k. The roof of this theorem is insired by the work of Beig and O Murchadha [9]. In the hyothesis of Theorem 5., the injectivity of the evaluation ma ker D h ψ ψ() Σ h M, is quite restrictive: it is fulfilled for instance when the sace of harmonic sinors is -dimensional if is not a zero of the harmonic sinor. In Theorem 4. we alied the result to the flat torus T n. Proof. For the roof we choose a non-zero ψ ker D h. Set ψ := ψ() Σ h M, by assumtion we have ψ 0. We will show that α g k (ψ ) tends to infinity.

8 8 BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT Let G k be the Green s function of D g k associated to ψ, that is G k is a distributional solution of D g k G k = δ ψ. In coordinates around we write (comare Proosition 2.) x G k = η ω n r n ψ + v g k (ψ ). (5) Here η is a cutoff function which is equal to near and has suort in U. We shorten notation by writing v k for the sinor field v g k (ψ ). Ste. We show that there are k M for which v k ( k ). Let the smooth function Ω : M \ {} (0, ] satisfy { r(x) if x B (ε), Ω(x) = if x M \ B (2ε). Note that Ω does not deend on k. We have 0 < ψ 2 = G k, D g k ψ dv g k M = M Ω n Ωn G k, D g k ψ dv g k Ω n dvg k Ω n G k D g k ψ. M As the integral is bounded and the last factor tends to zero as k, we conclude that Let k be oints for which lim k Ωn G k =. Ω n ( k )G k ( k ) = Ω n G k. Then ( ) Ω n ( k )G k ( k ) = Ω n x ( k ) η ω n r n ψ 0 ( k ) + Ω n ( k )v k ( k ), here the first term on the right hand side is bounded so the second term must tend to infinity. Since Ω n ( k )v k ( k ) v k ( k ) we conclude that v k ( k ) as k, and Ste is roven. To the sinor v k which is a section of Σ g k M the ma β g k h described in (2) associates a section w k := β g k h v k in the sinor bundle Σ h M. We decomose this section as w k = a k ϕ k + w k where ϕ k ker D h is normalized to have ϕ k L (Σ h M) =, a k R, and wk is orthogonal to ker D h. We choose large enough so that H (Σh M) embeds into C 0 (Σ h M). Ste 2. We show that a k. For a contradiction assume that the sequence a k is bounded. From (5) it follows that D g k x v k = gradη ω n r ψ n. This together

9 MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS 9 with the roerties of β g k h gives w k H C Dh w k L = C D h w k L = C (β g k h ) D h β g k h v k L = C D h g k v k L (6) C D g k v k L + C A h g k ( g k v k ) + Bg h k (v k ) L x C gradη ω n r n ψ L + Cε k w k H, here the first term is bounded and ε k 0 by our assumtion that g k h in the C -toology. By assumtion we also have Together this gives w k H a kϕ k H + w k H C + w k H. w k H C + Cε k + Cε k w k H, so wk H is bounded. We conclude that w k C0 is bounded, and the assumtion that a k is bounded then tells us that w k C 0 = v k C 0 is bounded. This contradicts Ste, so we have roved Ste 2. Ste 3. Conclusion. Set ω k := a k w k and ω k Then (6) tells us that ω k = ϕ k + ω k. := a k w k so that ω k H Ca k gradη x ω n r n ψ 0 L + Cε k ω k H, where the first term now tends to zero. Since the ϕ k are in ker D h and they are normalized in L (Σ h M) it follows that they are bounded in H (Σh M). From this we get ω k H ϕ k H + ω k H C + ωk H. It follows that ωk H o() + Cε k ω k H so ωk H 0 and ω k C0 0. Finally we have α g k (ψ ) = v k () = w k () = a k ω k () a k ( ϕ k () ω k () ) = a k ( ϕ k () + o()). By our assumtion that the evaluation ma of harmonic sinors at is injective we know that ϕ k () cannot tend to zero, so from Ste 2 we conclude that α g k (ψ ). This finishes the roof of Ste 3 and the theorem.

10 0BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT 6. Surgery and non-zero mass endomorhism Let M be obtained from M by surgery of codimension at least 2. We assume that M is not hit by the surgery, so we have M. As before R 0, (M) denotes the metrics with invertible Dirac oerator on M which coincide with the flat metric g flat on U and whose mass endomorhism at is not zero. The goal of this section is to rove that R 0, (M) imlies R 0, ( M). We start with a manifold M of dimension n and a oint M. We will erform a surgery of dimension k {0, n 2} on M. For this construction, we follow the beginning of Section 3 in [5] and use the same notation. So, we assume that we have an embedding i : S k M with a trivialization of the normal bundle of S := i(s k ) in M, which thus can be identified with S k R n k. The normal exonential ma then defines an embedding of a neighborhood of the zero section of the normal bundle of S, in other words for small R > 0 the normal exonential ma defines a diffeomorhism f from S k B n k (R) to an oen neighborhood of S, and f is an extension of S k {0} S k i M. Furthermore, for sufficiently small R > 0, the distance from f(x, y) to S = f(s k {0}) is y. As before we assume that U is an oen neighborhood of, on which a flat extendible metric g flat exists. We assume further that S, and by ossibly restricting U to a smaller oen set, we can also assume that U S =. Thus for small R > 0 one obtains U f(s k B n k (R)) =. As in Section of [5] we define ( ) M = M \ f(s k B n k (R)) ( B k+ S n k ) /, where identifies the boundary of B k+ S n k with f(s k S n k (R)) via the ma (x, y) f(x, Ry). Our constructions are carried out such that U is both a subset of M and M. The main result of this section is the following Theorem. Theorem 6.. If R 0, (M), then R 0, ( M). Proof. We assume the requirements for, U, f and k stated at the beginning of this section, and let g R 0, (M). The goal is to construct a metric ĝ R 0, ( M) following the constructions in [5]. Theorem.2 in [5] allows us to construct a metric ĝ on M with invertible Dirac oerator. We recall the scheme of the roof of this theorem. As in the beginning of Section 3 of [5] we define oen neighborhoods U S (r) by U S (r) := f(s k B n k (r)) for small r. Then we construct a family of metrics (g ρ ) ρ satisfying g ρ = g on M \ U S (R max ) for some small number R max. This family of metrics is constructed in two stes. First, we use Proosition 3.2 in [5] to assume that g has a roduct form in a neighborhood of S. Then, we do the construction of Section 3.2 in [5] to get g ρ. Once these metrics (g ρ ) are constructed, we roceed by contradiction. We take a sequence (ρ k ) k N tending to 0 and we assume that ker (D gρ k ) 0 for all k, that is k N, there exists a harmonic sinor ψ k 0 on ( M, g ρk ). (7)

11 MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS By showing that lim k ψ k converges in a weak sense to a non-zero limit sinor in ker D g, we will obtain a contradiction. So the metric ĝ := g ρ satisfies the requirements of Theorem.2 in [5] as soon as ρ is small enough. This roof actually allows us to require an additional roerty for the metrics g δ, and make weaker assumtions on the sinors ψ k. The number R max in the roof can be chosen arbitrarily small. So set δ = R max and choose ρ := ρ(δ) small enough so that g δ = g ρ has an invertible Dirac oerator. We obtain in this way a family of metrics (g δ ) δ (0,δ0) for some δ 0 > 0 such that all D g δ are invertible and such that g δ = g on M \ U S (δ). Let now (δ k ) k N be a sequence of ositive numbers going to 0. We make the following assumtion: k N, there exists a sinor ψ k on ( M, g δk ) and a sequence λ k converging to 0 such that D g δ k ψ k = λ k ψ k. Working with these sinors instead of the ones given by assumtion (7), the same contradiction is obtained. This roves that there is a uniform sectral ga for (g δ ) δ (0,δ0/2), or in other words that there exists a constant C 0 > 0 indeendent of δ (0, δ 0 /2) such that SecD g δ [ C 0, C 0 ] =. (8) Now, we rove that the metric ĝ := g δ for δ small enough satisfies the requirements of Theorem 6.. It is already clear that D g δ is invertible for δ small enough, and that g δ is flat on U for δ small enough. It remains to show that α g δ 0 for δ small enough. For this urose we show that α g δ α g as δ 0. Since we assume α g 0 this gives the desired result. So let us rove this fact. First, choose ψ 0 Σ g (M) = Σ g δ (M). To simlify the notation, set γ := G g ψ 0 and γ δ := G g δ ψ 0. The roof will be comlete if we rove that lim γ() γ δ() = 0. (9) δ 0 Note that the sinor γ γ δ, defined on M \ ({} U S (δ)), is smooth and extends smoothly to. Indeed, it is equal on U to v g (x)ψ 0 v g δ (x)ψ 0 (with the notations of Proosition 2. and Definition 2.2). Let η δ C ( M), 0 η δ be a cutoff function such that η δ = on M \ U S (3δ) and η δ = 0 on U S (2δ). Since on su(η δ ) M \ U S (2δ) = M \ U S (2δ) we have g δ = g we may assume that dη δ g = dη δ gδ 2 δ. (0) From Equation (8), we have C 2 0 M Dg δ ϕ δ 2 g δ dv g δ M ϕ δ 2 g δ dv g δ for all smooth non-zero sinors ϕ δ on ( M, g δ ). We evaluate this quotient for ϕ δ := η δ γ γ δ. Note that ϕ δ is well defined on ( M, g δ ) and smooth since γ is well defined on su(η δ ). Since γ and γ δ are harmonic, we have Dϕ δ = dη δ γ, and since g δ = g

12 2BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT on su(η δ ), we get from Equation (0) that D g δ ϕ δ 2 g δ dv g δ = dη 2 g γ 2 gdv g M M 4 δ 2 ( su γ(x) 2 ) Vol g (U S (3δ) \ U S (2δ)). x U S (3δ 0) We have that Vol g (U S (3δ) \ U S (2δ)) Cδ n k where we used the convention (used throughout this roof) that C is a ositive constant indeendent of δ. Since k n 2, this leads to D g δ ϕ δ 2 g δ dv g δ C. M Since η δ = on M \ U S (3δ) and since g δ = g on this set, it follows that ϕ δ 2 g δ dv g δ C. () M\U S (3δ) Now, we roceed as in ste 2 of the roof of Theorem.2 in [5]. Let Z > 0 be a large integer. By () the set {ϕ δ } δ>0 is bounded in L 2 (M \ U S (/Z)). By Lemma 2.2 in [5] it follows that {ϕ δ } δ>0 is bounded in C,α (M \U S (2/Z)) for all α. We aly Ascoli s Theorem and conclude there is a subsequence (ϕ δk ) of {ϕ δ } δ>0 which converges in C (M \U S (2/Z)) to a sinor Φ 0. Similarly we construct further and further subsequences of (ϕ δk ) converging to Φ i in C (M \U S (2/(Z+i))). Taking a diagonal subsequence of these subsequences, we obtain a subsequence (ϕ δk ) which converges in Cloc (M \ S) to a sinor Φ. As ϕ δ is D g -harmonic on (M \ U S (3δ)) the Cloc (M \ S)-convergence imlies that Dg Φ = 0 on M \ S. With () we conclude that Φ L 2 (M). Thus Φ is L 2 and smooth on M \ S. The equation D g Φ = 0 holds on M \ S. We now aly Lemmas 2. and 2.4 of [5] and conclude that Φ is smooth on (M, g) and DΦ = 0 on M. Since ker D 0 = 0, we get that Φ 0 and in articular Φ() = 0. This imlies Equation (9). 7. From existence to genericity The goal of this section is to rove the following Theorem. Theorem 7.. Let M be a comact sin manifold of dimension n, n 3, let M and let U be a neighborhood of. If R 0, (M) is non-emty then it is generic in R U,gflat (M). 7.. Continuity of the mass endomorhism. The goal of this subsection is to rove that the mass endomorhism deends continuously on g in the C -toology. Proosition 7.2. Equi R inv (M) with the C -norm. Then the ma is continuous. R inv (M) g α g End(Σ M) It follows that R 0, (M) is oen in R inv (M) and thus in R U,gflat (M). Proof. Let (g k ) k N be a family of metrics in R inv (M) such that g k g in the C -toology. For each k the oerator D g k g = (β g g k ) D g k β g g k

13 MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS 3 is invertible. We define P k := D g k g D g. Further, let G g k and G g be the Green s functions of D g k and D g. We define Q k := (β g g k ) G g k β g g k G g. Let ψ Σ M. Using the equation (5) for G g k and for G g and using the fact that g k U = g U = g flat we find that Q k ψ = (β g g k ) v g k β g g k ψ v g ψ. Therefore Q k ψ has a smooth continuation to all of M. The equation D g k G g k = D g G g = δ Id ΣM then tells us that Q k = (D g k g ) P k G g. (G g ψ)(x) becomes singular as x. However we may take a smooth function η which is equal to near and has suort in U and since g k U = g U = g flat we obtain P k (ηg g ψ) = D g (ηg g ψ) D g (ηg g ψ) = 0. It follows that P k G g ψ = P k ( η)g g ψ, where ( η)g g ψ is smooth on all of M. From (4) it follows that the sequence (D g k g ) k N converges to D g with resect to the norm of bounded linear oerators from C (Σ g M) to C 0 (Σ g M). Therefore P k G g ψ C 0 0 as k. Then it follows from [5, Thm. IV-.6] that ((D g k g ) ) k N converges to (D g ) with resect to the norm of bounded linear oerators from C 0 (Σ g M) to C (Σ g M). Therefore Q k ψ C 0 as k. Evaluating Q k at yields α g k α g. Thus the statement of the Proosition follows Analyticity of the mass endomorhism. In this section M is a closed sin manifold. Definition 7.3. Let ε > 0. A family (g t ) t ( ε,ε) of Riemannian metrics on M is called real analytic if there exist sections h k of the bundle of symmetric bilinear forms on M, k N, such that for all t ( ε, ε) and for all r N we have g t N k=0 tk h k C r 0 as N. Let r, s N. A holomorhic family ϕ in C r (Σ g M) is a ma ϕ: Ω C r (Σ g M), where Ω is an oen subset in C and ϕ is differentiable in the norm. C r (Σ g M). A holomorhic family P in the sace of bounded oerators B(C r (Σ g M), C s (Σ g M)) is a ma P : Ω B(C r (Σ g M), C s (Σ g M)), where Ω is an oen subset in C and P is differentiable in the oerator norm. B(C r (Σ g M),C s (Σ g M)). The terminology we use here is the same as in Kato s book [5]. In articular we may use that a family P : Ω B(C r (Σ g M), C s (Σ g M)) is holomorhic if and only if it is weakly holomorhic, i.e. if and only if for every fixed ϕ C r (Σ g M) the family P ϕ: Ω C s (Σ g M) is holomorhic. The restriction of a holomorhic family of sinors or oerators to a real interval will be called a real-analytic family. We first show that for every real-analytic family (g t ) t ( ε,ε) of Riemannian metrics on M the family (Dg gt ) t ( ε,ε) of Dirac oerators is real analytic. The authors of the article [0] state a similar result, however they consider the Dirac oerator as a closed oerator on the sace of L 2 -sinors. Lemma 7.4. Let M be a closed sin manifold and let (g t ) t ( ε,ε) be a real analytic family of Riemannian metrics in R inv (M). Then the family (Dg gt ) t ( ε,ε) is a real analytic family in B(C (Σ g M), C 0 (Σ g M)).

14 4BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT Proof. It is sufficient to show that for every fixed ϕ C (Σ g M) the family of sinors (Dg gt ϕ) t ( ε,ε) is real analytic. As in section 3.2 we define endomorhisms and a g h k, k N, of T M such that for all X, Y in T M we have a g g t g(a g g t X, Y ) = g t (X, Y ), g(a g h k X, Y ) = h k (X, Y ). Note that a g h k also exists if h k is not ositive definite. Let. be the norm on Σ g M induced by the inner roduct and let {e i } n i= be a local g-orthonormal frame. Since (g t ) t ( ε,ε) is real analytic it follows that su X T M, X = = su X T M, X = a g g t X N t k a g h k X k=0 n g(a g g t X, e i )e i i= n i= k=0 N t k g(a g h k X, e i )e i 0, N for all t ( ε, ε). From this calculation it follows that for every vector field X of length on M the vector field b g g t X is also given by a convergent ower series and the convergence is uniform in X. Furthermore for any vector fields X, Y the vector field gt XY is also given by a convergent ower series as can be seen in local coordinates. The assertion now follows from the formula (3) for D gt Proosition 7.5. If (g t ) t ( ε,ε) is a real-analytic family of metrics in R inv (M), then α gt is also real-analytic. Proof. It is sufficient to show that for every ψ Σ M the family of sinors Q t ψ := (β g g t ) G gt β g g t ψ G g ψ is a real analytic family in C 0 (Σ g M). As above we define P t := Dg gt D g and we obtain Q t = (Dg gt ) P t G g. By the revious lemma the family of oerators (Dg gt ) t ( ε,ε) is real analytic. It follows from [5, VII-.] that the family of oerators ((Dg gt ) ) t ( ε,ε) is also real analytic. As in the roof of Proosition 7.2 one concludes that for every t the sinor P t G g ψ is smooth on all of M. Thus the family (P t G g ψ) t ( ε,ε) is a real analytic family in C (Σ g M) and thus the family (Q t ψ) t ( ε,ε) is a real analytic family in C 0 (Σ g M). Consider a real analytic family (g t ) t (a,b) of Riemannian metrics on M. By unique continuation we immediately see: If there is a t 0 (a, b) with α gt 0 0, then the set S := {t (a, b) α gt = 0} is a discrete subset of (a, b). Two metrics in the same connected comonent of R inv (M) can be joined by a iecewise real-analytic ath of metrics. It follows that if a connected comonent of R inv contains at least one metric with non-zero mass endomorhism, then the metrics with non-zero mass endomorhism are dense in this comonent. In order to obtain Theorem 7., we still have to discuss families (g t ) t (a,b) where D gt is not invertible for some t. As the mass endomorhism is not defined for these t, we comlexify the arameter t and ass around the metric with non invertible D gt in the imaginary direction. This is discussed in the following subsection. g.

15 MASS ENDOMORPHISM, SURGERY AND PERTURBATIONS Analytic continuation in the imaginary direction. Again let (g t ) t (a,b) be a real-analytic family of metrics. We assume g t R U,gflat (M) for any t (a, b), but we do not assume that all D gt are invertible. Because of the real-analyticity of Dg gt, the family can be extended to a comlex-analytic family of oerators defined for t in an oen subset U (a, b) of C. In this comlexification the oerators Dg gt will no longer be self-adjoint, instead we have (Dg gt ) = D g t g. As the set of invertible oerators is oen, we can assume without loss of generality is invertible on U \ (a, b). In other words we assume that that D gt g T := {t U D gt g is not invertible} is contained in (a, b). The arguments from above also yield that t α gt is a holomorhic function on U \T. As U \T is connected, unique continuation imlies the following Proosition. Proosition 7.6. If the mass endomorhism α gt 0 is non-zero for any t 0 (a, b)\t, then {t (a, b) \ T α gt 0} is dense in (a, b). References [] B. Ammann, A sin-conformal lower bound of the first ositive Dirac eigenvalue, Diff. Geom. Al. 8 (2003), [2], A variational roblem in conformal sin geometry, Habilitationsschrift, Universität Hamburg, [3], The smallest Dirac eigenvalue in a sin-conformal class and cmc-immersions, Comm. Anal. Geom. 7 (2009), [4] B. Ammann, M. Dahl, and E. Humbert, Harmonic sinors and local deformations of the metric, Math. Res. Lett. 8 (20), [5], Surgery and harmonic sinors, Adv. Math. 220 (2009), [6] B. Ammann, J.-F. Grosjean, E. Humbert, and B. Morel, A sinorial analogue of Aubin s inequality, Math. Z. 260 (2008), [7] B. Ammann, E. Humbert, and B. Morel, Mass endomorhism and sinorial Yamabe tye roblems, Comm. Anal. Geom. 4 (2006), [8] C. Bär and M. Dahl, Surgery and the Sectrum of the Dirac Oerator, J. reine angew. Math. 552 (2002), [9] R. Beig and N. Ó Murchadha, Traed surfaces due to concentration of gravitational radiation, Phys. Rev. Lett. 66 (99), no. 9, [0] J.-P. Bourguignon and P. Gauduchon, Sineurs, oérateurs de Dirac et variations de métriques, Comm. Math. Phys. 44 (992), [] T. Friedrich, Dirac Oerators in Riemannian Geometry, Graduate Studies in Mathematics 25, AMS, Providence, Rhode Island, [2] A. Hermann, Generic metrics and the mass endomorhism on sin 3-manifolds, Ann. Glob. Anal. Geom. 37 (200), [3] A. Hermann, Dirac eigensinors for generic metrics, PhD thesis, Universität Regensburg, 202, arxiv [4] O. Hijazi, Première valeur rore de l oérateur de Dirac et nombre de Yamabe, C. R. Acad. Sci. Paris t. 33, Série I (99), [5] T. Kato, Perturbation theory for linear oerators, Grundlehren der mathematischen Wissenschaften, vol. 32, Sringer-Verlag, 966. [6] H. B. Lawson and M.-L. Michelsohn, Sin geometry, Princeton University Press, Princeton, 989. [7] J. M. Lee and T. H. Parker, The Yamabe roblem, Bull. Am. Math. Soc., New Ser. 7 (987), 37 9.

16 6BERND AMMANN, MATTIAS DAHL, ANDREAS HERMANN, AND EMMANUEL HUMBERT [8] S. Maier, Generic metrics and connections on sin- and sin c -manifolds, Comm. Math. Phys. 88 (997), [9] R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom. 20 (984), [20] S. Stolz, Simly connected manifolds of ositive scalar curvature, Ann. of Math. (2) 36 (992), no. 3, Bernd Ammann, Fakultät für Mathematik, Universität Regensburg, Regensburg, Germany address: bernd.ammann@mathematik.uni-regensburg.de Mattias Dahl, Institutionen för Matematik, Kungliga Tekniska Högskolan, Stockholm, Sweden address: dahl@math.kth.se Andreas Hermann, LMPT, Université de Tours,, Parc de Grandmont, Tours, France address: andreas.hermann@lmt.univ-tours.fr Emmanuel Humbert, LMPT, Université de Tours,, Parc de Grandmont, Tours, France address: emmanuel.humbert@lmt.univ-tours.fr