N-BLACK HOLE STATIONARY AND AXIALLY EQUATIONS GILBERT WEINSTEIN. Abstract. The Einstein/Maxwell equations reduce in the stationary

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1 N-LACK HOLE STATIONARY AND AXIALLY SYMMETRIC SOLUTIONS OF THE EINSTEIN/MAXWELL EQUATIONS GILERT WEINSTEIN Abstract. The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities ': R 3 n! H 2 C, where is a subset of the axis of symmetry, and H 2 C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic maps with prescribed singularities ': R n n! H, where is a submanifold of R n of codimension > 2, and H is a classical Riemannian globally symmetric space of noncompact type and rank one. This result, when applied to the black hole problem, yields solutions which can be interpreted as equilibrium congurations of multiple co-axially rotating charged black holes held apart by singular struts. 1. Introduction Let (M; g) be a four-dimensional Lorentzian manifold, and let F be a two-form on M. Consider the Einstein/Maxwell eld equations: (1.1) (1.2) (1.3) Ric g? 1 2 R g g = 2 T F F = da df = 0; where Ric g is the Ricci curvature tensor of the metric g, R g its scalar curvature, and T F is the energy-momentum-stress tensor of the electromagnetic eld F : (1.4) T F (X; Y ) = 1? ix F i Y F + i X F i Y F 2 = i X F i Y F? 1 2 jf j2 g(x; Y ); Here, i X denotes inner multiplication by the vector X, is the Hodge star operator, denotes the inner product, and jj 2 the norm of k-forms. This research was supported in part by NSF Grant DMS The author would like to express his thanks for the support and hospitality of the Erwin Schrodinger Institute where part of this work was completed. 1

2 2 GILERT WEINSTEIN These are given in local coordinates by: (1.5) (1.6) (1.7) 1 ::: 4?k = 1 k! 1::: k " 1 ::: k 1 ::: 4?k ; =? ( ^ ) = 1 k! 1::: k 1 ::: k ; jj 2 = = 1 k! 1::: k 1 ::: k ; where, according to the Einstein summation convention, repeated indices are summed from 0 to 3, and " is the volume form of g. Note that tr T F = 0, hence taking the trace of Equations (1.1) yields R g = 0. Thus, Equation (1.1) can be rewritten as (1.8) Ric g = 2 T F : We are using rationalized units in which 4G = 1, where G is the gravitational constant. We will seek asymptotically at, stationary and axially symmetric solutions of Equations (1.1){(1.3). These equations reduce, using an idea originally due to Ernst [7], to an axially symmetric harmonic map ': R 3 n! H 2 C, where = A S n N j=1 I j consists of the axis of symmetry A R 3 with N closed intervals I j removed, and H 2 C = SU(1; 2)=S(U(1)U(2)) is the complex hyperbolic plane, see [12, 3]. Each interval I j corresponds in (M; g) to a connected component of the event horizon. Thus N will denote, throughout this paper, the number of black holes. The space H 2 C has negative sectional curvature, and nite energy harmonic maps into spaces of negative sectional curvature are well understood, see [17, 18]. Nevertheless, this problem is analytically interesting due to the fact that the boundary conditions for ' on and as r! 1 are singular, and hence ' has innite energy. For the case N = 1, there is a family of solutions known in closed form, the Kerr-Newman black holes, see [2]. We note that the problem has 4N? 1 parameters: N masses m j, N? 1 distances d j, N angular momenta L j, and N electric charges q j. Special cases of the Ernst reduction have been used to prove non-existence results for the Einstein equations. In early work along this line, Weyl investigated the vacuum static case and showed that the equations reduce to a single linear equation. The case N = 1 was known to have the Schwarzschild solution. Superimposing two such solutions, Weyl obtained new solutions which could be interpreted as equilibrium congurations of a pair of black holes. However, with ach [1] in 1921, they showed that an obstruction arose as a conical singularity along the axis separating the two black holes. Having interpreted this singularity as the gravitational force, they computed its value and veried that the result was asymptotic to the Newtonian gravitational force in the appropriate limit. The reduction was also used by Robinson [16], following work of Carter [2], to prove that within the N = 1 vacuum case, the Kerr solutions were unique. Mazur [12], and unting [3] independently, generalized this uniqueness result to the charged N = 1 case.

3 EINSTEIN/MAXWELL EQUATIONS 3 In [20, 21], we used the Ernst reduction to construct vacuum solutions which could be viewed as nonlinear generalizations of the Weyl solutions. Clearly, it is of interest to nd out whether the obstruction found by Weyl in the vacuum nonrotating case persists also for all possible choice of the parameters, i.e., whether the force between rotating black holes is always positive. It has been conjectured for some time that this is so, see [15, Problem 14], but the evidence is limited. Indeed, it is unlikely that the angle deciencies can in general be computed exactly. We mention here a number of partial results, e.g., [10], where the small angular momentum case is treated, [22], where non-existence is proved in the extreme regime, and also [9], where Y. Li and G. Tian proved non-existence in the case when the solution admits an involutive symmetry. In the present paper, we begin the generalization of this work to the Einstein/Maxwell Equations. We prove the existence of a unique (4N? 1)-parameter family of solutions to the reduced problem. Our main tool is the study of harmonic maps with prescribed singularities into classical globally symmetric spaces of noncompact type and rank one [23], i.e., the real, complex, and quaternionic hyperbolic spaces, see [13]. However, the results of [23] do not apply directly to the problem considered here. In [23], we restricted our attention to bounded domains R n, and to singular sets compactly contained in, while here = R n and is unbounded. The necessary generalization is achieved in two steps. First, we consider the problem on a large ball R n, allowing, however, the singular set to Then, we let exhaust R n. In this second step, we need the boundary data at innity in R n to be given by a harmonic map which admits those singularities which lie along the unbounded components of. In the application to black holes, this map is obtained from the Kerr-Newman solutions. The main motivation for this generalization is an attempt to come to a better understanding of the force between rotating black holes, i.e., the magnitude of the conical singularities on the bounded components of the axis. An important question in this respect is the dependence of this force on the parameters. It should be pointed out that the force can vanish in the case of the Einstein/Maxwell Equations. Indeed, even in the static case, equilibrium can be achieved with masses and charges being equal [11, 14]. However, these congurations are extreme, i.e., all the event horizons are degenerate, see Section 2.3. Consequently, one could still pose the question of whether the force is positive for all nondegenerate Einstein/Maxwell black holes [15]. In a future paper, we will study the regularity properties of these maps. This is a necessary step in order to complete the application of the results presented here to black holes. The plan of the paper is as follows. In Section 2, we review the Ernst Reduction as it applies to the Einstein/Maxwell Equations. We carry out the reduction in two steps. First in Section 2.1, we reduce the equations in the presence of one Killing eld. Then, the second reduction, with two

4 4 GILERT WEINSTEIN commuting Killing elds, is carried out in Section 2.2. In the rest of Section 2, we examine the parameters of the problem, derive singular boundary conditions, and set-up a Reduced Problem for the stationary and axially symmetric Einstein/Maxwell Equations. In Section 3, we prove our main existence and uniqueness result. In Section 4, we address briey the problem of going back from a solution of the reduced problem to a solution of the Einstein/Maxwell Equations. 2. The Generalized Ernst Reduction for the Einstein/Maxwell Equations In this section, we describe the generalized Ernst reduction for the Einstein/Maxwell equations in the stationary and axially symmetric case. The main result in this section is Theorem 1, based on which we state the Reduced Problem for the stationary and axially symmetric Einstein/Maxwell Equations in terms of harmonic maps with prescribed singularities. The computations are carried out mostly in the exterior algebra formalism which is particularly well suited to the Maxwell Equations Axial Symmetry. We rst describe the connection between the axially symmetric Einstein/Maxwell Equations and harmonic maps into H 2 C. We note that the arguments given here could be applied to any spacelike Killing eld. Denition 1. Let (M; g) be an oriented 4-dimensional Lorentzian manifold, and let F be a two-form on M. We say that (M; g; F ) is SO(2)- symmetric if SO(2) acts eectively on (M; g) as a group of isometries leaving F invariant. The axis in an SO(2)-symmetric spacetime, is the set of points xed by the action of SO(2). We say that (M; g; F ) is axially symmetric if it is SO(2) symmetric, and it has a nonempty axis. Let (M; g; F ) be a simply connected SO(2)-symmetric solution of the Einstein/Maxwell Equations. Let be the Killing eld generator, then L F = 0. Note that if (M; g) is causal, i.e., admits no closed causal curves, then either is spacelike, or vanishes. Dene the one-forms = i F, and = i F. Then, we nd d =?i df + L F = 0. Thus, there is a function such that d =. Similarly d = 0, hence there is a function such that d =. Note that and are determined only up to constants. Now, dene the one-form = d? d, and observe that d = 2 ^. It is easy to see from (1.5) that for any k-form and one-form, we have: i = ( ^ ); where we have used the metric g to identify the tangent space T p M and its dual T p M. Using [L ; ] = 0, and = (?1) k+1 on k-forms, it follows that ( ^ ) = (?1) k+1 L + ^ ;

5 EINSTEIN/MAXWELL EQUATIONS 5 where = d is the divergence operator on forms. Thus, if we dene the twist of by! = (d ^ ), we nd d! = (d ^ ) =?L d + d ^ = d ^ : Now for any Killing eld, we have = 0, and (2.1) d =?2 tr r 2 = 2 i Ric g : Hence in view of Equation (1.8), we obtain d! = 4 i T F ^. Furthermore, in view of (1.4), i T F =?i F?(1=2) jf j 2. Thus, we have d! = 4 ^i F. On the other hand, = (F ^ ), hence It follows that (2.2) ( ^ ) =?i = ^ i F: d! = 2 d; i.e., the one-form!? 2 is closed. We conclude that there is a function v, also determined up to a constant, such that 2 dv =!? 2. Let M 0 = fx 2 M; jj 2 > 0g, and let u =? log jj on M 0. Then, we have i d =?di + L = 2e?2u du. Thus, since du =?u, we have (e?2u du) = 2e?2u jduj 2? e?2u u. On the other hand, i d =? (d ^ ), hence using Equations (2.1), (1.8), and (1.4), we obtain: (i d) =?i d + jdj 2 =?2? jj 2 + jj 2 + jdj 2 : Furthermore, since! = i d, we nd j!j 2 = (d ^ i d ^ ) + (d ^ d ^ i ) = 4e?4u jduj 2? e?2u jdj 2 i.e., jdj 2 = 4 e?2u jduj 2? e 2u j!j 2. We conclude that (2.3) Now,! = (d ^ d), and u? 1 2 e4u j!j 2? e 2u? jj 2 + jj 2 ) = 0: 2 du! = e 2u (d ^ i d ^ ) + e 2u (d ^ d ^ i ) =?2 du! + (d ^ d); i.e., 4 du! = (d ^d). Thus, we nd that (e 4u!) =?4e 4u du!+e 4u! = 0, which we write as (2.4) div(e 4u!) = 0; where we have put div =? for any one-form. In addition, since =? (F ^ ), we nd =? d(f ^ ) = d F, and 2 du =?e 2u (d ^ i F ^ )? e 2u (d ^ F ^ i ) = e 2u! + d F: It follows that (e 2u ) =?2e 2u du + e 2u =?e 4u!, i.e., (2.5) div(e 2u )? e 4u! = 0:

6 6 GILERT WEINSTEIN Similarly, (e 2u ) = e 4u!, i.e., (2.6) div(e 2u ) + e 4u! = 0: Substituting the denitions of v, and into Equations (2.3){(2.6), it follows that ' = (u; v; ; ) satises the following system of equations: u? 2e 4u jrv + r? rj 2? e 2u? (2.7)? jrj 2 + jr j 2 = 0 (2.8) div e 4u (rv + r? r) = 0 (2.9) (2.10) div? e 2u r? 2e 4u r (rv + r? r) = 0 div? e 2u r + 2e 4u r (rv + r? r) = 0: It is now clear that for every subset M 0, ' = (u; v; ; ) is a critical point of the functional: E (') = njruj 2 + e 4u jrv + r? rj 2 + e 2u? jrj 2 + jr j 2 o d g ; where d g is the volume element of the metric g. Thus, if we choose an `upper half-space model' for H 2 C, i.e., R4 with the metric given by the line element: (2.11) ds 2 = du 2 + e 4u (dv + d? d) 2 + e 2u? d 2 + d 2 ; then the map ': (M 0 ; g)! H 2 C is a harmonic map, see Section 3 and [23] Stationary and Axially Symmetric Solutions. We now turn to the case where (M; g; F ) admits an additional symmetry. Denition 2. Let (M; g) be an oriented and time-oriented 4-dimensional Lorentzian manifold and let F be a two-form on M. Let G be a group acting on M. The orbit of a point p 2 M is degenerate if the isotropy subgroup at p is nontrivial. The axis is the set of points whose orbits are degenerate. We say that (M; g; F ) is stationary and SO(2)-symmetric if the group G = R SO(2) acts eectively on (M; g) as a group of isometries leaving F invariant and such that the orbits of points not on the axis are timelike two-surfaces. We say that (M; g; F ) is stationary and axially symmetric if it is stationary and SO(2)-symmetric and has a nonempty axis. Assume that (M; g; F ) is stationary and axially symmetric, and let be the Killing eld generator of the SO(2)-symmetry normalized so that its orbits are closed circles of length 2 jj. Let be a linearly independent generator. Then, we have [; ] = 0, and as before, if (M; g) is causal, then is either spacelike or vanishes, i.e., is spacelike outside the axis. We will now prove that if (M; g; F ) is a stationary and axially symmetric solution of the Einstein/Maxwell Equations, then: (i) F and F vanish on the orbits, i.e., F (; ) = F (; ) = 0; (ii) the distribution of planes orthogonal to the orbits of G is integrable.

7 EINSTEIN/MAXWELL EQUATIONS 7 To see (i), note that [L ; i ] = i [;] = 0, hence we have di =?i d + L i F = 0: Since i = 0 on the axis it follows that F (; ) = i = 0 everywhere. Similarly, F (; ) = i = 0. To show (ii), it suces by Frobenius Theorem to show that: (2.12) ( ^ ^ d) = 0; ( ^ ^ d) = 0: However, in view of (2.2) and (i), we have d ( ^ ^ d) = di! =?i d!? L! = 4i ( ^ ) = 0; where! is the twist of. Thus, ( ^ ^d) is constant, but since it vanishes on the axis, it is identically zero. Similarly, ( ^ ^ d) = 0. Let Q be an integral surface of the distribution of planes orthogonal to the orbits, and let h be the metric induced by g on Q. Then, the quotient space M=G, with its quotient metric can be identied with (Q; h). We choose the orientation on Q so that (^) is positively oriented. The map ': M 0! H 2 C is invariant under G, hence reduces to a map, which we also denote by ', on the quotient Q 0 = Q \ M 0. Indeed, let be an arbitrary Killing eld generator in the Lie algebra of G. We have e?2u = 2g([; ]; ) = 0, and hence u = 0. Also, = i = F (; ) = 0, and similarly = 0. Finally, () = ()? () = 0, hence 2 v = i (!?2) = (d^^)?2() = 0. Dene = ^, then on M 0 we have jj 2 = jj 2 jj 2? ( ) 2 < 0. Let 2 =? jj 2, then is invariant under G, and thus reduces to a function on Q 0. It follows that for every subset Q 0, the map ': (Q 0 ; h)! H 2 C is a critical point of the functional: E 0 (') = n jruj 2 h + e4u jrv + r? rj 2 h + e2u? o jrj 2 h + jr j2 h d h ; where jj h is the norm with respect to the metric h, and d h is the volume element of the metric h. Indeed, if Q 0 is such a subset, and f is a function on M 0 invariant under G, then 2 R f d h = R G f d g, where G is the orbit of under G. Next, we show that h = 0. Since g =?1 div h ( r), it suces to prove that g =?1 jdj 2. To see this, note rst that 2 =?(; ) = i i. Thus, we obtain: (2.13) Therefore, we see that 2 d = di i = i i d =? (d ^ ): 2 g + 2 jdj 2 =?(2 d) = d(d ^ ) = d? jdj 2 ; or equivalently (2.14) g = 1 2 ( d? jdj2? 2 jdj 2 ):

8 8 GILERT WEINSTEIN On the other hand, since d = d ^? ^ d, we have, in view of Equation (2.12), that i d = (d ^ ) = 0, and similarly i d = 0. We deduce, using (2.13), that i.e., (2.15) 4 2 jdj 2 = (i d ^ d ^ i ) =? (d ^ d ^ i i ) =? 2 jdj 2 ; jdj 2 =?4 jdj 2 Furthermore, we claim that d = 0. Indeed, in view of Equation (2.1): d = d(i d? i d) = d ^ + ^ d = 2 (i Ric g ^ + ^ i Ric g ); and consequently, d = i i d = 2 jj 2 Ric g (; )? 4 ( ) Ric g (; ) + 2 jj 2 Ric g (; ): Introducing ~ = i F, and ~ = i F, we can use Equations (1.8) and (1.4) to write Ric g (; ) = jj 2 + jj 2 from which it follows that: (2.16) Ric g (; ) = ~ + ~ Ric g (; ) = j~j 2 + j ~ j 2 ; d = 2 jj 2? jj 2 + jj 2? 4 ( )? ~ + ~ + 2 jj 2? j~j 2 + In addition, we have: and hence, (2.17) (2.18) jj 2 F = ^ + i jj 2 F = ^? i ; jj 2 ~? ( ) = i i jj 2 ~? ( ) =?i i : A computation similar to the one leading to (2.15) yields: ji i j 2 = 2 jj 2 ; ji i j 2 = 2 jj 2 : ~ 2 : Thus, taking the norm squared of both sides of (2.17) and (2.18), and adding the results, we obtain: jj 4? j~j 2 + j ~ j 2? 2 jj 2 ( )? ~ + ~ + jj 2 jj 2? jj 2 + jj 2 = 0; which, in view of (2.16), implies d = 0 as claimed. Substituting this result back into (2.14), and taking into account (2.15), we obtain g =?1 jdj 2 as required. Therefore, if d 6= 0, the function can be used locally as a harmonic coordinate for the metric h on Q 0. Let z be a conjugate harmonic coordinate, i.e., a function on Q 0 such that jdzj 2 = jdj 2, dz d = 0, and d ^ dz is

9 EINSTEIN/MAXWELL EQUATIONS 9 positively oriented. Then, in the (; z)-coordinate system, the metric h takes the form: h ab dx a dx b = e 2 (d 2 + dz 2 ); where =? log jdj. Let ~ h = e?2 h, then h ~ is a at metric on Q 0, and since the functional E 0 is conformally invariant, we have that ' is a critical point of: ~E (') = njruj 2 ~ h + e 4u jrv + r? rj 2 ~ h + e 2u? jrj 2 ~ h + jr j 2 ~ h o d ~h : This gives a semilinear elliptic system of partial dierential equations for the four unknowns (u; v; ; ). We have obtained that the metric g must be locally of the form given by the line element: (2.19) ds 2 =? 2 e 2u dt 2 + e?2u? d? w dt 2 + e 2 (d 2 + dz 2 ); where w =?e 2u ( ). All the metric coecients can be determined from the map '. Indeed, u is obtained directly from ', and we will now obtain equations for the gradient of w and. These quadratures are to be integrated after the harmonic map system has been solved. Dene = + w, and observe that = 0, while jj 2 = =?e 2u 2. Furthermore, we have dw = 2 w du + e 2u i d = e 2u i d. Since d = 2 ^ du? e 2u ( ^!), and ^ =, we nd i d = e 2u (! ^ ), and hence (2.20) i i dw =? 2 e 4u! The operator?1 i i restricted to forms tangential to Q is the Hodge star operator of the metric h. Since in two dimensions, the star operator restricted to one-forms is conformally invariant,?1 i i is also the Hodge star operator? of the at metric ~ h restricted to one-forms. Now, on oneforms in two dimensions?? =?1, hence we can rewrite Equation (2.20) as: (2.21) dw = e 4u?!: To derive the equations for d, we must now use of the components of the Einstein/Maxwell Equations in the z-plane: Ric g (@ )? Ric g (@ z z ) = 2? F F? F F Ric g (@ z ) = 2 F F: A straightforward coordinate computation leads to:? u = u 2? u 2 z e4u (! 2?! z) 2 + e 2u ( 2? 2 z + 2? z) 2 (2.22) z? u z = 2 u u z e4u!! z + e 2u ( z + z ) :

10 10 GILERT WEINSTEIN Denition 3. We collect here a few denitions related to the notion of asymptotic atness. A data set ( M; g; k; E; ) for the Einstein/Maxwell Equations consists of a 3-manifold M, a Riemannian metric g on M, a symmetric twice covariant tensor k on M, and two vectors E and on M, satisfying the Constraint Equations: R? k ij k ij + (tr k) 2 = E ; ri kij? r j tr k = 2 " ijk E j k ; r i Ei = 0; r i i = 0; where " and r denote respectively the volume form and the covariant derivative of g. In addition, we require that R S g( ; n) = 0 for every closed 2-surface S non-homologous to zero in M, where n is the unit normal to S. A 3-manifold M is topologically Euclidean at innity, if there is a compact set K M, such that each connected component of M nk, henceforth called an end, is homeomorphic to the complement of a ball in R 3. The data set ( M; g; k; E; ) is strongly asymptotically at if M is topologically Euclidean at innity, and if there is on each end a coordinate system (x 1 ; x 2 ; x 3 ), in which g, k, E,, take the asymptotic forms: g ij = 1 + 2m r ij + o 2 (r?3=2 ); kij = o 1 (r?5=2 ); E i = q x i r 3 + o 1(r?5=2 ); i = o 1 (r?5=2 ); where m and q are constants. Here, a function f is o k (r?l ) j f = o(r?l?j ) for j = 0; : : :; k, and r = P (x i ) 2. A spacetime is globally hyperbolic if it admits a Cauchy hypersurface, i.e., a hypersurface which intersects every inextendible causal curve exactly once. Given a data set ( M; g; k; E; ) which is strongly asymptotically at, the Cauchy Problem for the Einstein/Maxwell Equations, consists of nding a globally hyperbolic spacetime (M; g), a twoform F, and an embedding : M! M such that (M; g; F ) is a solution of (1.1){(1.3), g and k are respectively the rst and second fundamental forms of, and such that E = i F, and = i F, where is the future pointing unit normal of M in M. The triple (M; g; F ) is called the Cauchy development of the data ( M; g; k; E; ). A triple (M; g; F ) will be called asymptotically at if it is the Cauchy development of strongly asymptotically at data. A domain of outer communications in an asymptotically at spacetime (M; g) is a maximal connected open set O M such that from each point p 2 O there are both future and past directed timelike curves to asymptotically at regions of (M; g). An event horizon is the boundary of a domain of outer communications. If (f M; g) is asymptotically at and globally hyperbolic, so is any domain of outer communications M in (f M; g), since M is then the intersection of future and past sets. We will restrict our attention to such a domain. We may then assert that (M; g) is causal. We assume that M is simply connected. If in addition, (M; g; F ) is a solution of the Einstein/Maxwell

11 EINSTEIN/MAXWELL EQUATIONS 11 Equations which is stationary and axially symmetric, then there is a unique Killing eld generator, which is future directed, and such that jj 2!?1 at spacelike innity. Dening now and as before, it can be shown, using h = 0, that has no critical points, and hence can be used as a harmonic coordinate on all of Q 0. Furthermore, the event horizon H M f can be characterized by the conditions: jj 2 > 0 and = 0. If jdj 2 vanishes identically on some component H 0 of H, then we say that H 0 is degenerate. If a component H 0 of H is nondegenerate, then by (2.15), we have jdj 2 = jdzj on H 0. We will see in Section 2.3 that each component H 0 of H is a Killing horizon in the sense that the null generators of H 0 are tangent to a Killing vector 0 which is non-zero on H 0, see [2, 4]. Furthermore, e 2u jdj 2 tends on? H 0 to a constant 0, the surface gravity of H 0, dened by the equation d j 0 j 2 =?20 0 which holds on H 0. Thus, H 0 is degenerate if and only if 0 = 0 and the denition of a degenerate horizon adopted here coincides with the standard one, see [2]. If a component H 0 of H is nondegenerate, then H 0 \ Q is an interval on the boundary of Q; otherwise, it is a point. Indeed, the intersection of H 0 with a spacelike hypersurface must have the topology of a 2-sphere, see [5]. We will assume that the event horizon H consists of N nondegenerate components, which we denote by H j, 1 6 j 6 N. Let A be the z-axis in R 3, and consider Q 0 SO(2) = R 3 na with the at metric 2 d d + h. ~ Let = A S N n j=1 I j where I j are the open intervals on the boundary of Q 0 corresponding to the N components H j of the event horizon H. Since jj 2 > 0 on I j, the map ' can be extended continuously across I j. Let x 2 I j, then since I j is of codimension 2, one can show, using a cut-o as in Lemma 6, that ' is weakly harmonic and has nite energy in a suciently small ball centered at x. It then follows from regularity theory for harmonic maps [17, 18] that ' is smooth in that ball. Thus, is an axially symmetric harmonic map. We have proved: ': R 3 n! H 2 C Theorem 1. Let (M; g; F ) be a solution of the Einstein/Maxwell Equations which is stationary and axially symmetric. Assume that M is simply connected, that (M; g) is the domain of outer communications of an asymptotically at globally hyperbolic spacetime, and that every component of the event horizon in (M; g) is nondegenerate. Dene ' = (u; v; ; ), and as above. Then ': R 3 n! H 2 C is an axially symmetric harmonic map. In addition, the metric g is of the form (2.19), and the metric coecients w and satisfy Equations (2.21) and (2.22) Physical Parameters. Let S be an oriented spacelike two-sphere in M. We dene the mass m(s), angular momentum L(S), and charge q(s),

12 12 GILERT WEINSTEIN contained in S by: (2.23) (2.24) (2.25) m(s) =? 1 8 L(S) = 1 16 q(s) =? 1 4 S S S d; d; The mass and the angular momentum are standard Komar integrals associated with the Killing elds and, see [2], and the denition of the charge is classical. We will always perform the integration over a submanifold of revolution, so that the integral can be computed on the quotient Q. Let 1 6 k 6 3, let be a k-form and let be a k-dimensional submanifold in M which is the orbit of a (k? 1)-dimensional submanifold? Q under the F: subgroup SO(2) G, i.e., = SO(2)?, then we nd: (2.26) =?2 It is well-known that Stokes' Theorem can used in (2.23){(2.25) to relate the conserved quantities of two spheres S and S 0 which are homologous in M. Indeed, suppose that S and S 0 = S?S 0 for some 3-dimensional submanifold, then: (2.27) q(s)? q(s 0 ) =? 1 4? i ; d F = 0; so that the charge q(s) actually depends only on the homology class of S. In our situation, the second homology group is generated by spacelike crosssections S j of the N components H j of the event horizon H. Hence q(s) is determined by the homology class of S and by the N parameters q j = q(s j ), j = 1; : : :; N, which we may call the electric charges of the N black holes. y (2.26), we nd: q j = 1 2 I j d : In view of (2.27), is constant along each component of. Let j, j = 1; : : :; N + 1, denote the N + 1 connected components of in order of increasing z. We may assume that q 0 = j 1 = 0, and j j = q j?1 for j = 2; : : :; N + 1. According to Equation (1.2), we also have: 1 2 I j d = 1 4 S j F = 1 j A = 0; which simply expresses the absence of magnetic charge. As a consequence, is constant throughout, and we may as well assume j = 0.

13 EINSTEIN/MAXWELL EQUATIONS 13 In analogy, we dene the angular momenta: (2.28) L j = L(S j ) = 1 d = 1! = 1 16 S j 8 I j 4 I j (dv + ): However, since the electromagnetic eld carries angular momentum L(S) is not determined by these parameters. Indeed, = S? S 0 and if = SO(2), then: L(S)? L(S 0 ) = 1 d d = 1 ^ : 16 2 Nevertheless, since vanishes on, it is easy to see from (2.28), that v is Rconstant on each component of. We introduce the R N parameters j = I j dv, and note that L j = ( j + l j )=4 where l j = I j. We may assume that 0 = vj 1 = 0, and vj j = j?1 for j = 2; : : :; N + 1. Finally, dene the N masses: m j = m(s j ) = 1 4 I j i d: Equation (2.21) implies that w is constant on each I j. The N constants w j = wj Ij may be interpreted as the angular velocities of the N black holes. Now, in view of (2.13), we have: and therefore, (2.29) dz =?d = 1 2 d; i d =?2 dz + (d ^ ): Since d = 2 ^ du? e 2u ( ^!), we have (2.30) (d ^ ) =?2? du? w!: Combining Equations (2.29) and (2.30), we obtain: i d =?2 dz? w!? 2? du: The last term vanishes on I j, and thus we obtain: m j = 1 4 I j (2 dz + w!) = j + 2w j L j ; R where 2 j = I j dz is the length of I j in the metric ~ h. Again, m(s) is not determined by these, since the electromagnetic eld carries energy. = S? S 0 where = SO(2), then we nd: m(s)? m(s 0 ) = 1 4 = 1 2 d(w! + 2? du) n e 2u (jj 2 + jj 2 ) d dz + 2w ^ For future reference, we also introduce the N? 1 parameters r j = R j dz, j = 2; : : :; N. o :

14 14 GILERT WEINSTEIN We conclude this section by verifying as mentioned earlier that H j is degenerate if and only if its surface gravity vanishes. If we let j = + w j, then j is a Killing eld tangent to H j, which is null but non-zero on H j. Thus, H j is a Killing horizon. Let j be the surface gravity of H j, then? d j j j j j j 2! 2 j ; on H j, see [2, p. 150]. However, j j j 2 =? 2 e 2u + e?2u (w? w j ) 2, hence in view of (2.21), d? j j j 2 2 = 4 2 e 4u jdj 2 + O( 4 ): Therefore we conclude that e 2u jdj 2! j on H j as claimed in Section 2.1, see also [2, p. 192] oundary Conditions. In order to complete the reduction, and formulate a reduced problem for the stationary and axially symmetric Einstein/Maxwell Equations, we need to derive boundary conditions for ' on and as r! 1 in R 3, where r = p 2 + z 2. Since u behaves like log near and as r! 1, these boundary conditions will be singular. They may be viewed as prescribed singularities for the map ' on and at innity. In particular, ER ~ 3(') the total energy of ' will be innite. Let u 0 be the potential of a uniform charge distribution of, normalized so that u 0 + log! 0 as r! 1 in R 3. The map (u 0 ; 0; 0; 0) is the solution of the corresponding problem when j = q j = 0, i.e., the Weyl solution. Dene N + 1 singular harmonic maps ' j = (u 0 ; j?1 ; 0; q j?1 ): R 3 n! H 2 C, j = 1; : : :; N + 1. Note that each maps into a geodesic j (t) =? t; j ; 0; q j ) of H 2 C. Furthermore, the distance dist('(x); ' j(x)) remains bounded in a neighborhood of j. Indeed, if x 2 j is not an end point of j, then in a ball small enough about x, the function u + log is bounded, and the functions v,, and are even functions of. It is straightforward now using (2.11) to check that dist('; ' j ) is bounded on. A similar argument can be used at the end points. Finally, in order to get a uniform estimate on, one uses the asymptotic atness of (M; g; F ), see the boundary conditions in [2]. This motivates the following denition [23]. Denition 4. Let H be a real, complex, or quaternionic hyperbolic space, let R n be a smooth bounded domain, let be a closed smooth submanifold of codimension at least 2, possibly with boundary, let 0, and let '; ' 0 : n! H be harmonic maps. We say that ' and ' 0 are asymptotic near 0 if there is a neighborhood 0 of 0 such that dist('; ' 0 ) 2 L 1 ( 0 n 0 ). Thus, in this terminology, ' is asymptotic to ' j near j. Now let e = 1 [ N+1 be the union of the unbounded components of. R 3 n e is the domain of a harmonic map ~' corresponding to a Kerr-Newman

15 EINSTEIN/MAXWELL EQUATIONS 15 solution with one black hole, such that ~' is asymptotic to ' 1 near 1, and asymptotic to ' N+1 near N+1. The map ~' can be written down explicitly. Introduce the global parameters q,, m, and a dened by: q = NX j=1 q j ; 2 = 2 NX j=1 j + N?1 X j=1 r j ; ma = 1 4 NX j=1 j ; m 2? a 2? q 2 = 2 : Here q is the total charge, 2 is the length, in the Euclidean metric of R 3, of the interval of the z-axis from the south pole of I 1 to the north pole of I N+1, m and a are the mass and angular momentum per mass of the corresponding Kerr-Newman spacetime. Dene the Schwarzschild-type elliptical coordinates (s; ) on R 3 dened by: 2 = (s? m + )(s? m? ) sin 2 ; z = (s? m) cos ; where we have assumed without loss of generality that the point z = 0 on A is the center of the interval ~ I = A n e. Then, ~' = (~u; ~v; ~; ~ ) is given by: (2.31) ~u = log sin log s 2 + a 2 + a 2 (2ms? q 2 )(s 2 + a 2 cos 2 )?1 sin 2 ~v = ma cos (3? cos 2 )? a2 (q 2 s? ma 2 sin 2 ) cos sin 2 s 2 + a 2 cos 2 + 2ma ~ =?qas(s 2 + a 2 cos 2 )?1 sin 2 ~ = q(s 2 + a 2 cos 2 )?1 (s 2 + a 2 ) cos + q: Furthermore, the asymptotic atness of (M; g; F ) implies that dist('; ~')! 0 as r! 1 in R 3, see [2]. Denition 5. Assume that R n, and let '; ' 0 : R n n! H be harmonic maps. We say that ' and ~' are asymptotic at innity, if dist('; ~')! 0 as x! 1 in n. We can now formulate the reduced problem for the stationary and axially symmetric Einstein/Maxwell Equations. Let 4N? 1 parameters (2.32) ( 1 ; : : :; N ; r 1 ; : : :; r N?1 ; 1 ; : : :; N ; q 1 ; : : :q N ) be given, with j ; r j j=1 I j, where I j are N intervals on the z-axis, of lengths 2 j, distance r j apart, and let e = 1 [ N+1. Dene the harmonic maps ' j > 0. Construct the set = A n S N = (u 0 ; j ; 0; q j ), and the map ~': R 3 n e! H according to Equations (2.31). We associate with the given parameters (2.32), the data (' 1 ; : : :; ' N+1 ; ~') consisting of these N +2 singular harmonic maps. The 4N? 1 physical parameters, m j, d j, L j, and q j, are determined only a posteriori. Note that the parameters (2.32) are nonetheless geometric invariants. Reduced Problem. Given a set of 4N? 1 parameters as in (2.32), prove the existence of a unique axially symmetric harmonic map ': R 3 n! H 2 C

16 16 GILERT WEINSTEIN such that ' is asymptotic to ' j near j for each j = 1; : : :; N + 1, and such that ' is asymptotic to ~' at innity. This problem will be solved in Section 3, see the Main Theorem and its Corollary. Clearly, of at least equal importance is the converse: given a solution of the Reduced Problem, is there a corresponding solution of the Einstein/Maxwell Equations which is stationary and axially symmetric, and which is a simply connected domain of outer communications of a globally hyperbolic and asymptotically at spacetime with N nondegenerate components to the event horizon? This question will be examined briey in Section Harmonic Maps with Prescribed Singularities In this section, we solve the Reduced Problem for the stationary and axially symmetric Einstein/Maxwell Equations. We state a somewhat more general result, whose proof, however, requires no additional eort. Let R n, and let be a smooth closed submanifold of R n of codimension at least 2. Let H be one of the classical globally symmetric space of noncompact type and rank one, i.e., either real, complex, or quaternionic hyperbolic space of real dimension m. For our purpose, we take a harmonic map ' = (' 1 ; : : :; ' m ): n! H to mean a map ' 2 C 1 ( n ; H), which is for each 0 n a critical point of the energy functional: E 0 = 0 jd'j 2 ; P where jd'j 2 n = k=1 h@ k 'i is the energy density, and h; i denotes the metric of H. This is equivalent to requiring ' to satisfy in n the following elliptic system of nonlinear partial dierential equations: ' a + nx k=1? a k ' k ' c = 0; where? a bc are the Christoel symbols of H. Denition 6. Let R n be a smooth domain, and let be a smooth closed submanifold of of codimension at least 2. Let be a geodesic in H. We say that a harmonic map ': n! H is a -singular map into if (i) '( n ) (R) (ii) '(x)! (+1) as x! (iii) There is a constant > 0 such that jd'(x)j 2 > dist(x; )?2 in some neighborhood of. We will assume that we are given N + 1 disjoint smooth closed connected submanifolds j R n of codimension at least 2, possibly with boundary. Let J = f1 6 j 6 N + 1: j boundedg, and let e = Sj =2J j be the union of the unbounded components of. For p 2 H, and a geodesic in H, let dist(p; ) denote the distance from p to (R), i.e., inf t dist? p; (t).

17 EINSTEIN/MAXWELL EQUATIONS 17 Denition 7. Singular Dirichlet data for the harmonic map problem with prescribed singularities on R n n into H consists of N + 2 harmonic maps (' 1 ; : : :; ' N+1 ; ~') satisfying: (i) ' j : R n n! H is -singular into a geodesic j of H. (ii) dist(p0 ; ' j )? dist(p 0 ; ' j 0) 6 C, for all 1 6 j; j0 6 N + 1, where p 0 is some xed point of H. (iii) ~': R n n e! H is asymptotic to ' j near j for j =2 J. (iv) dist( ~'(x); j )! 0 as x! j, for j =2 J outside a compact subset of R n. The following result is our main existence theorem. Main Theorem. Let (' 1 ; : : :; ' N+1 ; ~') be singular Dirichlet data on R n n. Then there exists a unique harmonic map ': R n n! H, such that for each j = 1; : : :; N + 1, ' is asymptotic to ' j near j, and such that ' is asymptotic to ~' at innity. In other words, given a harmonic map ~' with the correct asymptotic behavior at innity, we can modify it to a harmonic map ' with prescribed singular behavior near the compact components of. It is easily seen that the data (' 1 ; : : :; ' N+1 ; ~') dened in Section 2.4 constitute singular Dirichlet data on R 3 n. Thus, as an immediate corollary to the Main Theorem, we obtain: Corollary. The Reduced Problem for the stationary and axially symmetric Einstein/Maxwell Equations has a unique solution. We note that the axial symmetry of the solution ' follows immediately from the uniqueness statement in the Main Theorem. corresponding to the vacuum equations was In [21], the case H = H 2 R studied, and in [23] we proved a version of this theorem for maps ': n! H where is bounded and is compactly contained in. The proof of our Main Theorem, which combines the ideas in both these papers, is divided into three steps. In step one we consider the problem on a ball R n, but allow to extend esides some minor technical points there are no new diculties in this step. In step two, we use the known map ~' to obtain pointwise a priori bounds on d('; ' j ) and d('; ~') which are independent of the size of. This allows us to complete the proof in step three Preliminaries. The primary tools for the proof of the Main Theorem are the usemann functions on Cartan-Hadamard manifolds, see [6, 8]. We note that H is such a manifold with pinched sectional curvatures?4 6 6?1. Let be a geodesic in H, then f (p), the usemann function associated with evaluated at p 2 H, is the renormalized distance between p and the ideal point (+1) It is dened by: f (p) = lim t!1? dist? p; (t)? t :

18 18 GILERT WEINSTEIN It is not dicult to see that the limit is obtained uniformly for p in compact subsets of H. It is well-known that f is analytic, convex, and has gradient of constant length one. The level sets, which we denote by S (t) = fp 2 H; f (p) = tg, are called horospheres. They are dieomorphic to R m?1. We will also use the horoballs dened by (t) = fp 2 H; f (p) 6 tg, and the geodesic balls R (p) = fq 2 H;? dist(p; q) 6 Rg. Two geodesics and 0 are said to be asymptotic if dist (t); 0 (t) is bounded for t > 0. This is clearly an equivalence relation. The boundary of H is dened to be the set of equivalence classes of geodesics in H. We will denote the equivalence class of in H by (+1). Let be a geodesic in H. We denote the reverse geodesic t 7! (?t) by?, and we also write (?1) =?(+1). We will introduce an analytic coordinate system in H adapted to. Let u = f?. Let v 0 : S? (0)! R m?1 be an analytic coordinate system on S? (0) centered at (0). The integral curves of rf? are geodesics parameterized by arclength. We `drag' the coordinate system v 0 along these integral curves. More precisely, let t be the analytic ow generated by this vector eld, then?t maps S? (t) to S? (0), hence v t = v 0?t is an analytic coordinate system on S? (t). Dene v : H! R m?1 by vj S?(t) = v t, and let = (u; v): H! R m, then is an analytic coordinate system on H. In these coordinates, the metric of H reads: ds 2 = du 2 + Q p (dv); where for each p 2 H, Q p is a positive quadratic form on R m?1. We note that the convexity of f? implies that for each non-zero X 2 R m?1, and each xed v 2 R m?1, Q?1 (u;v)(x) is a positive increasing function of u. The following lemmas were proved in [23]. They are quoted here without proofs. Except for Lemma 5, they depend only on the bound?4 6 6?1 for the sectional curvatures of H. Lemma 1. For any (u; v) 2 R m and any X 2 R m?1, there holds:? Q?1(u;v)(X) 6 4Q?1(u;v)(X): (3.1) Lemma 2. Let be a geodesic in H. Then for any t 0 2 R, and any T > 0, we have where R = T + log 2. (?t 0 + T ) \? (t 0 + T ) R ((t 0 )); Lemma 3. Let and 0 be geodesics in H, such that (?1) = 0 (?1) and (+1) 6= 0 (+1). Then for some d 2 R, we have: lim t!1 lim t!?1? f?? f? f? + f 0 (t) = d 0 (t) = 0:

19 EINSTEIN/MAXWELL EQUATIONS 19 Lemma 4. Let be a geodesic in H. Then, for any t 0 2 R, and any T > (1=2) log 2, we have hrf (p); rf? (p)i > 0; 8p 2 S (?t 0 + T ) n? (t 0 + T ) where h; i denotes the metric on H. Lemma 5. Let be a geodesic in H. Then there is an analytic coordinate system = (u; v): H! R m with u = f? such that in these coordinates, the metric of H is given by: and satises the following conditions: ds 2 = du 2 + Q p (dv); (i) Let R > 0, t 0 2 R, and let 0 be a geodesic in H with 0 (?1) = (?1). Then, there exists a constant c > 1 such that for all t > t 0, and all p 2 R? 0 (t), there holds: 1 c Q 0 (t)(x) 6 Q p (X) 6 c Q 0 (t)(x); 8X 2 R m?1 : (ii) For all t; t 0 2 R, the set S? (t) \ 0(t 0 ) is star-shaped in these coordinates with respect to its `center', the point where 0 intersects S? (t). The following lemma is a slight generalization of Lemma 8 in [23]. Lemma 6. Let = R R n be the ball of radius R centered at the origin, and let be a smooth closed submanifold of R n of codimension at least 2. Suppose that u 2 C 1 ( n ) satises u > 0, and 0 6 u 6 1 in n. Then, for any R 0 < R, there holds (3.2) R 0 jruj 2 6 C; where C is a constant that depends only on R, R 0, and n. Furthermore, we have: (3.3) Proof. If 2 C 0;1 sup u 6 sup u: 0 ( n ) with , then integrating by parts the inequality 2 u u > 0, and using 0 6 u 6 1, we obtain: (3.4) Dene the cut-o function " by: " = 2 jruj jrj 2 : 8 >< 2? log r= log " if " 2 6 r 6 " >: 0 if r 6 " 2 1 if r > ",

20 20 GILERT WEINSTEIN where r(x) = dist(x; ), and let 0 2 C 1 0 () be another cut-o satisfying and 0 = 1 on R 0. Now, take = " 0 in (3.4), and then let "! 0 to get: ( 0 ) 2 jruj r 0 2 ; see [23, Lemma 8], The estimate (3.2) follows immediately. Now, the same argument shows that u > 0 weakly throughout, hence, by the maximum principle, we obtain (3.3) over R 0 instead of. Finally, let R 0! R to get (3.3) Step One. In this step, we restrict our attention to a ball R n large enough so that (iv) in Denition 7 holds We prove the existence of a solution ': n! H in a manner similar to the proof of [23, Theorem 1]. We will assume all the hypotheses of the Main Theorem. Some of the elements in the proof of Proposition 1 will be used also in the next steps. Proposition 1. Let R n be a large enough ball. Then there is a unique harmonic map ': n! H such that ' is asymptotic to ' j near j for each j = 1; : : :; N + 1, and ' = ~' n. Proof. The proof follows closely that of Proposition 2 in [23]. We begin with the uniqueness. Let ' and ' 0 be two such maps, then clearly dist('; ' 0 ) 2 L 1 ( n ), and ' = ' 0 n. Consider the function u = dist('; ' 0 ) 2. We have that u > 0 on n, see [19], u is bounded on n, and = 0. Thus, by Lemma 6, it follows that u = 0 and hence ' = ' 0 on n. For the existence proof, we set-up a variational approach. Pick a smooth open cover of R n which separates the j 's, i.e., a collection f 0 j gn+1 j=1 of smooth open sets such that S j 0 j = R n, j 0 j, and j \ 0 j 0 = ; for j 6= j 0. It can easily be arranged that if j 2 J then j. Let j = 0 j \. Let f jg N+1 j=1, be a partition of unity subordinate to the cover f j g N+1 j=1 of such that j = 1 near j. We may assume, without loss of generality, that all the geodesics j have the same initial point Indeed, if this is not the case, we can write ' j = j U j, where U j is a harmonic function on n. Since H has strictly negative curvature, there are geodesics 0 j with a common initial point p 1 = 0 j (?1) such that j and 0 j are asymptotic. Let '0 j = 0 j U j, then it is easy to see that ' j and ' 0 j are asymptotic near j, so that we may replace ' j by ' 0 j. We may also assume that the U j's are all equal. Indeed, they dier at most by constants, since, by (ii) of Denition 7, U j? U j 0 is a bounded harmonic function on R n. So we may replace ' 0 j by '00 j = 0 j u 0, where U j = u 0 + c j for some N + 1 constants c j. Again ' 00 j is asymptotic to ' j near j. Note that the other conditions in Denition 7 are not aected by these changes, and that the singular Dirichlet data (' 00 1 ; : : :; '00 N+1 ; ~') is

21 EINSTEIN/MAXWELL EQUATIONS 21 equivalent to (' 1 ; : : :; ' N+1 ; ~'). We will assume these changes have already been carried out. Let = (u; v) be the coordinate system on H with u = f?1 given in Lemma 5. We will identify any map ': n! H with its parameterization ' = (u; v) whenever no confusion can arise. We have ' j = (u 0 ; w j ) where w j 2 R m?1 are constants. We write u 0 = P N+1 j=1 u j where u j is a harmonic function on R n which is singular only on j. Without loss of generality, we may assume that for j 2 J, u j (x) = 0 and in particular u j > 0 in. For any domain, dene the norms: (3.5) (3.6) kuk = kvk 'j ; = n u 2 + jruj 2o 1=2 ; n 1=2 jvj 2 + Q 'j (rv)o ; where Q 'j (rv)j x = P n k=1 Q ' j (x)? rk v(x) for x 2 n j. We will use the Hilbert spaces: H 1 () = fu:! R; kuk < 1g H ' j 1 (; Rm?1 ) = fv :! R m?1 ; kvk 'j ; < 1g; and the subspaces H 1;0 () and H ' j 1;0 (; Rm?1 ), dened to be the closure in H 1 () of C0 1(), and the closure in H ' j 1 (; Rm?1 ) of C0 1( n j; R m?1 ), respectively. We write ~' = (~u; ~v), and introduce the function ~v 0 obtained by truncating ~v near the bounded components of and setting it to its prescribed values w j 2 R m?1 : ~v 0 = Y 1 X? j ) A ~v + j2j(1 j2j j w j : Note that ~v 0 = w j near j for j 2 J, hence clearly ~v 0?w j 2 H ' j 1 ( j; R m?1 ). We also decompose u 0 = U 0 + e U0 into U 0 = P j2j u j, the contribution from the bounded components of, and e U0 = P j62j u j, the contribution from the unbounded components. Clearly, U 0 = e U0 = 0 on n and U 0 = 0 Since ~' is a harmonic map, ~u is subharmonic on R n n, e hence (~u? U0 e ) > 0 on R n n. e Furthermore, j~u? U0 e j is bounded on the ball R+1. Indeed, on the one hand, for j 62 J, we have j~u? U0 e j 6 j~u? u 0 j+u 0 6 dist( ~'; ' j ) + C j on 0 j \ R+1 for some constant C j. On the other hand, for j 2 J, we have that both ~u and e U0 are bounded on j. Hence it follows from Lemma 6 that ~u? e U0 2 H 1 (). In addition, ~u satises the equation ~u = (@=@u)? Q ~' (r~v), and Lemma 1 implies that: Q ~' (r~v) (~u? e U0 );

22 22 GILERT WEINSTEIN in R+1 n e. Multiplying by a cut-o as in the proof of Lemma 6 and integrating over, we conclude, after taking the appropriate limit, that (3.7) Q ~' (r~v) < 1: Dene H to be the space of maps ' = (u; v): n! H satisfying: 8 >< >: For maps ' 2 H dene: u? ~u? U T 0 2 H 1;0 (); N+1 v? ~v 0 2 j=1 H ' j 1;0 (; Rm?1 ); dist('; ' j ) 2 L 1 ( j n j ); 8j = 1; : : :; N + 1: F (') = n o jr(u? u 0 )j 2 + Q ' (rv) : For R > 0, dene H R to be the space of maps ' 2 H such that dist('; ' j ) 6 R for a.e. x 2 j n j for j = 1; : : :; N +1. We will need the following lemma which is a direct consequence of Lemma 5. Lemma 7. Let R > 0, then there is a c > 1 such that for all ' 2 H R, there holds for each j = 0; : : :; N and for all X 2 R m?1 : 1 c Q ' j (x)(x) 6 Q '(x) (X) 6 c Q 'j (x)(x); for a.e. x 2 j n j. R We already know that j Q 'j (r~v 0 ) < 1 for j 2 J, and it follows from Lemma 7 and (3.7) that the same holds also for j 62 J. Indeed, since ~' is asymptotic to ' j near j for j =2 J, there is a constant c > 1 such that j Q 'j (r~v) 6 c j Q ~' (r~v) < 1: Since j is smooth and r j = 0 near e, we obtain that R j Q 'j (r~v 0 ) < 1 as claimed. Now, let ' = (u; v) 2 H, then there is a constant c > 1 such that: (3.8) Q ' (rv) 6 c NX j=0 j Q 'j (rv) < 1: Since also u? u 0 = (u? ~u? U 0 ) + (~u? e U0 ) 2 H 1 (), we conclude that F is nite on H. It is straightforward to check that a minimizer ' 2 H of F is a harmonic map on n, see [23]. Hence, by the regularity theory for harmonic maps [17, 18], we have ' 2 C 1 ( n ; H). Furthermore, ' is asymptotic to ' j near j for j = 1; : : :; N + 1 by construction, and ' = ~' n. Thus, to prove Proposition 1, it suces to show that F admits a minimizer in H. It can be shown, exactly as in [23, Proposition 1], that F admits a minimizer in H R. The main steps are as follows. Consider a minimizing sequence ^' k = (^u k ; ^v k ) in H R. Then for some subsequence k 0, ^u k 0? u 0 converges

23 EINSTEIN/MAXWELL EQUATIONS 23 weakly in H 1 () and pointwise a.e. in. Furthermore, using the bound dist( ^' k ; ' j ) 6 R in j, it is shown that, perhaps along a further subsequence, ^v k 0? ~v 0 also converges weakly in each H ' j 1;0 (; Rm?1 ), and pointwise a.e. in. Let ' = (u; v) 2 H R be the weak limit. Then clearly we have (3.9) jr(u? u 0 )j 2 6 lim inf Furthermore, Q ' is equivalent to Q 'j (3.10) where Q ' (rv) = lim 6 lim inf k = Q ' (rv; r^v k 0) " jr(^u k 0? u 0 )j 2 : on j, hence we have k 0Q ' (rv) 1=2 ( Q ' (r^v k )=Q ^'k (r^v k ) if ^v k 6= 0 1 otherwise: 1=2 # Q ^'k 0 (r^v k 0) ; However k 0! 1 pointwise a.e. in, and k 0 is bounded. Thus, we conclude that lim and therefore (3.10) implies: Q ' (rv) 6 k 0Q ' (rv) = Q ' (rv) 1=2 lim inf Combining this with (3.9) we obtain Q ' (rv); F (') 6 lim inf F ( ^' k 0) = inf H R F; Q ^'k 0 (r^v k 0) 1=2 : and it follows that ' is a minimizer in H R. Thus, it remains to prove that for some R > 0 large enough, inf H F = inf HR F. This is the content of the next lemma. Lemma 8. There is a constant R > 0 such that for every " > 0, and every ' 2 H, there is ' 0 2 H R such that F (' 0 ) 6 F (') + ". Proof of Lemma 8. Let H be the space of maps ' = (u; v) 2 H such that v = w j in a neighborhood of j for j 2 J, ' = ~' outside some compact set K, and v = w j in a neighborhood of j \ K for j 62 J. Lemma 8 will follow immediately from: (i) for each ' 2 H and each " > 0 there is ' 0 2 H such that F (' 0 ) 6 F (') + "; and (ii) for each ' 2 H there is ' 0 2 H R such that F (' 0 ) 6 F ('). The proof of (i), a standard approximation argument, is practically unchanged from [23, Lemma 11]. We turn to the proof of (ii). The bound dist('; ' j ) 6 R will be achieved consecutively on each j beginning with all j 62 J. The case j 2 J is simpler. Assume without loss of

24 24 GILERT WEINSTEIN generality that 1 62 J. Introduce a `dual' coordinate system = (u; v) with u = f 1 as given by Lemma 5. Write ds 2 = du 2 + Qp (dv): for the metric of H in these coordinates. Set u 0 =?u 1 + P N+1 j=2 u j. Let ' 2 H and write ' = (u; v). Then, since jruj 2 +Q ' (rv) = jruj 2 + Q' (rv), we nd that in n : jr(u? u 0 )j 2 + Q ' (rv) = jr(u? u 0 )j 2 + Q' (rv)? 2 r(u + u) ru 1? 2 N+1 X j=2 r(u? u) ru j + 4 N+1 X j=2 ru j ru 1 : The idea is now to integrate this identity over, use the right hand side to truncate u? u 0, and the left hand side to truncate u? u 0. The convexity of the usemann functions u and u on H, expressed in the monotonicity of Q and Q, ensures that these truncations do not increase F. This can be viewed as a weak form of a variational maximum principle. Some care, however, needs to be taken because of the singularities on. Integrating rst over n ", where " = fx 2 ; dist(x; ) 6 "g, we obtain: n" n jr(u? u 0 )j 2 + Q ' (rv) where: (3.11) I " (') = o = n"?2 div? (u + u) ru1 X N+1? 2 j=2 n" n jr(u? u 0 )j 2 + Q ' (rv)? N+1 X div (u? u) ru j + 4 j=2 o ru j ru 1 9 = ; : + I " ('); A tedious but straightforward verication shows that I " (')! C as "! 0, where C is a constant independent of ' 2 H, yielding an integral identity: n o (3.12) F (') = jr(u? u 0 )j 2 + Q ' (rv) + C: Note that if " S > 0 is small enough, we can n " ) into a N+1 disjoint union j ) [ (@ n " ). Then, for example, the rst term in (3.11) can be integrated by parts: (3.13)?2? 0 div (u + u) ru 1 + n " 1 + N+1 X " j 1 A (u + 1 For the rst integral on the right hand side of (3.13), we have, as "! 0: (3.14) (u + @! (u + :