SINGULAR HARMONIC MAPS

INTO HYPERBOLIC SPACES

AND APPLICATIONS TO GENERAL RELATIVITY

BY LUC L. NGUYEN

A dissertation submitted to the

Graduate School—New Brunswick
Rutgers, The State University of New Jersey
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
Graduate Program in Mathematics

Written under the direction of

YanYan Li
and approved by

New Brunswick, New Jersey

May, 2009
 ABSTRACT OF THE DISSERTATION

Singular harmonic maps

into hyperbolic spaces

and applications to general relativity

by Luc L. Nguyen
Dissertation Director: YanYan Li

Harmonic maps with singular boundary behavior from a Euclidean domain into hyper-
bolic spaces arise naturally in the study of axially symmetric and stationary spacetimes
in general relativity. In particular, the study of multi-black-hole configurations and the
force between co-axially rotating black holes requires, as a first step, an analysis on the
boundary regularity of the “next order term” of those harmonic maps. We carry out
this analysis by considering those harmonic maps as solutions to some homogeneous
divergence systems of partial differential equations with singular coefficients. We then
apply our result to study the regularity of axially symmetric and stationary electrovac
spacetimes, which extends previous works by Weinstein [22], [23] and by Li and Tian
[10], [11], [12]. This dissertation is based on a preprint of the author [16].

ii
 Acknowledgements

I would like to thank Professor YanYan Li who suggested me this problem as the subject
of a Ph.D. dissertation and has given much advice since. I wish to thank Professors
Gang Tian and Gilbert Weinstein for many insightful discussions and for their support
and encouragement. I am grateful to Professors Piotr Chruściel and Michael Kiessling
whose comments on the draft help improve the presentation of this dissertation. I am
indebted to Professors Abbas Bahri, Haı̈m Brezis and Zheng-Chao Han for their interest
in this work and their willingness to listen to me several times. I thank Professor Penny
D. Smith for letting me know of [20]. Finally, I dedicate this dissertation to my parents,
Bô´ Minh and Me. Sen, and my wife, Duyên, without whose supports this work could
not have been done.

This research was funded in part by a grant from the Vietnam Education Foundation
(VEF)1 and the Rutgers University and Louis Bevier Dissertation Fellowship.

1
The opinions, findings, and conclusions stated herein are those of the author and do not necessarily
reflect those of VEF.

iii
 Table of Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2. The model PDE problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3. The case of a single linear equation . . . . . . . . . . . . . . . . . . . . . 14

4. Hölder regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Appendix A. Hyperbolic spaces . . . . . . . . . . . . . . . . . . . . . . . . . 41

1,p
Appendix B. The space WΣ (Ω, w) . . . . . . . . . . . . . . . . . . . . . . . 43

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

iv
 1

Chapter 1

Introduction

A spacetime in general relativity is a 4-manifold M equipped with a Lorentzian metric

g which satisfies the Einstein field equations,
1
Rab − R gab = 2 Tab .
2
Here Rab and R denote respectively the Ricci and scalar curvature of M , and Tab is
the energy-momentum tensor which represents matter. It is of interest to study the
regularity of spacetimes in equilibrium consisting of more than one body. In [1], Bach
and Weyl showed that any axially symmetric and static vacuum spacetime outside
two bodies possesses a conical singularity along the axis connecting the bodies. Later,
Bunting and Massood-ul-Alam [3] cleverly used the positive mass theorem of Schoen
and Yau to show that there does not exist any regular static “multiple-body vacuum
spacetimes”. For non-static spacetimes, much less is known. Regardless of regularity,
Weinstein [22],[23],[27] used the harmonic map reduction, known as the Ernst-Geroch
reduction, to construct a family of multi-black-hole axially symmetric and stationary
vacuum/electrovac solutions, of which a regular one must be a member. In the vacuum
case, Li and Tian [11], [12] and Weinstein [22], [23], [24] independently proved some reg-
ularity results for the reduced harmonic maps and then used them to show that within
the solutions constructed by Weinstein, there is a continuum of irregular solutions. In
this paper we bridge the methods used by Li and Tian and by Weinstein to extend the
above regularity results to the axisymmetric stationary electrovac case.

To describe the reduction used by Weinstein, we first introduce the notion of singular
harmonic maps (see [25]). Let Γ be a subset of the z-axis in R3 obtained by removing
some bounded line segments. Let h be the Newtonian potential created by a charge
distribution of strictly positive density along Γ. Note that h is a harmonic function on
 2

R3 \ Γ and h ‘behaves like’ some negative multiple of log ρ near an interior point of Γ
where ρ is the distance to the z-axis. Let H be either the real or the complex hyperbolic
plane. For HR , we use the standard half-plane model {(X, Y ) : X > 0} with the metric

ds2 = X −2 (dX 2 + dY 2 ).

For HC , we model it by R4 = {(u, v, χ, ψ)} with the metric

ds2 = du2 + e4u (dv − ψ dχ + χ dψ)2 + e2u (dχ2 + dψ 2 ).

(See Appendix A for a derivation of this line element from the standard disk model
of HC .) Then given a geodesic ζ in H, ζ ◦ h is a harmonic map from R3 \ Γ into H.
Moreover, as x → Γ, ζ ◦ h(x) approaches the ideal point ζ(+∞) ∈ ∂H. Recall that a
map Φ : Ω ⊂ R3 → H is harmonic if it satisfies in local coordinates the equations

∆Φα + Γαβγ (Φ)DΦβ · DΦγ = 0

where Γαβγ are the Christoffel symbols of H.

Definition 1.1 Let Γ be a subset of the z-axis in R3 obtained by removing n bounded

line segments. To each component Γj of Γ, 1 ≤ j ≤ n+1, we associate an ideal point pj
∈ ∂H. Pick any normalized geodesic ζj so that ζj (+∞) = pj . Let h be the Newtonian
potential created by a line charge distribution of strictly positive density on Γ. We say
that a map Φ : R3 \ Γ → H is a singular harmonic map controlled by the distribution
h and the ideal points pj if Φ is harmonic and near each component Γj the hyperbolic
distance between Φ and ζj ◦ h is bounded. Sometimes, we simply say that Φ is a singular
harmonic map controlled by h.

The Ernst-Geroch reduction formulation states that every axially symmetric, sta-
tionary exterior solution of the Einstein vacuum equation can be conveniently put in
the form (see [22] or [8], for example)

ρ2 2 1 2µ 2
ds2 = − dt + X(dϕ − p dt)2 + e (dρ + dz 2 )
X X

where ρ is the distance to the z-axis, ϕ is the cylindrical angle around the z-axis, and X,
p and µ are functions on R3 \ Γ which are determined by the following four conditions.
 3

(i) X, p and µ are independent of the angle variable ϕ.

(ii) X is the first component of some axially symmetric singular harmonic map (X, Y )
from R3 \Γ into the real hyperbolic plane HR which is controlled by the Newtonian
potential created by a uniform line charge distribution h of unit density and some
ideal points on ∂HR . In particular, (X, Y ) satisfies the harmonic map equations

|DX|2 − |DY |2
∆X = , (1.1)
X
2DX · DY
∆Y = . (1.2)
X

(iii) p satisfies
ρ ρ
dp = − Yz dρ + 2 Yρ dz. (1.3)
X2 X

(iv) µ satisfies

ρ ρ
dµ = (X 2 − Xz2 + Yρ2 − Yz2 ) dρ + (Xρ Xz + Yρ Yz ) dz. (1.4)
4X 2 ρ 2X 2

Notice that the harmonic map equations (1.1) and (1.2) give the integrability conditions
needed to integrate (1.3) and (1.4).

Using this reduction and a variational approach, Weinstein proved the existence
of a family of (possibly singular) spacetimes which can be interpreted as equilibrium
configurations of asymptotically flat co-axially rotating vacuum black holes ([22], [23]).
Moreover, he showed that this family is uniquely parametrized by 3n − 1 parameters (n
is the number of black holes) which are interpreted as the masses and angular momenta
of the black holes, and the distances between them. This result implies in particular
Robinson’s theorem on the uniqueness of the Kerr solutions [18], [19].

After a first look at the reduction, one might have the impression that given any
solution to the singular harmonic map equations (with a right rate of singularity, of
course), one could easily cook up a solution to the Einstein vacuum equations. This is
in fact not the case, which was probably first observed by Bach and Weyl. As one tries
to construct a spacetime out of a singular harmonic map, one might possibly introduce
a conical singularity along Γ. A necessary and sufficient condition for regularity is

lim (µ − log X + log ρ) = 0 along Γ. (1.5)

ρ→0
 4

Bach and Weyl showed in [1] that this limit is nonzero in the static setting, i.e. Y ≡
0. Their method does not seem welcoming of the general setting since it relies on the
explicit form of the solution. Even though there has been some progress in getting the
explicit form of solutions for multiple-body spacetimes, e.g. the famous double Kerr
solutions of Kramer and Neugebauer [9], the dependence of the validity or invalidity of
(1.5) on the parameters seems still unclear in general.

Concerning the equation (1.5) itself, some regularity structure across the symmetry
axis Γ of the harmonic map (X, Y ) is required in order to make sense of the limit on the
left hand side. Note that because of the singularity of h, (X, Y ) necessarily approaches
∂HR near Γ. Therefore, the equations (1.1)-(1.2) are not satisfied across Γ, and so
one cannot apply directly well-known results on the regularity of harmonic maps into
negatively curved targets to obtain the required regularity for X and Y near Γ. This
regularity issue was settled independently by Weinstein and by Li and Tian. Weinstein
showed that, about any interior point of Γ, any singular harmonic maps corresponding
to the spacetimes he constructed in [22] and [23] can be “decomposed” into an explicit
singular part and a C ∞ -regular part. Independently, using a different approach, Li
and Tian [11], [12] proved a weaker version: C k,α regularity for the regular part, which
suffices to justify the limit on the left hand side of (1.5). On the other hand, their result
remains valid even if the harmonic map (X, Y ) is not axisymmetric or the singular
control h is the potential of a uniform charge distribution of arbitrary positive density
along Γ. To describe their results more precisely, we define the following weighted
spaces.

Definition 1.2 Let Ω be a domain in Rn and Σ a subset of Ω. Let w be a positive

measurable function in Ω. We denote by Lp (Ω, w) (1 ≤ p ≤ ∞) the space of p-integrable
functions with respect to the measure w(x) dx, equipped with its standard Banach space
structure, i.e. with the norm
Z 1/p
kgkLp (Ω,w) = |g|p w dx .
Ω

The spaces WΣ1,p (Ω, w) and W0,Σ

1,p
(Ω, w) are respectively the completions of Cc∞ (Ω̄\Σ)
 5

and Cc∞ (Ω \ Σ) with respect to the norm

Z h i 1/p
p p
kgkW 1,p (Ω,w) = |g| + |Dg| w dx .
Σ
Ω

1 (Ω, w) for W 1,2 (Ω, w) and W 1,2 (Ω, w).

When p = 2, we also write HΣ1 (Ω, w) and H0,Σ Σ 0,Σ

Note that these two spaces are Hilbert spaces. Also, if w is bounded from below by a
strictly positive number, they are naturally embedded into the familiar Sobolev spaces
H 1 (Ω) and H01 (Ω), respectively.

Li and Tian [11], [12] and Weinstein [22], [23] proved the following.

Theorem A (Y.Y. Li - G. Tian; G. Weinstein) Let Γ be the z-axis in R3 with

some line segments removed and ρ the distance function to the z-axis. Let h be the
potential of a uniform line charge distribution of density γ > 0 along Γ. If (X, Y ) is
a singular harmonic maps from R3 \ Γ into HR controlled by h and some ideal points
on ∂HR with log X + h ∈ H 1 (R3 ), Y ∈ H 1 (R3 , e2h )), then log X + h and Y are C k,α
across the interior of Γ for any integer k and α ∈ (0, 1) with k + α < 4γ and C ∞ in
R3 \ Γ. Also, near any compact subset of the interior of any component of Γ, Y has the
asymptotic expansion
Y = C + O(ρk+α )

where C is a constant (see [11],[12]).

Moreover, if (X, Y ) is axially symmetric about the z-axis and γ = 1, (log X + h, Y )

is everywhere C ∞ except possibly at the endpoints of Γ (see [22],[23]).

Remark 1.1 In the above theorem, we can allow k + α = 4γ. See Corollary 3.1 and
Remark 3.2.

After settling the regularity of the reduced harmonic maps, Li and Tian [11], [12],
[10] and Weinstein [24] went forward to study if there is a conical singularity in an
axially symmetric stationary vacuum spacetime. As mentioned above, such conical
singularity exists if and only if equation (1.5) is violated. Using Theorem A, the two
works showed that the limit on the left hand side of equation (1.5) exists and depends
continuously on the masses, angular momenta and distances between two bodies. More
 6

importantly, they proved that there are several continuous sets of parameters which give
rise to spacetimes violating (1.5). Unfortunately, their results do not reveal whether
there is any set of parameter that realizes (1.5) except those corresponding to the Kerr
spacetimes, i.e. one-body spacetimes.

Later, in an attempt to shed more insight into the problem, Weinstein started
analyzing a generalization of the problem for Einstein-Maxwell equations ([25], [26],
[27]). It should be noted that there exists regular axially symmetric stationary charged
spacetimes which have more than one black hole: the Papapetrou-Majumdar solutions
[17], [13], [7]. However, these have degenerate event horizons. It is not known if there
are any regular charged spacetimes having non-degenerate event horizons and more
than one body.

Similar to the case of vacuum, by the Ernst-Geroch formulation, one can write the
metric of any axially symmetric stationary charged spacetime in the form (see [27], [8])

ds2 = −ρ2 e2u dt2 + e−2u (dϕ − p dt)2 + e2µ+2u (dρ2 + dz 2 )

where ρ is again the distance to the z-axis, ϕ is the cylindrical angle around the z-axis
and u, p and µ are determined by

(i) u, p and µ are independent of the angle variable ϕ;

(ii) u is the first component of some axially symmetric singular harmonic map (u, v, χ, ψ)
from R3 \ Γ into the complex hyperbolic plane HC , i.e. it satisfies

∆u − 2 e4u |Dv − ψ Dχ + χ Dψ|2 − e2u (|Dχ|2 + |Dψ|2 ) = 0, (1.6)

div e4u (Dv − ψ Dχ + χ Dψ) = 0,

 
(1.7)

div (e2u Dχ) − 2e4u Dχ · (Dv − ψ Dχ + χ Dψ) = 0, (1.8)

div (e2u Dψ) + 2e4u Dψ · (Dv − ψ Dχ + χ Dψ) = 0; (1.9)

moreover, it is singular near Γ and the singular rate is controlled by the Newtonian
h 1
potential creaated by a uniform line charge distribution 2 of density 2 and some
ideal points on ∂HC ;
 7

(iii) p satisfies

dp = −ρ e4u (vz − ψ χz + χ ψz )dρ + ρ e4u (vρ − ψ χρ + χ ψρ ) dz; (1.10)

(iv) and µ satisfies

e4u
dµ = ρ[(u2ρ − u2z ) + ((vρ − ψ χρ + χ ψρ )2 − (vz − ψ χz + χ ψz )2 )
4
+ e2u (χ2ρ − χ2z + ψρ2 − ψz2 )]dρ
e4u
+ 2ρ[uρ uz + (vρ − ψ χρ + χ ψρ )(vz − ψ χz + χ ψz )
4
+ e2u (χρ χz + ψρ ψz )]dz. (1.11)

Again, the harmonic map equations (1.6)-(1.9) are the integrability conditions for (1.10)
and (1.11). Moreover, in the absence of charge, i.e. χ = ψ = 0, the system (1.6)-(1.9)
reduces to the system (1.1)-(1.2) via the transformation X = e−2u and Y = 2v.

The existence problem for the asymptotically flat case was settled by Weinstein him-
self. Physically speaking, this solution represents (possibly singular) asymptotically flat
co-axially rotating electro-vacuum, or charged, black holes in gravitational equilibrium.
Moreover, he also showed that this family of solutions is parametrized by 4n − 1 pa-
rameters, n being the number of black holes, which can be interpreted as the masses,
angular momenta and charges of the black holes, and the distances between them. Es-
pecially, when n = 1, he recovered the uniqueness of the Kerr-Newman solutions which
was proved independently by Mazur and Bunting ([14], [15], [2]).

Here we would like to address the regularity of the harmonic maps obtained by
performing the Ernst-Geroch reduction to the solutions constructed by Weinstein. In
this work, we prove:

Theorem B Let Γ be the z-axis in R3 with some bounded line segments removed and
let ρ be the distance function to the z-axis. Let h be the potential of a uniform charged
distribution of density γ > 0. If (u, v, χ, ψ) is a singular harmonic map from R3 \ Γ into
h
the complex hyperbolic plane HC controlled by 2 and some ideal points on ∂HC and if
h
u− 2 ∈ H 1 (R3 ), v ∈ H 1 (R3 , e2h ), and χ, ψ ∈ H 1 (R3 , eh ), then u − h2 , v χ, ψ ∈ C k,α
 8

across the interior of Γ for any k + α < 2γ. In addition, near any compact subset of
the interior of any component of Γ, v, χ and ψ admit the asymptotic expansion

v = C1 − C2 χ + C1 ψ + O(ρ2k+2α ),

χ = C2 + O(ρk+α ),

ψ = C2 + O(ρk+α ),

where C1 , C2 and C3 are constants.

Moreover, if (u, v, χ, ψ) is axially symmetric about Γ and γ = 1, u − h, v, χ, ψ are

everywhere C ∞ except possibly at the endpoints of Γ.

Remark 1.2 By letting u = − 21 log X, v = 1

2Y , and χ = ψ = 0 in Theorem B, we
recover Theorem A.

Remark 1.3 The conclusions of Theorem B can be extended to singular harmonic

maps with values in any real, complex or quaternionic hyperbolic spaces, i.e. symmetric
spaces of rank one. See Remark 2.1.

As an immediate consequence of Theorem B, we also prove:

Theorem C The metric components of the co-axially rotating, stationary, multiple-

black-hole, charged spacetimes constructed by Weinstein in [27] are C ∞ outside the
event horizons.

Again, we emphasize that this theorem does not rule out the possibility of having
conical singularity along the symmetry axis. For the metric to be regular across the
axis, one needs
lim (µ + 2u + log ρ) = 0 along Γ. (1.12)
ρ→0

In view of Theorem B, one can verify easily that the limit on the left hand side exists.
It is then possible to carry out an analysis similar to that in [11], [12] and [24] to
prove nonexistence of regular spacetimes corresponding to certain class of parameters.
However, we will not pursue this direction in the present work.
 9

The rest of this dissertation is organized as follows. In Chapter 2, we carry out

some preliminary analysis on the harmonic map equations (1.6)-(1.9) which allows us
to consider it as a special case of a broader class of singular quasilinear elliptic systems.
In Chapter 3 we consider the case when the model problem is a single linear equation.
In Chapter 4, we return to the study of the model problem in the general setting. The
proofs of Theorems B and C are carried out in Chapter 5. Finally, for completeness, we
include in Appendix A a quick review on hyperbolic spaces and in appendix B a brief
study the weighted Sobolev and Lebesgue spaces WΣ1,p (Ω, w) and Lp (Ω, w) defined in
Definition 1.2.
 10

Chapter 2

The model PDE problem

In R3 , let Γ be the z-axis with some line segments removed. Let h be the Newtonian
potential created by a uniform line charge distribution of density γ > 0 along Γ. We
would like to understand the regularity of a singular harmonic map (u, v, χ, ψ) from
R3 \ Γ into HC controlled by h. Recall that (u, v, χ, ψ) satisfies (1.6)-(1.9),

∆u − 2 e4u |Dv − ψ Dχ + χ Dψ|2 − e2u (|Dχ|2 + |Dψ|2 ) = 0,

div e4u (Dv − ψ Dχ + χ Dψ) = 0,

 

div (e2u Dχ) − 2e4u Dχ · (Dv − ψ Dχ + χ Dψ) = 0,

div (e2u Dψ) + 2e4u Dψ · (Dv − ψ Dχ + χ Dψ) = 0.

Since HC has negative sectional curvature, standard harmonic map theory implies that
(u, v, χ, ψ) is smooth away from the singularity set Γ. Thus we only need to study its
behavior near Γ. Also, as we are only interested in the interior of Γ, it suffices to restrict
our attention to a subset Ω of R3 such that Σ = Ω ∩ Γ has only one component whose
endpoints lie on ∂Ω. Also, we only need to pick one controlling ideal point p ∈ ∂HC .
Using an isometry of HC , we can assume without loss of generality that p is the +∞
point sitting on the u-axis of HC . We pick the geodesic going to p to be

t 7→ ζ(t) = (t, 0, 0, 0) ∈ HC .

As noted earlier, ζ( h2 ) is a singular harmonic map from Ω \ Σ into HC . Also, as h

2 and
p control (u, v, χ, ψ), we must have
 
dHC ( 12 h, 0, 0, 0), (u, v, χ, ψ) < C < ∞ in Ω.

After some calculation, this turns out to be equivalent to

u − h  + e2h |v| + eh (|χ| + |ψ|) < C < ∞ in Ω.

 
2
 11

We have shown:

Lemma 2.1 Let Ω be a subset of R3 and Σ be a curve in Ω. Let h be the Newtonian

potential created by a uniform line charge distribution of density γ > 0 along Σ. Then
(u, v, χ, ψ) is a singular harmonic map from Ω \ Σ into the complex hyperbolic plane
h
HC controlled by 2 and the ideal point (+∞, v0 , χ0 , ψ0 ) ∈ ∂HC if and only if it satisfies
the harmonic map equations (1.6)-(1.9) and satisfies the estimate

u − h  + e2h |v − v0 − ψ0 χ + χ0 ψ| + eh (|χ − χ0 | + |ψ − ψ0 |) < C < ∞ in Ω.

 
(2.1)
2

This a priori estimate shows that the harmonic map equations (1.6)-(1.9) is a sin-
gular semilinear elliptic system. As h ‘behaves like’ −2γ log ρ near Σ, the singularity
types are negative powers of the distance function to a line.

Next, we observe that our target manifold is HC , a symmetric space. By Noether’s

theorem, the harmonic map equations (1.6)-(1.9) induce several conservation laws, each
for an isometry of HC . In fact, there are even enough symmetries to turn (1.6)-(1.9)
into divergence form:
h i
div Du − 2e4u v Dv + (2e4u v ψ − e2u χ)Dχ − (2e4u v χ + e4u ψ)Dψ = 0, (2.2)
h i
div e4u (Dv − ψ Dχ + χ Dψ) = 0, (2.3)
h i
div − 2e4u ψ Dv + (e2u + 2e4u |ψ|2 )Dχ − 2e4u χ ψ Dψ = 0, (2.4)
h i
div 2e4u χ Dv − 2e4u χ ψ Dχ + (e2u + 2e4u |χ|2 )Dψ = 0. (2.5)

(In the vacuum case, this divergence structure is more apparent (cf. [21], Chapter 7).)

Write u1 = u − h/2, u2 = v, u3 = χ, u4 = ψ, and note that h is harmonic, the above

system can be rewritten in the form

div (cαβ (x, u) Duβ ) = 0, 1≤α≤4

where cαβ is a 4 × 4 matrix of coefficients given by

 
4u 4u 2u −2e4u v χ + e2u ψ 
 1 −2e v 2e v ψ − e χ
 
 0 e4u −e4u ψ e4u χ 
.
 

 0 −2e4u ψ e2u + 2e4u |ψ|2 −2e4u χ ψ 
 
 
0 2e4u χ −2e4u χ ψ e2u + 2e4u |χ|2
 12

The drawback of this way of writing the harmonic map equations is that the coefficients
cαβ might not satisfy the Legendre-Hadamard condition. Indeed, for ξ = (ξ 1 = a, ξ 2 =
1, ξ 3 = 0, ξ 4 = 0) ∈ R4 ,
cαβ ξ α ξ β = a2 − 2e4u v a + e4u .

It is readily seen that when e2u v > 1, the right hand side changes sign as a varies in R.
Nevertheless, when the a priori estimate (2.1) holds, the Legendre-Hadamard condition
can be recovered (for a different but equivalent set of coefficients). To this end we
rewrite (2.2)-(2.5) in the form

div (aαβ (x, u) Duβ ) = 0, 1≤α≤4

where aαβ is a 4 × 4 matrix of coefficients given by

 
 1 −2e4u v −2e4u v ψ e2u χ+ −2e4u v χ e2u ψ 
 
 0 2le4u −2le4u ψ 2le4u χ 
 
 
 0 −2le4u ψ le2u + 2le4u |ψ|2 −2le4u χ ψ 
 
 
0 2le4u χ −2le4u χ ψ le2u + 2le4u |χ|2

and l is some constant to be determined shortly. Then, for ξ ∈ R4 ,

aαβ (x, u) ξ α ξ β = |ξ 1 |2 + 2le4u |ξ 2 − ψξ 3 + χξ 4 |2 + le2u (|ξ 3 |2 + |ξ 4 |2 )

− [2e4u v(ξ 2 − ψξ 3 + χξ 4 ) − e2u ψξ 3 + e2u χξ 4 ]ξ 1

In view of (2.1), we can always pick l sufficiently large and positive λ, Λ such that

λ[|ξ 1 |2 + e4u |ξ 2 |2 + e2u (|ξ 3 |2 + |ξ 4 |2 )] ≤ aαβ (x, u) ξ α ξ β

≤ Λ[|ξ 1 |2 + e4u |ξ 2 |2 + e2u (|ξ 3 |2 + |ξ 4 |2 )] for any ξ ∈ R4 .

We are thus led to study the following problem:

Problem 2.1 Let Ω be a domain in Rn and Σ be a (n − k)-submanifold of Ω, k ≥ 2.

Consider the system

div (aαβ (x, u) Duβ ) = 0 in Ω \ Σ, 1 ≤ α ≤ m, (2.6)

 13

where aαβ : (Ω \ Σ) × Rm → R are given smooth functions and u : Ω → Rm is the

unknown.

Let d = dΣ denote the distance function to Σ. Assume that there exist constants
k(1), k(2), . . . , k(m) ≥ 0 such that for any given R > 0, there exist λ = λ(R) > 0 and
Λ = Λ(R) such that
m
X m
X
λ d(x)−2k(α) |ξ α |2 ≤ aαβ (x, u) ξ α ξ β ≤ Λ d(x)−2k(α) |ξ α |2 (2.7)
α=1 α=1

for any x ∈ Ω \ Σ, ξ ∈ Rm and u ∈ Rm satisfying d(x)−k(α) |uα | ≤ R, 1 ≤ α ≤ m.

Assume that u solves (2.6) in some appropriate sense. Determine how regular u is!

Remark 2.1 Analogous to the case of harmonic maps into the complex hyperbolic
plane, regularity for harmonic maps with prescribed singularity into any real, complex,
or quaternionic hyperbolic space in the sense described in [25] can always be recast as
a part of Problem 1. Consequently, the result in Theorem B extends parallelly to those
cases.

We now describe what we mean by a solution to (2.6). The singularity of the

coefficients aαβ and the aforementioned existence result established by Weinstein makes
it reasonable to assume that uα ∈ HΣ1 (Ω, d−2k(α) ). In addition, in a compactly supported
open subset of Ω \ Σ, aαβ behaves nicely and so (2.6) should hold in the usual weak
sense, i.e.
Z
aαβ (x, u) Duα Dξ β dx = 0, ξ ∈ Cc∞ (Ω \ Σ, Rm ).
Ω

This suggests the following definition.

Definition 2.1 A measurable function u : Ω → Rm with uα ∈ HΣ1 (Ω, d−2k(α) ) is said

to be a weak solution of (2.6) if
Z
aαβ (x, u) Duα Dξ β dx = 0
Ω

1 (Ω, d−2k(α) ).
for any ξ α ∈ H0,Σ
 14

Chapter 3

The case of a single linear equation

In this chapter, we consider a special case of Problem 1 where the unknown is a scalar
map and the coefficient is independent of the unknown. Recall that Ω is an open subset
of Rn and Σ is a (n−k)-dimensional submanifold of Ω with k ≥ 2. Let d be the distance
function to Σ and w be a weight that satisfies

λ d(x)−γ ≤ w(x) ≤ Λ d(x)−γ ,

for some γ > 0, 0 < λ ≤ Λ < ∞. We are interested in the regularity of weak solutions
of


div (w(x)Du = 0 in Ω \ Σ. (3.1)

Recall that by weak solution we mean a function u ∈ HΣ1 (Ω, d−γ ) such that
Z
1
w(x) Du Dξ dx = 0 for any ξ ∈ H0,Σ (Ω, d−γ ).
Ω

This problem has been studied extensively in the literature. When γ < k − 2, w
belongs to the Muckenhoupt class A2 and the result of Fabes, Kenig and Serapioni
[4] implies that u is Hölder continuous across Σ. Unfortunately, in our application to
the problem from general relativity, w might get too singular in a way that it is not
even integrable across Σ, and so their work does not apply directly. Moreover, for
other purposes, we are also interested in whether w vanishes along Σ and how fast it
decays there. A result in this direction was given by Li and Tian in [11] when Σ has
codimension 2. We generalize their work to the case where Σ is a general submanifold
of any codimension k ≥ 2 and sharpen the decay estimate to its optimal form.

Theorem 3.1 Assume that w ∈ C 2 (Ω \ Σ) and

λ d−γ ≤ w ≤ Λ d−γ
 15

for some γ ≥ 0 and 0 < λ ≤ Λ < ∞. Moreover, assume that |D(dγ w)| = o(d−1 ) in a
neighborhood of Σ.

Then any u ∈ HΣ1 (Ω, w) which solves

div (w Du) = 0

in Ω \ Σ is locally Hölder continuous in Ω and enjoys

+ γ
sup d−(γ−k+2) |u| ≤ C kd− 2 ukL2 (Ω)
ω

for any ω compactly supported in Ω. The constant C depends only on Λ, λ and the
modulus of continuity of d|D(dγ w)|.

Remark 3.1 To see that the decay estimate is optimal, consider the case where Σ is
an n − k hyperplane and w = d−γ .

When γ > k−2, (3.1) has a special solution u = dγ−k+2 which belongs to HΣ1 (Ω, d−γ ).
This solution vanishes along Σ and exhibits Hölder continuity along Σ.

When γ < k−2, the above special solution does not belong to HΣ1 (Ω, d−γ ). Moreover,
constant functions are solutions which are in HΣ1 (Ω, d−γ ) and, of course, they need not
vanish along Σ. Nevertheless, as mentioned above, w is in the Muckenhoupt class A2
and so u is Hölder continuous across Σ.

Before proving Theorem 3.1, let us give an application which improves Theorem A.

Corollary 3.1 Under the hypotheses of Theorem A, (log X + h, Y ) is of class C k,α

where k is the biggest integer smaller than 4γ and α = 4γ − k. Moreover, near any
compact subset of the interior of any component of Γ, Y admits the expansion

Y = C + O(ρ4γ )

for some constant C.

Proof. Focus our attention to a small neighborhood Ω of an interior point of Γ, we can

assume that Y ∈ H 1 (Ω, d−4γ ) and Y |Σ = 0 where Σ = Γ ∩ Ω (see Lemma 2.1). This
implies that Y ∈ HΣ1 (Ω, d−4γ ),
 16

Rewrite (1.2) as
div (X −2 DY ) = 0

and apply Theorem 3.1, we get the required decay for Y and then its C k,α regularity.
To get the regularity for X, we rewrite (1.1) as

|DY |2
∆(log X + h) = −
X2

and apply a simple estimate for the Poisson equation. 

Remark 3.2 To see that the C k,α regularity in Corollary 3.1 is optimal, consider for
example the case where Γ is the whole z-axis, h = −2γ log ρ, γ > 0 and

ρ2γ
X= ,
ρ4γ + 1
ρ4γ
Y = 4γ .
ρ +1

This example also shows that the C k,α regularity in Theorem B is almost optimal as a
harmonic map into the real hyperbolic plane is a special harmonic map into the complex
hyperbolic plane.

We will prove Theorem 3.1 through a sequence of lemmas. Also, to avoid technical-
ity, we will assume that Σ is a (n − k)-hyperplane and w = d−γ . In fact, if w̄ = dγ w is
not constant, the proofs below require minor changes.

Also, we will frequently use the following inequality (see Appendix B) without ex-
plicitly mentioning,
Z Z
−γ−2
d 2
|f | dx ≤ C [|f |2 + d−γ |Df |2 ] dx
ω Ω

for any f ∈ HΣ1 (Ω, d−γ ) and ω compactly supported in Ω.

Lemma 3.1 Theorem 3.1 holds for 0 ≤ γ < 2(k − 2).

Proof. For 0 ≤ γ ≤ k − 2, the assertion is a consequence of the aforementioned result

of Fabes et al. [4]. Assume that k − 2 < γ < 2(k − 2). Set v = d−γ+k−2 u and w̃ =
d−2(k−2)+γ . Then v ∈ H 1 (Ω, w̃) and satisfies in Ω \ Σ the equation

div (w̃ Dv) = 0.

 17

Since γ < 2(k − 2), w̃ belongs to the Muckenhoupt class A2 and the above equation
is satisfied across Ω. Again, the result of Fabes et al. in [4] applies showing that v is
locally bounded in Ω and for any ω compactly supported in Ω,

γ γ
sup |v| ≤ C kd 2 −k+2 vkL2 (Ω) ≤ C kd− 2 vkL2 (Ω) .
ω

Lemma 3.2 Assume that γ ≥ 2(k − 2) and u is as in Theorem 3.1. Then u satisfies

γ γ
sup d− 2 |u| ≤ C kd− 2 ukL2 (Ω)
ω

for any ω compactly supported in Ω.

γ
Proof. Write v = d− 2 u. Then v ∈ H 1 (Ω) ∩ L2 (Ω, d−2 ) and satisfies in Ω \ Σ the
equation
∆v − c v = 0

where
γ γ 1
c= ( − k + 2) 2 .
2 2 d
Note that c is positive. We therefore can apply De Giorgi or Moser techniques to show
that v is locally bounded and

sup |v| ≤ C kvkL2 (Ω) .

ω

The lemma is proved. 

Lemma 3.3 Assume that u ∈ H 1 (Ω)∩C 0 (Ω̄) solves the following inhomogeneous equa-
tion in Ω \ Σ
div (w(x)Du) = f, (3.2)

where f is smooth away from Σ and |f | ≤ C d−γ−2+µ in a neighborhood of Σ for some

0 < µ < λ − k + 2. Assume in addition that u vanishes along Σ. Then for any λ ≤ µ,
d−λ u is locally bounded and for any ω compactly supported in Ω,

sup d−λ |u| ≤ C sup |u|.

ω Ω
 18

Proof. Fix some ω compactly supported in Ω. Fix some 0 < h < 1. Assume for the
moment that d−c u is bounded. We will show that if c + h ≤ µ then

sup d−c−h |u| ≤ C sup d−c |u|.

ω ω

This obviously implies our result.

Fix x0 ∈ Ω. Consider the function

θ(x) = dc+h (x) + dc (x) |x − x0 |2 .

We compute

Dθ = (c + h) dc+h−1 Dd + c dc−1 |x − x0 |2 Dd + 2 dc (x − x0 ),

div (d−γ Dθ) = (c + h)(c + h − γ + k − 2) d−γ+c+h−2

+ c(c − γ + k − 2) d−γ+c−2 |x − x0 |2

+ 2(2c − γ) d−γ+c−1 Dd · (x − x0 ) + 2n d−γ+c .

Since h < 1, this implies that

div (d−γ Dθ) ≤ C(c + h)(c + h − γ + k − 2) d−γ+c+h−2 ≤ −C |f |.

in some neighborhood U of Σ. Take another neighborhood V of Σ such that V̄ ⊂ U .

Set δ = dist (V, ∂U ) > 0. For x0 ∈ V with |x − x0 | > δ, we have

|u| ≤ kd−c ukL∞ (U ) dc ≤ kd−c ukL∞ (U ) δ −2 θ

Also, θ vanishes on Σ. Hence, by the maximum principle,

|u| ≤ kd−c ukL∞ (U ) δ −2 θ in Bδ (x0 ).

In particular, for x = x0 , we have |u(x0 )| ≤ kd−c ukL∞ (U ) δ −2 dc+h (x0 ). Since x0 is

arbitrary, this proves the lemma. 

Lemma 3.4 Let v ∈ H 1 (Ω) ∩ L2 (Ω, d−2 ) solve the following equation in Ω \ Σ:

Dd 1
∆v − p · Dv − q 2 v = 0 (3.3)
d d

where p and q are two real numbers satisfying

 19

• q ≥ 0,

• (p − k + 2)2 + 4q − p2 > 0,

2q 4q 2 4q(p−k+2)
• and p − k + 2 ≤ k−2 or (k−2)2
− k−2 + p2 − 4q < 0.

Then v ∈ L∞
loc (Ω) and for any ω compactly supported in Ω,

sup |v| ≤ C kvkL2 (Ω) .

ω

The constant C depends continuously on p and q.

Remark 3.3 The classical De Giorgi - Nash - Moser estimate does not apply here as
d−1 ∈
/ L1,n−1+ and d−2 ∈
/ L1,n−2+ for any  > 0.

Proof. We can assume that Ω is the unit ball B1 . We will show that v is bounded in
B1/2 . The idea is to adapt De Giorgi’s proof exploiting the fact that q is negative.

Let A(k, ρ) = {x ∈ Bρ : u(x) > k} and

Z
Φ(k, ρ) = |u − k|2 dx.
A(k,ρ)

Let η be a standard cut-off function supported in Bρ . Observe that we can take

η 2 (u − k)+ as a test function for (3.3). This yields
Z Z
2 2 Dd
η |Du| dx ≤ { η 2 |Du|2 + C |u − k|2 |Dη|2 − p η 2 Du (u − k)+
A(k,ρ) A(k,ρ) d
1
− q η2 |u − k|2 }dx. (3.4)
d2

On the other hand, by integrating by parts, we see that

Z Z
Dd 2 + 1 Dd
± η Du (u − k) dx = ± η2 D(|u − k|2 ) dx
A(k,ρ) d 2 A(k,ρ) d
Z
1 Dd Dd
=∓ [η 2 div ( ) + 2 η Dη ] |u − k|2 dx
2 A(k,ρ) d d
k−2
Z
1
≤ −(± + 0 ) η 2 2 |u − k|2 dx
2 A(k,ρ) d
Z
+ C0 |u − k|2 |Dη|2 dx.
A(k,ρ)
 20

Substituting the above estimate into (3.4), we arrive at

Z Z
Dd
η 2 |Du|2 dx ≤ { η 2 |Du|2 + C |u − k|2 |Dη|2 − (p − b) η 2 Du (u − k)+
A(k,ρ) A(k,ρ) d
(k − 2)b 1
− (q − − 0 ) η 2 2 |u − k|2 }dx
2 d
2q
where b is any real number. If we requires that b < k−2 , this implies
Z Z
2 2
η |Du| dx ≤ {α η 2 |Du|2 + C |u − k|2 |Dη|2 }dx
A(k,ρ) A(k,ρ)

where
(p − b)2
α= (k−2)b
+ .
4(q − 2 − 0 )
2q
Our hypotheses on p and q are exactly to allow us to find some b < k−2 so that α < 1.
We have therefore shown that
Z Z
2 2
η |Du| dx ≤ C |u − k|2 |Dη|2 dx. (3.5)
A(k,ρ) A(k,ρ)

Using this estimate, we can run through De Giorgi’s proof for local boundedness and
obtain
sup u ≤ C ku+ kL2 (B1 ) .
B1/2

The lemma follows. 

Proof of Theorem 3.1. The case 0 ≤ γ < 2(k − 2) is taken care by Lemma 3.1. We
hence assume that γ ≥ 2(k − 2).
γ
By Lemma 3.2, d− 2 u is locally bounded. Thus, by elliptic theory, u is continuous
in Ω and vanishes along Σ. Lemma 3.3 hence shows that d−λ u is locally bounded for
any λ < γ − k + 2. By elliptic theory, we infer that d−λ+1 |Du| is locally bounded for
the same λ’s.

Take a sequence λm < γ − k + 2 converging to γ − k + 2. Set vm = d−λm u.

Since λm < γ − k + 2, the above estimates on the sizes of u and Du show that vm ∈
H 1 (Ω) ∩ L2 (Ω, d−2 ). Moreover, in Ω \ Σ, vm satisfies

Dd 1
∆vm − pm · Dvm − qm 2 vm = 0
d d
 21

where pm = γ − 2λm and qm = λm (γ − k + 2 − λm ) > 0. An application of Lemma 3.4

shows that, for any ω compactly supported in Ω,

sup d−λm |u| = sup |vm | ≤ Cm (ω) kvm kL2 (Ω) = Cm (ω) kd−λm ukL2 (Ω) .
ω ω

More importantly, as λm → γ − k + 2, the constant Cm does not blow up as

(pm − k + 2)2 + 4qm − p2m → (γ − k + 2)2 − (γ − 2k + 4)2 > 0

and
2qm
pm − k + 2 − → −γ + k − 2 < 0.
k−2
As a consequence, if ω ⊂⊂ ω 0 ⊂⊂ Ω,

sup d−γ+k−2 |u| ≤ C kd−γ+k−2 ukL2 (ω0 ) ≤ C sup d−γ+k−2+ |u|

ω ω0
− γ2 − γ2
≤ C sup d |u| ≤ C kd ukL2 (Ω) .
ω0

In any case, the theorem is ascertained. 

 22

Chapter 4

Hölder regularity

We next switch our attention back to the system (2.6), i.e.

div (aαβ (x, u) Duβ ) = 0, 1 ≤ α ≤ m.

Recall that uα ∈ HΣ1 (Ω, d−2k(α) ) is a weak solution of (2.6) if

Z
aαβ (x, u) Duα Dξ β dx = 0, ∀ ξ α ∈ H0,Σ
1
(Ω, d−2k(α) ).
Ω

Throughout the chapter, the coefficients aαβ are assumed to satisfy the ellipticity con-
dition (2.7), i.e. for any R > 0, there exists λ = λ(R) > 0 and Λ = Λ(R) < ∞ such
that
m
X m
X
−2k(α)
λ d(x) α 2
|ξ | ≤ aαβ (x, u) ξ ξ ≤ Λ α β
d(x)−2k(α) |ξ α |2
α=1 α=1

for any x ∈ Ω \ Σ and u ∈ Rm verifying d(x)−k(α) uα ≤ R. Here k(1), k(2), . . . , k(m)

are non-negative numbers. Let τ be the smallest integer such that k(τ ) > 0. (If there
is no such index, set τ = m + 1.)

In general, without any additional assumption on the coefficients aαβ or the un-
knowns uα , one does not expect uα to be regular, or even Hölder continuous, across Σ.
For example, when the coefficients aαβ satisfies the classical ellipticity, i.e. k(α) = 0,
it is well-known that u satisfies (2.6) in Ω in the weak sense and so is regular at any
point x ∈ Σ at which
Z
1
lim inf |Du|2 dx = 0.
δ→0 δ n−2 Bδ (x)

(See [5], Chapter 4, for example.) If aij

αβ do not depend on u and are smooth across Σ,

this set is empty and so u is regular in Ω. Nonetheless, if aij

αβ is allowed to depend on

u, the points where regularity fails might constitute a non-vacuous subset of Σ (see [5],
Chapter 2).
 23

We show that the above phenomenon also persists for the problem we are consider-
ing. For any Bδ (x) ⊂ Ω, we define
Z m
1 X
Eδ (x) = d(z)−2k(α) |Duα (z)|2 dz. (4.1)
δ n−2 Bδ (x) α=1

Also, define

bαβ (x, ũ) = d(x)−[k(α)+k(β)] a(x, d(x)−k(γ) ũγ ), x ∈ Ω \ Σ, ũ ∈ Rm .

By (2.7), the functions bi,j α m

αβ (x, ũ) remain bounded as long as {ũ }α=1 stays bounded.

Theorem 4.1 Let Ω be a domain in Rn and let Σ be a (n − k)-submanifold of Ω, k ≥

2. Assume that aαβ are smooth functions on (Ω \ Σ) × Rm which satisfy the ellipticity
condition (2.7) with k(α) being either zero or bigger than k − 2. Let τ be the smallest
index so that k(τ ) > k − 2. (If all k(α) vanish, we set τ = m + 1.) Assume that

bαβ (x, ũ1 , . . . , ũτ −1 , 0) = 0 unless α = β. (4.2)

Let uα ∈ HΣ1 (Ω, d−2k(α) ) be a weak solution of (2.6).

If uα ∈ L∞ (Ω, d−k(α) ), then there exists ε0 > 0 such that if Eδ (x0 ) ≤ ε0 for some
0 < δ < dist (x0 , ∂Ω)/4, then

Eσ (x) ≤ C σ 2λ , x ∈ Bδ/2 (x0 ), σ < δ/4,

for some λ = λ(k(α), kuα kL∞ (Ω,d−k(α) ) ) > 0 and C = C(k(α), kuα kL∞ (Ω,d−k(α) ) ) ≥ 0. In
particular, u is Hölder continuous in Bδ/4 (x0 ) and for α ≥ τ , d−k(α)−λ |uα | is bounded
in Bδ/4 (x0 ).

Remark 4.1 The requirement that k(τ ) > k − 2 is probably merely technical. It would
be interesting to remove this extra hypothesis. However, in doing so, one must be careful
with the assumption that d−k(α) uα is bounded. For, in case of a single linear equation,
if k(α) < k − 2, there are examples that uα may not vanish along Σ as fast as dk(α)
(see Remark 3.1).

In the context of harmonic maps, an -regularity result is usually established using

some monotonicity formula. For example, this is the case in the work of Li and Tian
 24

[11]. It seems that their proof does not apply in our context. However, as demonstrated
in Chapter 2, our harmonic map problem has an equivalent homogeneous divergence
form. This special structure makes it apparent to obtain a Caccioppoli inequality which
we will state shortly. When restricted to the vacuum case in [11], this Caccioppoli
inequality implies the monotonicity formula therein.

Observe that, as long as we stay away from Σ, (2.6) is an elliptic system for u with
bounded smooth coefficients, so classical elliptic theory tells us that a Caccioppoli in-
equality holds there. However, as we move towards Σ, the coefficients of these equations
behave badly signifying some possible deterioration in the structure of the Caccioppoli
inequality. This is indeed true as stated in the following result.

Proposition 4.1 (Caccioppoli inequality) Let uα ∈ HΣ1 (Ω, d−2k(α) )∩L∞ (Ω, d−k(α) )
be a weak solution of (2.6). Let x0 be a point in Ω and δ0 the distance from x0 to ∂Ω.
There exists C = C(Λ, λ) > 0 such that for δ ≤ δ0 /2 and b ∈ Rm whose last m − τ + 1
components vanish when B2δ (x0 ) ∩ Σ 6= ∅,
Z m Z m
X
−2k(α) C X
d α 2
|Du | dz ≤ 2 d−2k(α) |uα − bα |2 dz.
Bδ/2 (x0 ) δ Bδ (x0 )
α=1 α=1

Proof. Fix ε > 0 and δ < δ0 . Let η : Rn → R be a cut-off function satisfying η(z) = 0
if |z − x0 | ≥ δ and η(z) = 1 if |z − x0 | ≤ δ/2, and |Dη| ≤ 4δ −1 .

Observe that our constraints on b imply that uα − bα ∈ HΣ1 (Ω, d−2k(α) ). Thus, by
Corollary B.1 in Appendix B, we can take ξ = η 2 (u − b) as a test function for (2.6).
Using this special choice of ξ and recalling (2.7) yields
Z m
X Z
2 −2k(α) α 2
η d |Du | dz ≤ C η 2 aαβ Duα Duβ dz
Ω α=1 Ω
Z
= −C ηaαβ Duβ Dη (uα − bα ) dz
Ω
m h
Z X i
≤C η 2 d−2k(α) |Duα |2 + |Dη|2 d−2k(α) |uα − bα |2 dz.
Ω α=1

This implies the inequality in question. 

We next study the limiting system when doing a blow-up analysis and then prove
that smallness in energy density implies regularity. We first recall a well known result
(see [5], Chapter 4).
 25

Lemma 4.1 Let aij

αβ(h) be a sequence of measurable functions satisfying

λ|p|2 ≤ aij α β 2
αβ(h) (x, v)pi pj ≤ Λ|p| , (x, v, p) ∈ Ω × Rm × Rmn

for some 0 < λ ≤ Λ < ∞ and converging in L2 (B1 (0)) to aij

αβ verifying the same ellip-

ticity bound. Let f(h) be a sequence in L2 (B1 (0); Rm ) converging weakly in L2 (B1 (0))
1 (B (0); Rm ) ∩ L2 (B (0)) such that
to f . Let u(h) be a sequence in Hloc 1 1

β
∂ ij
∂u(h)

α
a = f(h)
∂xi αβ(h) ∂xj

in the weak sense in B1 (0).

If u(h) converges weakly in L2 (B1 (0); Rm ) to u then u ∈ Hloc

1 (B (0); Rm ) and
1

• for any ρ < 1, u(h) converges strongly to u in L2 (Bρ (0); Rm ) and Du(h) converges
weakly to Du in L2 (Bρ (0); Rm ),

• and more importantly, u satisfies in the weak sense in B1 (0) the system

∂ β
ij ∂u

a = f.
∂xi αβ ∂xj

Lemma 4.2 Let uα ∈ HΣ1 (Ω, d−2k(α) ) ∩ L∞ (Ω, d−k(α) ) be a weak solution to (2.6). Let
d(x(h) )
B(h) = Bδ(h) (x(h) ) be a sequence of balls in Ω such that the ratios δ(h) are all less
than some fixed κ. Let Σ(h) be the image of Σ under the map x 7→ (x − x(h) )/δ(h) and
d(h) the distance function to Σ(h) . Assume that Σ(h) stabilizes to some Σ∗ , i.e. d(h)
converges uniformly away from Σ∗ to d∗ , the distance function to Σ∗ .

Set ε(h) = Eδ(h) (x(h) )1/2 . Define

1
uα(h) (x) = k(α)
[uα (x(h) + δ(h) x) − ūα(h) ], x ∈ B1 (0),
ε(h) δ(h)

where 
 the average of uα over B if k(α) = 0,
(h)
ūα(h) =
 0 otherwise.

If ε(h) → 0, there exists uα∗ ∈ Hk(α)v̄

1
0 (B1/2 (0)) such that, up to extracting a subse-
∗

quence,

• uα(h) converges weakly in H 1 (B1/2 (0)) and strongly in L2 (B1/2 (0)) to uα∗ ,
 26

−k(α) −k(α) −k(α)

• d(h) uα(h) converges strongly to d∗ uα∗ in L2 (B1/2 (0)) and d(h) Duα(h) con-
−k(α)
verges weakly to d∗ Duα∗ in L2 (B1/2 (0)).

Moreover, u∗ satisfies
−[k(α)+k(β)]
bαβ∗ Duβ = 0


div d∗

in the weak sense in B1/2 (0) \ Σ∗ , where bαβ∗ are constant and satisfy
m m
−2k(α) α 2 −[k(α)+k(β)] −2k(α)
X X
ν1 d∗ |p | ≤ d∗ bαβ∗ pα pβ ≤ ν2 d∗ |pα |2 ,
α=1 α=1

and the ratio ν2 /ν1 does not depend on the sequence of balls Bδ(h) (x(h) ). Additionally,
bαβ∗ can be written in the form

bαβ∗ = bαβ (x∗ , ũ1∗ , . . . , ũ∗τ −1 , 0)

for some accumulation point x∗ of the sequence x(h) .

Proof. Let’s assume for the moment that the above convergences have been established.
−k(α) −k(α)
Passing to a subsequence, we can assume that xh → x∗ , d(h) uα(h) → d∗ uα∗ a.e.
Also, as
Z
−2k(α) α 2 C
δ(h) |ū(h) | ≤ n d−2k(α) |uα |2 dz ≤ C,
δ(h) B(h)

−k(α)
we can assume that δ(h) ūα(h) → ũα∗ . It follows that

−k(α) −k(α) k(α)

ũα (x(h) + δ(h) x) = d(h) (x) δ(h) [ε(h) δ(h) uα(h) (x) + ūα(h) ] → ũα∗ d∗ (x)−k(α) a.e.,

Therefore, if we define
 
−[k(α)+k(β)]
aαβ(h) (x) = δ(h) aαβ x(h) + δ(h) x, u(x(h) + δ(h) x)
 
−[k(α)+k(β)]
= d(h) bαβ x(h) + δ(h) x, ũ(x(h) + δ(h) x) .

then
 
aαβ(h) (x) → d∗ (x)−[k(α)+k(β)] bαβ x∗ , ũα∗ d∗ (x)−k(α) a.e.

By the Lebesgue dominated convergence theorem, this can be regarded as convergence

in L2loc (B1 (0) \ Σ∗ ). An application of Lemma 4.1 yields the second half of the Lemma.
 27

It remains to check the convergences. We have

Z m Z m
X −2k(α) 1 X
d(h) |Duα(h) |2 dx = n−2 2 d−2k(α) |Duα |2 dx = 1. (4.3)
B1 (0) α=1 δ(h) ε(h) B(h) α=1

Here we have used the assumption that Eδ(h) (x(h) ) = ε2(h) . In particular, this implies
that the L2 norm of Duα(h) over B1 (0) is uniformly bounded for α < τ . The standard
Poincaré inequality then implies that the H 1 norm of uα(h) over B1 (0) is also uniformly
bounded for α < τ . For α ≥ τ , we use Corollary B.1 in Appendix B to show that the
L2 norm and so the HΣ1 norm of uα(h) over B3/4 (0) is uniformly bounded. One can then
modify the proof of Proposition B.2 in Appendix B to get the required convergences.
The details work out in exactly the same manner and hence are omitted. 

Lemma 4.3 Let uα ∈ HΣ1 (Ω, d−2k(α) ) ∩ L∞ (Ω, d−k(α) ) be a weak solution to (2.6). Let
d(x(h) )
B(h) = Bδ(h) (x(h) ) be a sequence of balls in Ω such that the ratios δ(h) are all at least
some fixed κ.

Set ε(h) = Eδ(h) (x(h) )1/2 and define

1
uα(h) (x) = k(α)
[uα (x(h) + δ(h) x) − ūα(h) ], x ∈ B1 (0),
ε(h) d(x(h) )
0
where ūα(h) is the average of uα over B(h) = Bκδ(h) /2 (x(h) ).

If ε(h) → 0, there exists u∗ ∈ H 1 (Bκ/2 (0)) such that, up to extracting a subsequence,

u(h) converges weakly in H 1 (Bκ/2 (0)) and strongly in L2 (Bκ/2 (0)) to u∗ . Moreover, u∗
satisfies
div aαβ∗ (x) Duβ = 0

in the weak sense in B1 (0) \ Σ∗ , where aαβ∗ are smooth and satisfy
m
X m
X
α 2 α β
ν1 |p | ≤ aαβ∗ p p ≤ ν2 |pα |2 ,
α=1 α=1

and the ratio ν2 /ν1 does not depend on the sequence of balls Bδ(h) (x(h) ) and does not
exceed C κ−2 max k(α) .

Proof. The proof of this lemma is similar to but easier than that of Lemma 4.2 and is
omitted. 
 28

Having a Caccioppoli inequality at hand and some understanding of the limiting

system, we are ready to go forward with our proof of Theorem 4.1, i.e. smallness
in density Eδ (x) implies regularity. Recall we require here that any limiting system
decouples into m independent equations by asking (4.2), namely

bαβ (x, ũ1 , . . . , ũτ −1 , 0) = 0 unless α = β.

Lemma 4.4 Let uα ∈ HΣ1 (Ω, d−2k(α) ) ∩ L∞ (Ω, d−k(α) ) be a weak solution to (2.6).
Assume in addition that (4.2) holds and all the k(α) are either zero or bigger than
k − 2. Let x0 be a point in Ω and δ0 = dist (x0 , ∂Ω). There exists ε0 and λ0 such that
for any x ∈ Bδ0 /2 (x0 ) and δ ∈ (0, δ0 /4) satisfying Eδ (x) ≤ ε20 there holds Eλ0 δ (x) ≤
Eδ (x)/2.

Proof. Arguing indirectly, we assume that the conclusion fails. Fix some λ0 and κ for
the moment. They will be determined by a set of constraints which will be formulated
in the sequel. We can find sequences ε(h) → 0, x(h) ∈ Bδ0 /2 (x0 ), and δ(h) ∈ (0, δ0 /4)
such that Eδ(h) (x(h) ) = ε2(h) but Eλ0 δ(h) (x(h) ) > ε2(h) /2. In addition, we can assume that
d(x(h) )
one of the following two cases occurs: All the ratios δ(h) are less than κ or all are
not.
d(x(h) )
Case 1: δ(h) < κ for all h.

Define d(h) , uα(h) , d∗ , Σ∗ , uα∗ and bαβ∗ as in Lemma 4.2.

In the ball B1/2 (0), uα∗ satisfies

div (d−2k(α) Duα ) = 0.

Hence, by Theorem 3.1,

|uα∗ (x) − uα∗ (0)| ≤ C|x| α<τ

and
|uα∗ (x)| ≤ C d(x)2k(α)−k+2 α≥τ

for some C independent of the balls Bδ(h) (x(h) ).

 29

Therefore, by the Caccioppoli inequality,

τ −1
Eλ0 δ(h) (x(h) )
Z
C nX
k(α)
≤ 2 |uα − ūα(h) − ε(h) m(h) uα∗ (0)|2
ε2(h) ε(h) (λ0 δ(h) )n B2λ0 δ(h) (x(h) ) α=1
m
X o
+ d−2k(α) |uα |2 dz
α=τ
Z τ −1 m
C nX X −2k(α)
o
≤ |uα(h) − uα∗ (0)|2 + d(h) |uα(h) |2 dz
λn0 B2λ0 (0) α=τ
α=1
Z m
C X −k(α) −k(α)
≤ |d(h) uα(h) − d∗ uα∗ |2 dz
λn0 B2λ0 (0) α=1
Z m
C n
2
X 2k(α)−2k+4
o
+ n |z| + d∗ dz
λ0 B2λ0 (0) α=τ
Z m
C X −k(α) −k(α)
≤ |d(h) uα(h) − d∗ uα∗ |2 dz
λn0 B2λ0 (0) α=1

+ C1 λ20 + C2 (κ + 2λ0 )2k(τ )−2k+4 .

Hence if
1
C1 λ20 + C2 (κ + 2λ0 )2k(τ )−2k+4 ≤ , (4.4)
8
ε2(h) ε2
we infer that 2 < Eλ0 δ(h) (x(h) ) ≤ 4 for h large enough, a contradiction.
d(x(h) )
Case 2: δ(h) ≥ κ for all h.

Let uα(h) , uα∗ and aαβ∗ as in Lemma 4.3.

In Bκ/2 (0), u∗ satisfies

div aαβ∗ (x)Duβ = 0

where the aαβ∗ are smooth and satisfy

m
X m
X
ν1 |pα |2 ≤ aαβ∗ pα pβ ≤ ν2 |pα |2 ,
α=1 α=1

with ν2 /ν1 ≤ C κ−2 max k(α) . Also, the L2 norm of u∗ in Bκ/2 (0) is universally bounded.
Therefore
m
X n+A
|uα∗ (x) − uα∗ (0)| ≤ C κ− 2 |x|, x ∈ Bκ/4 (0)
α=1

for some A > 0. Like case 1, the constant C is independent of the balls Bδ(h) (x(h) ).
 30

We again invoke the Caccioppoli inequality and obtain

Eλ0 δ(h) (x(h) )

ε2(h)
Z m
C X
≤ d−2k(α) |uα − ūα(h) − ε(h) d(x(h) )k(α) uα∗ (0)|2 dz
ε2(h) (λ0 δ(h) )n B2λ0 δ (x(h) ) α=1
(h)
Z m
C X
≤ n |uα(h) − uα∗ (0)|2 dz
λ0 B2λ (0)
0 α=1
Z m Z
C X
α α 2 C
≤ n |u(h) − u∗ | dz + n+A n |z|2 dz
λ0 B2λ (0) κ λ0 B2λ (0)
0 α=1 0
Z m
C X
≤ n |uα(h) − uα∗ |2 dz + C3 κ−n−A λ20 .
λ0 B2λ (0)
0 α=1

Similar to case 1, if we can choose λ0 and κ such that

1
C3 κ−n−A λ20 ≤ , (4.5)
8

this will lead us to an absurdity.

To complete the proof, we need to furnish (4.4) and (4.5). This is always doable by
1 1
first picking λ0 small enough such that λ0 ≤ √1 1
and (8C3 λ20 ) n+A ≤ ( 16C ) B − 2λ0 ,
4 C1 2

and then picking κ in between the last two figures. Here B = 2k(τ ) − 2k + 4 > 0. 

Proof of Theorem 4.1. Let uα ∈ HΣ1 (Ω, d−2k(α) ) ∩ L∞ (Ω, d−k(α) ) be a weak solution
of (2.6). Let ε0 be as in Lemma 4.4. Assume that Eδ (x0 ) ≤ ε0 for some 0 < δ <
dist (x0 , ∂Ω)/4.

It follows from Lemma 4.4 and standard iteration techniques (cf. [6]) that there
exists positive constants C and λ depending only on k(α) and the L∞ (Ω, d−k(α) )-norm
of uα such that
Z m
1 X
Eσ (x) = d−2k(α) |Duα |2 dz ≤ C σ 2λ , x ∈ Bδ/2 (x0 ), σ < δ/4.
σ n−2 Bσ (x) α=1

It follows from Morrey’s lemma that u is Hölder continuous in Bδ/2 (x0 ).

Moreover, if k(α) > 0, the above estimate implies a rate of vanishing of uα near Σ.
To see this, observe that if σ < d(x)/2, then
Z
|Duα |2 dz ≤ C d(x)2k(α) σ n−2+2λ , 1 ≤ α ≤ m,
Bσ (x)
 31

which implies
osc uα ≤ C d(x)k(α) σ λ .
Bσ/2 (x)

(See Theorem 7.19 in [6], for example.) Since uα vanishes along Σ when k(α) > 0, this
implies that dk(α)−λ uα is bounded in Bδ/4 (x0 ). The proof is complete. 
 32

Chapter 5

Applications

We now turn to the proof of Theorem B and C. We first consider a special case of
Problem 1 in which the smallness assumption in Theorem 4.1 is fulfilled. The condition
we impose additionally on the system (2.6) is a common feature that harmonic map
problems into hyperbolic spaces share. This was first used by Li and Tian in [11] in the
case of real hyperbolic spaces. We doubt that this phenomenon has some connection
to the underlying geometry of the target manifolds, but we have very limited evidence.

Proposition 5.1 Suppose all the assumptions of Theorem 4.1 hold. Assume in addi-
tion that τ = 2. If

aβ0 = δβ0 and a0β = −l(α) aαβ uα , 2 ≤ β ≤ m, (5.1)

for some l(1) = 0, l(2), . . . , l(m) > 0, then the uα are Hölder continuous across Σ.
Moreover, for α > 1, |uα | ≤ C dk(α)+λ for some λ > 0.

First observe that, due to (5.1), the first equation in the system (2.6) can be rewrit-
ten as
X
∆u1 = l(α) aαβ Duα Duβ . (5.2)
2≤α,β≤m

Note that the right hand side, which we will abbreviate as F (x, u, Du), is always non-
negative by ellipticity.

Lemma 5.1 Let uα ∈ HΣ1 (Ω, d−2k(α) ) ∩ L∞ (Ω, d−k(α) ) be a weak solution to (2.6). Let
x0 be a point in Ω and δ0 = dist (x0 , ∂Ω). Then there exists K such that if δ < δ0 /4
and x ∈ Bδ0 /2 (x0 ),
Z m
X
|x − z|−n+2 d−2k(α) |Duα |2 dz ≤ K,
Bδ (x) α=1
 33

and so
Z m
1 X
Eδ (x) = d−2k(α) |Duα |2 dz ≤ K.
δ n−2 Bδ (x) α=1

Proof. Note that u1 satisfies equation (5.2) in the sense of distributions throughout Ω.

For ε sufficiently small, let ηε : [0, ∞) → R be a cut-off function satisfying


 0 if t ≤ ε or t ≥ 2δ,
ηε (t) =
 1 if 2ε ≤ t ≤ δ,

and |ηε0 |, |ηε00 | ≤ C in [δ, 2δ], and |ηε0 | ≤ Cε−1 , |ηε00 | ≤ Cε−2 in [ε, 2ε].

Let R be a positive number such that |u1 | ≤ R. Let Gx (z) = |x − z|−n+2 and set
Gεx (z) = ηε (|x − z|)Gx (z). Inserting Gεx (z)(u1 + R + 1) as a test function into equation
(5.2) and note that F is nonnegative, we get
Z Z
Gεx (z) F (z, u, Du) dz ≤ F (z, u, Du) Gεx (z) (u1 + R + 1) dz
B2δ (x) B2δ (x)
Z
=− Du1 D[Gεx (z) (u1 + R + 1)] dz
B2δ (x)
Z h i
=− Gεx (z)|Du1 |2 + (u1 + R + 1) Du1 DGεx dz.
B2δ (x)

Hence
Z m
X
Gx (z) d−2k(α) |Duα |2 dz
Bδ (x)\B2ε (x) α=1
Z h i
≤C Gx (z) F (z, u, Du) + |Du1 |2 dz
Bδ (x)\B2ε (x)
Z
≤ −C (u1 + R + 1) Du1 DGεx dz.
B (x)
Z 2δ
=C (u1 + R + 1)2 ∆[ηε (|x − z|) Gx (z)] dz
B (x)
Z 2δ h i
≤C |ηε00 (|x − z|)| |x − z|−n+2 + |ηε0 (|x − z|)| |x − z|−n+1 dz
B2δ (x)

≤ K.

The conclusion then follows from Lebesgue’s monotone convergence theorem. 

Proposition 5.2 (Smallness of density) Let uα ∈ HΣ1 (Ω, d−2k(α) ) ∩ L∞ (Ω, d−k(α) )
be a weak solution to (2.6). Let x0 be a point in M and δ0 = dist (x0 , ∂M ). Let K be
 34

the constant obtained in Lemma 5.1. For given ε > 0, δ < δ0 /4, and x ∈ Bδ0 /2 (x0 ),
there exists σ = σ(ε, δ, x) between e−2K(n−2)/ε δ and δ such that

Eσ (x) ≤ ε.

Proof. Fix ε > 0. For δ 0 = e−2K(n−2)/ε δ, we have

Z δ Z δ Z m
Eσ (x) −n+1
X
dσ = σ d−2k(α) |Duα |2 dz dσ
δ0 σ δ0 Bσ (x) α=1
δ
Z m
X 
−n+2 −2k(α) α 2
= (−n + 2)σ d |Du | dz 

Bσ (x) α=1 0
δ
Z δ Z m
X
+ (n − 2) σ −n+2 d−2k(α) |Duα |2 dS(z) dσ
δ0 ∂Bσ (x) α=1
δ
Z m
X 
−n+2 −2k(α) α 2
= (−n + 2)σ d |Du | dz 

Bσ (x) α=1 0
δ
Z m
X
+ (n − 2) |x − z|−n+2 d−2k(α) |Duα |2 dz
Bδ (x)\Bδ0 (x) α=1

Hence, by Lemma 5.1

Z δ
Eσ (x)
dσ ≤ 2K(n − 2).
δ0 σ
The assertion follows immediately from an argument by contradiction. 

Proof of Proposition 5.1. By Proposition 5.2,

lim inf Eσ (x) = 0 ∀ x ∈ Ω.

σ→0

Theorem 4.1 then applies yielding the assertion. 

Finally, we consider Theorem B. Recall that u, v, χ and ψ satisfy (1.6)-(1.9), i.e.

∆u − 2 e4u |Dv − ψ Dχ + χ Dψ|2 − e2u (|Dχ|2 + |Dψ|2 ) = 0,

div e4u (Dv − ψ Dχ + χ Dψ) = 0,

 

div (e2u Dχ) − 2e4u Dχ · (Dv − ψ Dχ + χ Dψ) = 0,

div (e2u Dψ) + 2e4u Dψ · (Dv − ψ Dχ + χ Dψ) = 0.

 35

We will use various equations derivable from (1.6)-(1.9). The first set gives the formulas
for the Laplacians of u, χ and ψ.

h
∆(u − ) = ∆u = 2e4u |Dv − ψ Dχ + χ Dψ|2 + e2u (|Dχ|2 + |Dψ|2 ), (5.3)
2
∆χ = −2Du · Dχ + 2e2u Dψ · (Dv − ψ Dχ + χ Dψ), (5.4)

∆ψ = −2Du · Dψ − 2e2u Dχ · (Dv − ψ Dχ + χ Dψ), (5.5)

The second makes way to apply Lemma 3.3.

div (e4u Dv) = e4u Du · (ψ Dχ − χ Dψ) + e4u (ψ ∆χ − χ ∆ψ), (5.6)

div (e2u Dχ) = 2e4u Dψ · (Dv − ψ Dχ + χ Dψ), (5.7)

div (e2u Dψ) = −2e4u Dχ · (Dv − ψ Dχ + χ Dψ). (5.8)

The proof consists of two parts. In the first part, we prove the general case where
γ is arbitrary. In the second part, we consider the physical case where γ = 1 and
(u, v, χ, ψ) is axially symmetric around the z-axis.

For the first part, we apply Theorem 4.1 and Corollary 5.1 to obtain Hölder conti-
nuity. Unlike proving Corollary 3.1, we cannot apply Theorem 3.1 due to the coupling
of the unknowns. Instead, we repeatedly apply Lemma 3.3, a primitive of Theorem 3.1,
to (5.6)-(5.8) to get C k,α regularity.

For the second part, we follow the reversed Ernst-Geroch reduction scheme to prove
smoothness under axial symmetry. This was used by Weinstein in his work on the
vacuum case. His idea is the following. The system (1.1)-(1.2) was obtained by applying
the Ernst-Geroch formulation for axisymmetric stationary vacuum solutions, which was
done in the order of the axial Killing vector field being applied first and the stationary
Killing vector field second. This choice of order made it more convenient to establish
existence and uniqueness but regularity. If one applies the reduction scheme in the
reversed order, i.e. to the stationary Killing vector field first and then to the axial Killing
vector field, one obtains a harmonic map problem from R3 into the real hyperbolic plane
in the usual sense. Similarly, if one applies the reversed Ernst-Geroch reduction scheme
to the Einstein-Maxwell equations, the system acquired is, though no longer a harmonic
map problem, still elliptic. To finish one needs to prove some initial regularity for the
 36

acquired system. In the vacuum case, this can be done much simpler by a clever use
of the De Giorgi-Nash-Moser regularity result for scalar elliptic equation (see [22]). It
seems unlikely that this technique applies in the electrovac case. It is precisely at this
point that we need to use the C k,α regularity obtained in the general case to go forward.

Proof of Theorem B. Note that since the complex hyperbolic plane is a manifold of
negative sectional curvature, (u, v, χ, ψ) is C ∞ in R3 \ Σ. Thus we only need to consider
the regularity question around Γ. Without loss of generality, we assume that the origin
is an interior point of Γ, h = −2γ log ρ and that the controlling ideal point on ∂HC is
(+∞, 0, 0, 0) as in Chapter 2.

Step 1: C k,α regularity in the general case.

Let ū = u − h. Then, as shown in Chapter 2, (ū, v, χ, ψ) satisfies the hypotheses of

Theorem 4.1 and Proposition 5.1. Therefore, (ū, v, χ, ψ) is locally Hölder continuous in
Ω and there is some λ > 0 so that
Z h
1 2 −4γ 2 −2γ 2 2
i
|Dū| + ρ |Dv| + ρ (|Dχ| + |Dψ| ) ≤ C δλ.
δ n−2 Bδ
It follows that |Dū| = O(ρ−1+λ ), |Dv| = O(ρ−1+2γ+2λ ), and |Dχ|+|Dψ| = O(ρ−1+γ+λ ).
Thus, by (5.3), |∆u| = O(ρ−2+2λ ).

The above estimates show that |Dψ| |Dv − ψ Dχ + χ Dψ| = O(ρ−2+3γ+2λ ). Thus,
γ
if λ < 2, an application of Lemma 3.3 to (5.7) shows that |χ| = O(ργ+2λ ). Similarly,
|ψ| = O(ργ+2λ ). Applying basic gradient estimates for Poisson equations to (5.7) and
(5.8), we deduce that |Dχ| + |Dψ| = O(ρ−1+γ+2λ ). Using equations (5.4) and (5.5)
we infer that |∆χ| + |∆ψ| = O(ρ−2+γ+2λ ). Hence the right hand side of (5.6) is at
most O(ρ−2−2γ+4λ ). Lemma 3.3 applies again, but to (5.6), yielding |v| = O(ρ2γ+4λ ).
Gradient estimates for Poisson equations then show that |Dv| = O(ρ−1+2γ+4λ ).

Repeating the argument in the previous paragraph we see that |v| = O(ρ2γ+2λ ) and
|χ| + |ψ| = O(ργ+λ ) for any λ < γ. The regularity result in the general case follows.

Step 2: Smoothness when (u, v, χ, ψ) is symmetric about Γ and γ = 1.

It suffices to show that (u, v, χ, ψ) is C ∞ in Bδ (0) for some δ > 0 small enough.

Step 2(a): Construction of new functions.

 37

We first exploit the symmetry to construct new functions defined in a neighborhood

of the origin. Let D2 and div2 denote the gradient and divergence operators in the
half plane {(ρ, z) ∈ R2 |ρ > 0}. Then, in terms of cylindrical coordinates, (1.6)-(1.9) are
equivalent to

div2 (ρ D2 u) − 2 e4u ρ|D2 v − ψ D2 χ + χ D2 ψ|2 − e2u ρ(|D2 χ|2 + |D2 ψ|2 ) = 0, (5.9)

div2 e4u ρ (D2 v − ψ D2 χ + χ D2 ψ) = 0,

 
(5.10)

div2 (e2u ρ D2 χ) − 2e4u ρ D2 ψ · (D2 v − ψ D2 χ + χ D2 ψ) = 0, (5.11)

div2 (e2u ρ D2 ψ) + 2e4u ρ D2 χ · (D2 v − ψ D2 χ + χ D2 ψ) = 0. (5.12)

Set ω = (ωρ , ωz ) = D2 v − ψ D2 χ + χ D2 ψ. Using (5.10)-(5.12), we define p, χ̃ and ψ̃ by

dp = −2e4u ρ ωz dρ + 2e4u ρ ωρ dz, (5.13)

dχ̃ = −(e2u ρ ψz + p χρ )dρ + (e2u ρ ψρ − p χz )dz, (5.14)

dψ̃ = (e2u ρ χz − p ψρ )dρ − (e2u ρ χρ + p ψz )dz. (5.15)

Note that they are well-defined up to a constant. Also, by the C k,α regularity result
from the general case, p, χ̃ and ψ̃ are locally bounded.

Set

1  −4u −1 2 
ω̃ = (ω̃ρ , ω̃z ) = (e ρ p + ρ)pz − 4ρ p uz , −(e−4u ρ−1 p2 + ρ)pρ + 4ρ p uρ + 2p
2
1 2 
= 2p ωρ + ρ pz − 4ρ p uz , 2p2 ωz − ρ pρ + 4ρ p uρ + 2p .
2

A straightforward calculation using (5.9), (5.13), (5.14) and (5.15) shows that

2(ω̃ρ,z − ω̃z,ρ ) = 4p(pz ωρ − pρ ωz ) − 4p [(ρ uz )z + (ρ uρ )ρ ]

+ 2p2 (ωρ,z − ωz,ρ ) + ρ2 [(ρ−1 pz )z + (ρ−1 pρ )ρ ] − 4ρ(pz uz + pρ uρ )

= −4e2u ρ p(|D2 χ|2 + |D2 ψ|2 ) + 4(e4u ρ2 + p2 )(ψρ χz − χρ ψz )

= 4(ψ̃ρ χ̃z − χ̃ρ ψ̃z ). (5.16)

Thus there exists ṽ such that

dṽ = (ω̃ρ + ψ̃ χ̃ρ − χ̃ ψ̃ρ )dρ + (ω̃z + ψ̃ χ̃z − χ̃ ψ̃z )dz. (5.17)
 38

Like χ̃ and ψ̃, ṽ is defined up to a constant but remains locally bounded.

Finally, define ũ by

1 1
ũ = − log(e2u ρ2 − e−2u p2 ) = − log(e2ū − e−2ū ρ2 p2 ). (5.18)
2 2

Recall that ū = u − h = u + γ log ρ. Since ū and p are locally bounded, ũ is bounded

in B2δ (0) for δ small enough.

Step 2(b): Equations governing (ũ, ṽ, χ̃, ψ̃).

By construction, (ũ, ṽ, χ̃, ψ̃) is smooth in Bδ (0) \ Γ. We claim that it satisfies the
following equations in Bδ (0) \ Γ:

∆ũ − 2 e4ũ |Dṽ − ψ̃ Dχ̃ + χ̃ Dψ̃|2 + e2ũ (|Dχ̃|2 + |Dψ̃|2 ) = 0, (5.19)

div e4ũ (Dṽ − ψ̃ Dχ̃ + χ̃ Dψ̃) = 0,

 
(5.20)

div (e2ũ Dχ̃) + 2e4ũ Dψ̃ · (Dṽ − ψ̃ Dχ̃ + χ̃ Dψ̃) = 0, (5.21)

div (e2ũ Dψ̃) − 2e4ũ Dχ̃ · (Dṽ − ψ̃ Dχ̃ + χ̃ Dψ̃) = 0. (5.22)

In cylindrical coordinates, these equations take the form

div2 (ρ D2 ũ) − 2 e4ũ ρ |ω̃|2 + e2ũ ρ(|D2 χ̃|2 + |D2 ψ̃|2 ) = 0, (5.23)

div2 (e4ũ ρ ω̃) = 0, (5.24)

div2 (e2ũ ρ D2 χ̃) + 2e4ũ ρ D2 ψ̃ · ω̃ = 0, (5.25)

div2 (e2ũ ρ D2 ψ̃) − 2e4ũ ρ D2 χ̃ · ω̃ = 0. (5.26)

Define
e−2u p e−2u p
p̃ = = . (5.27)
e2u ρ2 − e−2u p2 e−2ũ
Then
dp̃ = −2e4ũ ρ ω̃z dρ + 2e4ũ ρ ω̃ρ dz, (5.28)

which shows that (ũ, ṽ, χ̃, ψ̃) satisfies (5.24).

By (5.14), (5.15) and (5.27),

dχ = −(e2ũ ρ ψ̃z − p̃ χ̃ρ )dρ + (e2ũ ρ ψ̃ρ + p̃ χ̃z )dz. (5.29)

 39

This implies that

−(e2ũ ρ ψ̃z + p̃ χ̃ρ )z = (e2ũ ρ ψ̃ρ − p̃ χ̃z )ρ .

In view of (5.28), (5.26) follows. Similarly, (5.25) holds due to

dψ = (e2ũ ρ χ̃z + p̃ ψ̃ρ )dρ − (e2ũ ρ χ̃ρ − p̃ ψ̃z )dz. (5.30)

We next verify (5.23). In the interior of {p̃ = 0}, p and ω̃ vanish, and ũ = −ū, so (5.23)
follows immediately from (5.9), (5.14) and (5.15). Thus, by continuity, it suffices to
consider the region where p̃ 6= 0. We note that

e−2ũ p̃ e−2ũ p̃
p= = .
e−2u e2ũ ρ2 − e−2ũ p̃2

In view of (5.13), this implies that

1  −4ũ −1 2 
ω= (e ρ p̃ + ρ)p̃z − 4ρ p̃ ũz , −(e−4ũ ρ−1 p̃2 + ρ)p̃ρ + 4ρ p̃ ũρ + 2p̃ ,
2

and so, by (5.16) and (5.27),

ωρ,z − ωz,ρ = 4e4ũ ρ p̃ |ω̃|2 − 2p̃ [(ρ ũz )z + (ρ ũρ )ρ ] + 2(e4ũ ρ2 + p̃2 )(ψ̃ρ χ̃z − χ̃ρ ψ̃z ).

On the other hand, by (5.29) and (5.30)

ωρ,z − ωz,ρ = 2(ψρ χz − χρ ψz )

= 2e2ũ ρ p̃(|D2 χ̃|2 + |D2 ψ̃|2 ) + 2(e4ũ ρ2 + p̃2 )(ψ̃ρ χ̃z − χ̃ρ ψ̃z ).

The above relations imply that

4e4ũ ρ p̃ |ω̃|2 − 2p̃ [(ρ ũz )z + (ρ ũρ )ρ ] = 2e2ũ ρ p̃(|D2 χ̃|2 + |D2 ψ̃|2 ).

Since p̃ is nonzero, (5.23) follows.

Step 2(c): Smoothness of (ũ, ṽ, χ̃, ψ̃).

We will use the following lemma, which is easy to prove.

Lemma 5.2 Assume that f is a regular function in B1 (0) \ Γ and that |f | = O(ρ−1+α )
for some α ∈ (0, 1). If N f be the Newtonian potential of f with respect to B1 (0), i.e.
Z
N f (x) = Φ(x − y)f (y) dy
B1 (0)
 40

where Φ is the fundamental solution of the Laplace equation, then N f is C 1,α in B1 (0).

As a consequence, if u ∈ L1 (B1 (0)) is a weak solution of

∆u = f

in B1 (0), then u is C 1,α in B1 (0).

We have shown that (5.19)-(5.22) hold in Bδ (0) \ Γ. Moreover, by the regularity

result for the general case, ũ, ṽ, χ̃ and ψ̃ belong to H 1 (Bδ (0)) ∩ C 0,α (B1 (0)) for any α
∈ (0, 1). Hence, since Γ is of codimension 2, they satisfy (5.19)-(5.22) in Bδ (0) in the
weak sense.

On the other hand, using the regularity result for the general case, we can show
that ũ, ṽ, χ̃ and ψ̃ are C 0,α in Bδ (0) for any α ∈ (0, 1). Moreover,

|Dṽ| + |Dχ̃| + |Dψ̃| = O(ρα−1 ).

Applying Lemma 5.2 to (5.19), (5.21), (5.22) and then (5.20), we can show that ũ,
ṽ, χ̃ and ψ̃ are C 1,α . This allows us to bootstrap in between (5.19)-(5.22) to obtain
smoothness for ũ, ṽ, χ̃ and ψ̃ in Bδ (0).

Step 2(d): Smoothness of (ū, v, χ, ψ).

The smoothness of χ and ψ follows from (5.29) and (5.30). By (5.28), p̃ and so
e−2u p = e−2ũ p̃ are smooth. That of e−2u and of ū follows from the identity

ρ2 e−2ū = e−2u = e2ũ ρ2 − e−2ũ p̃2 .

Next, since ρ2 p is smooth and p ∈ C 0,α , p is smooth too. The smoothness of v follows
from (5.13). The proof of the theorem is complete. 

Proof of Theorem C. The theorem follows immediately from Theorem B and the
results in [27].
 41

Appendix A

Hyperbolic spaces

In this appendix, we present some relevant facts about the complex hyperbolic plane
HC ∼
= U (1, 2)/(U (1) × U (2)).

In the disk model, the hyperbolic plane HC is modeled by the disk D = {ζ =

(ζ 1 , ζ 2 ) ∈ Cn : |ζ|2 < 1} with the metric element
n δij ζ i ζ̄ j o i j |dζ|2 |ζ · dζ|2
ds2 = + d ζ̄ dζ = + .
1 − |ζ|2 (1 − |ζ|2 )2 1 − |ζ|2 (1 − |ζ|2 )2

The action of a matrix A = (aij ) ∈ U (1, 2) on D is given by

a + a22 ζ1 + a23 ζ2 a31 + a32 ζ1 + a33 ζ2 
21
A·ζ = , .
a11 + a12 ζ1 + a13 ζ2 a11 + a12 ζ1 + a13 ζ2

The isotropy group at the origin is U (1) × U (2).

The geodesics at 0 are precisely the R-lines through 0. All other geodesics can be
obtained from those by some motion in U (1, 2). This implies that

|1 − ζ · ζ̂|
cosh dist(ζ, ζ̂) = p q .
1 − |ζ|2 1 − |ζ̂|2

We now derive the (u, v, χ, ψ) parametrization of HC from the disk model. See [25]
for a geometric interpretation.

Define

|1 + ζ1 |
u = log p ,
1 − |ζ|2
1
v = Im w1 ,
2
χ = Re w2

ψ = Im w2 ,
 42

where

1 − ζ1
w1 = ,
1 + ζ1
ζ2
w2 = .
1 + ζ1

Note that
1 − |ζ|2
e−2u = = Re w1 − |w2 |2 .
|1 + ζ1 |2
Hence, for a given (u, v, χ, ψ), we can recover ζ by

1 − w1
ζ1 = ,
1 + w1
2w2
ζ2 = ,
1 + w1

and

w1 = e−2u + χ2 + ψ 2 + 2i v,

w2 = χ + i ψ.

By a direct calculation, we see that the line element in terms of (u, v, χ, ψ) is

ds2 = du2 + e4u (dv − ψ dχ + χ dψ)2 + e2u (dχ2 + dψ 2 ).

 43

Appendix B
1,p
The space WΣ (Ω, w)

We will study the space WΣ1,p (Ω, w) defined in the introduction. Most of the results
here are probably known to readers. However we show them for completeness.

Throughout this appendix, we will frequently make the following assumptions on

Ω, Σ and w.

(C1) Σ is a (n − k)-dimensional submanifold of Rn (possibly with or without a bound-

ary) and Ω is a bounded domain in Rn .

(C2) ϕ is a positive C 2 function in a punctured neighborhood of Σ such that ϕ(x) →

0 as x → Σ.

(C3) w is a positive measurable function defined on Σ such that w = w(ϕ) in the

domain of ϕ.

(C4) w is bounded on any compact subset of Ω̄ \ Σ.

(C5) w is bounded from below by a positive number on any compact subset of Ω̄ \ Σ.

Lemma B.1 Let 1 < p < ∞. Assume (C1)-(C3). Let α and β be functions on (0, ∞)
which are locally bounded measurable functions on compact subsets of (0, ∞) and satisfy

∆ϕ(x)
α(t) ≥ sup 2
,
ϕ(x)=t |Dϕ(x)|

β(t) ≥ sup |Dϕ(x)|−(p−2)/(p−1) .

ϕ(x)=t

Let A be an anti-derivative of α and define

Z t −p
p µ A(t) −1/(p−1) −1/(p−1) µ A(s)
w̃µ (t) = e w (t) β(s) w (s) e ds
0

and w̃µ (x) = w̃µ (ϕ(x)).

 44

(i) If β w−1/(p−1) e−A/(p−1) is locally integrable near 0, then there exists δ > 0 such
that d(Ωc , Σ) < δ implies
Z Z
p 1,p
|f | w̃−1/(p−1) dx ≤ C |Df |p w dx f ∈ W0,Σ (Ω, w).
Ω Ω

The constant C depends only on p.

(ii) If, in addition, α is positive and the level surfaces of ϕ are regular hypersurfaces,
then there exists δ > 0 such that d(Ωc , Σ) < δ implies
Z Z
p 1,p
|f | w̃µ dx ≤ C |Df |p w dx f ∈ W0,Σ (Ω, w),
Ω Ω

for any µ for which β w−1/(p−1) eµ A is locally integrable near 0. The constant C
depends only on p and µ.

Proof. It suffices to consider non-negative f ∈ Cc∞ (Ω \ Σ).

(i) Let Bt = {d(x, Σ) ≥ t}. For t sufficiently small, f vanishes outside of Bt . Thus,
→
if F is a C 1 vector-valued function on Ω \ Σ then
Z Z
→ → →
p
0= div (f F ) dx = [f p div (F ) + p f p−1 Df · F ] dx.
Bt Bt
→
Hence, if div (F ) is positive,
→
| F |p
Z Z
→
f p div (F ) dx ≤ C p
|∇f | → dx. (B.1)
Bt Bt (div (F ))p−1

Define
Z t −p+1
−A(t) −1/(p−1) −A(s)/(p−1)
F (t) = −e β(s) w (s) e ds < 0,
0

so that
F 0 + α F = (p − 1) β w−1/(p−1) |F |p/(p−1) .
→
Hence, if F (x) = F (ϕ(x))Dϕ(x) then
→ ∆ϕ
div (F ) = |Dϕ|2 F 0 + F ≥ |Dϕ|2 (F 0 + α F )


|Dϕ|2

= (p − 1) |Dϕ|2 β w−1/(p−1) |F |p/(p−1)

≥ (p − 1) w−1/(p−1) |F |p/(p−1) |Dϕ|p/(p−1)

→
= (p − 1) w−1/(p−1) | F |p/(p−1) = (p − 1)w̃−1/(p−1) .
 45

Also, this implies

→
| F |p
→ ≤ (p − 1)p−1 w.
(div (F ))p−1
(i) then follows from (B.1).

(ii) We localize the proof of (i). Since α is positive, A is strictly increasing. Let tj be
an increasing sequence of positive real numbers such that 0 ≤ A(tj+1 ) − A(tj ) ≤
1. Let Sj = {d(x, Σ) = tj } and Aj = {tj+1 ≤ d(x, Σ) ≤ tj }. Similar to (i), any
→ →
vector-valued function F which is differentiable in Aj such that div (F ) > 0 gives
rise to
→
| F |p
Z Z
1 p
→
p
−Dj + f div (F ) dx ≤ C |∇f | → dx. (B.2)
2 Aj Aj (div (F ))p−1
where
Z Z
→ →
p
Dj = f F ·νj dσ − f p F ·νj+1 dσ.
Sj Sj+1

Here νj denotes the normal vector to Sj in the direction of increasing distance.

For tj ≤ t ≤ tj+1 , define

(Z )−p+1
t
Fj (t) = −e−A(t) β(s) w−1/(p−1) (s) e−A(s)/(p−1) ds + Kj < 0,
tj

→
where Kj is a constant to be specified later. Set Fj (x) = Fj (ϕ(x)) Dϕ(x). Again,

Fj0 (t) + α Fj (t) = (p − 1)β w−1/(p−1) (t) |Fj (t)|p/(p−1) ,

→
→ → |F j |p
which implies div (Fj ) ≥C w−1/(p−1) | Fj |p/(p−1) and → ≤ Cw.
(div (Fj ))p−1

On the other hand, recalling the definition of Fj and the sequence {tj }, we have
(Z )−p
t
|Fj |p/(p−1) (t) = e−p A(t)/(p−1) β(s)w−1/(p−1) (s) e−A(s)/(p−1) ds + Kj
tj
(Z )−p
t
= β(s)w−1/(p−1) (s) e(A(t)−A(s))/(p−1) ds + eA(t)/(p−1) Kj
tj
(Z )−p
t
−1/(p−1) −µ(A(t)−A(s)) A(t)/(p−1)
≥C β(s)w (s) e ds + e Kj
tj
(Z )−p
t
= Cep µ A(t) β(s)w−1/(p−1) (s) eµ A(s) ds + e(µ+1/(p−1))A(t) Kj
tj
(Z )−p
t
p µ A(t) −1/(p−1) µ A(s) (µ+1/(p−1))A(tj )
≥ Ce β(s)w (s) e ds + e Kj .
tj
 46

Hence, by setting
Z tj
Kj = e−(µ+1/(p−1))A(tj ) β(s) w−1/(p−1) (s) eµ A(s) ds
0

we arrive at
w−1/(p−1) |Fj |p/(p−1) ≥ C w̃µ (t).

The above estimates allow us to rewrite (B.2) as

Z Z
−Dj + C1 f p w̃ dx ≤ C2 |Df |2 w dx
Aj Aj

where C1 and C2 is independent of j. Summing over j and recall that f ∈

Cc∞ (Ω \ Σ), we get the assertion.

Remark B.1 (i) It happens in many cases that w = o(w̃µ ).

(ii) For ϕ(x) = d(x, Σ), we can take α(t) = k/t and β(t) = 1. Here k is the codimen-
sion of Σ. In that case,
Z t −p
pkµ −1/(p−1) −1/(p−1) kµ
w̃µ (t) = t w (t) w (s) s ds .
0

In particular, if w(x) = d(x, Σ)−γ , we can take w̃(x) = d(x, Σ)−γ−p .

(iii) The proof is actually valid if we only assume that f ∈ WΣ1,p (Ω, w) and f = 0 at
points on ∂Ω where the normal vector is not perpendicular to Dϕ.

Proposition B.1 Let 1 < p < ∞. Assume (C1)-(C4). Let δ and µ be as in Lemma
B.1. Define 
 w̃µ (x) if d(x, Σ) < δ,
w̃(x) =
 1 otherwise.

1,p
(i) For any f ∈ W0,Σ (Ω, w), f ∈ Lp (Ω, w̃) and
Z Z
|f |p w̃ dx ≤ C [|f |p + |Df |p w] dx.
Ω Ω
 47

(ii) For any f ∈ WΣ1,p (Ω, w), f ∈ Lploc (Ω, w̃) and
Z Z
|f |p w̃ dx ≤ C [|f |p + |Df |p w] dx
ω Ω

for any ω compactly supported in Ω.

Proof. Part (i) follows immediately from Lemma B.1 and the classical Poincare in-
equality.

For part (ii), it suffices to consider ω for which there is some r such that each
connected component of ω ∩{d(x, Σ) ≤ r} is bounded by the hypersurface {d(x, Σ) = r}
and two arbitrary hypersurfaces tangential to Dϕ. Also, we can assume that f ∈
Cc∞ (Ω̄ \ Σ). We will show that
Z Z
p
|f | w̃ dx ≤ C [|f |p + |Df |p w] dx (B.3)
ω ω

Let η be a standard cut-off function which vanishes in {d(x, Σ) > r} and is identically
1 in {d(x, Σ) < r/2}. Split f = f1 + f2 = f η + f (1 − η). Then f1 vanishes on
{x ∈ ∂ω|d(x, Σ) = r}. The remainder of ∂ω is tangential to Dϕ. Thus by the remark
following Lemma B.1,
Z Z
p
|f1 | w̃ dx ≤ C |D(f η)|p w dx
ω
Zω Z
p
≤C |f | w dx + C |Df |p w dx
{r/2<d(x,Σ)<r} ω
Z Z
p
≤ C(r) |f | dx + C |Df |p w dx.
ω ω

For f2 , we compute
Z Z Z
p p
|f2 | w̃ dx = |f2 | w̃ dx ≤ C(r) |f |p dx.
ω {r/2<d(x,Σ)<r} ω

The result follows from Minkowski inequality. 

Proposition B.2 Let 1 < p < ∞. Assume (C1)-(C5). Define w̃ as in Proposition

B.1. Let w̄ be a positive measurable weight in Ω such that w̄ = o(w̃) in a neighborhood
of Σ and w̄ is bounded on compact subsets of Ω̄ \ Σ. Then WΣ1,p (Ω, w) is compactly
embedded into Lploc (Ω, w̄).
 48

Proof. Let fn be a bounded sequence in WΣ1,p (Ω) and ω compactly supported in Ω.

By Proposition B.1 and our assumption on w̄, fn ∈ Lp (Ω, w̄). We will show that, up
to extracting a subsequence, fn converges in Lp (ω, w̄). Indeed, by (C5), we can assume
that fn converges to f pointwise and in Lploc (Ω̄ \ Σ). We claim that fn converges to f
in Lp (ω, w̄). To this end it suffices to show that
Z Z
p
lim |fn | w̄ dx = |f |p w̄ dx.
n→∞ ω ω

Fix a ε > 0. We observe that, |fn |p w̄ converges to |f |p w̄ in L1 (ω ∩ {d(x, Σ) ≥ t})

for any t, so
Z Z
lim |fn |p w̄ dx = |f |p w̄ dx.
n→∞ ω∩{d(x,Σ)≥t} ω∩{d(x,Σ)≥t}

On the other hand, by Proposition B.1,

Z Z
p w̄(s)
|fn | w̄ dx ≤ sup |fn |p w̃ dx
ω∩{d(x,Σ)≤t} 0≤s≤t w̃(s) ω∩{d(x,Σ)≤t}
Z
w̄(s)
≤ C(ω) sup [|fn |p + |Dfn |p w] dx ≤ ε
0≤s≤t w̃(s) Ω

for t sufficiently large. By Fatou’s lemma, this implies

Z
|f |p w̄ dx ≤ ε.
ω∩{d(x,Σ)≤t}

Combining the above estimates, we infer that

Z Z 
p p
 
lim sup  |fn | w̄ dx − |f | w̄ dx ≤ 3ε.

n→∞ ω ω

Letting ε → 0, the assertion follows. 

Remark B.2 Using the inequality (B.3) instead of Proposition B.1, we can show that
WΣ1,p (Ω, w) embeds compactly into Lp (Ω, w̄) whenever there is some r such that each
component of Ω ∩ {d(x, Σ) ≤ r} is a set bounded by {d(x, Σ) = r} and two hypersurfaces
tangential to Dϕ.

Proposition B.3 Let 1 < p < ∞. Assume (C1)-(C5) and that Ω intersects Σ. Define
w̃ as in Proposition B.1. Let w̄ be a positive measurable weight in Ω such that w̄ =
o(w̃) in a neighborhood of Σ and w̄ is bounded on compact subsets of Ω̄ \ Σ. Assume in
 49

addition that w̄ is bounded from below. If w̃ is nowhere locally integrable along Σ, then
for any ω compactly supported in Ω,
Z Z
p
|f | w̄ dx ≤ C |Df |p w dx.
ω Ω

Proof. It suffices to show that

Z Z
p
|f | w̄ dx ≤ C |Df |p w dx
ω ω

for any ω compactly supported in Ω such that, for some r, ω ∩ {d(x, Σ) ≤ r} is a set
bounded by {d(x, Σ) = r} and two hypersurfaces tangential to Dϕ.

Arguing indirectly, we assume that there exists fh such that

Z Z
p
1= |fh | w̄ dx ≥ h |Df |p w.
ω ω

Since w̄ is bounded from, fh is uniformly bounded in Lp (ω), and so in WΣ1,p (ω, w) (as
h → ∞). Hence, by Remark B.2, we can assume that there exists f ∈ WΣ1,p (ω, w) such
that fh converges weakly in WΣ1,p (ω, w) and strongly in Lp (ω, w̄) to f . This results in
Z Z
p
|Df | w dx ≤ lim inf |Dfh |p w dx = 0,
ω h→∞ ω

which show that f is constant in ω. But as |f |p w̃ is locally integrable in ω (cf. Propo-

sition B.1) and w̃ is nowhere locally integrable long Σ, f must vanish identically. This
contradicts
Z Z
p
|f | w̄ dx = lim |fh |p w̄ dx = 1.
ω h→∞ ω

The proof is complete. 

Finally, we restate the above results for the case w = d(x, Σ)−γ .

1,p
Corollary B.1 (i) For f ∈ W0,Σ (Ω, d−γ ),
Z Z
−γ−p
|f | d p
, dx ≤ C |Df |p d−γ dx.
Ω Ω

(ii) For f ∈ WΣ1,p (Ω, d−γ ) and ω compactly supported in Ω,

Z Z
|f |p d−γ−p dx ≤ C [|f |p + |Df |p d−γ ] dx.
ω Ω
 50

If, in addition, Ω intersects Σ and γ > k − 2, then

Z Z
0
|f |p d−γ dx ≤ C |Df |p d−γ dx,
ω Ω

where 0 ≤ γ 0 < γ + p. If Ω and ω are fixed concentric balls and Σ is straight, the
constant C depends only on p, γ, γ 0 and the distance from the center to Σ. The
bigger the distance, the bigger the constant.

0
(iii) The embedding WΣ1,p (Ω, d−γ ) ,→ Lploc (Ω, d−γ ) is compact for any 0 ≤ γ 0 < γ + p.
 51

References

[1] R. Bach and H. Weyl, Neue Lösungen der Einsteinschen Gravitationsgleichun-

gen, Math. Z., 13 (1922), pp. 134–145.

[2] G. Bunting, Proof of the uniqueness conjecture for black holes, PhD thesis, Univ.
of New England, Armadale N.S.W., 1983.

[3] G. Bunting and A. Massood-ul-Alam, Nonexistence of multiple black holes

in asymptotically Euclidean static vacuum space-times, Gen. Rel. Grav., 19 (1987),
pp. 147–154.

[4] E. Fabes, C. Kenig, and R. Serapioni, The local regularity of solutions of

degenerate elliptic equations, Comm. in PDEs, 7 (1982), pp. 77–116.

[5] M. Giaquinta, Multiple Integrals in the Calculus of Variations and Nonlinear

Elliptic Systems, Princeton Univ. Press, Princeton, 1983.

[6] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second

Order, Springer, Berlin Heidenberg, 2001.

[7] J. Hartle and S. Hawking, Solutions of the Einstein-Maxwell equations with

many black holes, Comm. Math. Phys., 26 (1972), pp. 87–101.

[8] M. Heusler, Black Hole Uniqueness Theorems, Cambridge Univ. Press, Cam-
bridge, 1996.

[9] D. Kramer and G. Neugebauer, The superposition of two Kerr solutions, Phys.
Lett. A, 75 (1980), pp. 259–261.

[10] Y. Li and G. Tian, Nonexistence of axially symmetric, stationary solution of ein-

stein vacuum equation with disconnected symmetric event horizon, Manu. Math.,
73 (1991), pp. 83–89.

[11] , Regularity of harmonic maps with prescribed singularities, Comm. Math.

Phys., 149 (1992), pp. 1–30.

[12] , Regularity of harmonic maps with prescribed singularities, in Differential

Geometry: PDE on manifolds, R. Greene and S. Yau, eds., vol. 54 of Proc. Symp.
Pure Math., Part 1, Providence, 1993, Amer. Math. Soc., pp. 317–326.

[13] S. Majumdar, A class of exact solutions of Einstein’s field equations, Phys. Rev.,
72 (1947), pp. 930–938.

[14] P. Mazur, Proof of uniqueness of the Kerr-Newman black holes, J. Phys. A, 15

(1982), pp. 3173–3180.
 52

[15] , Black hole uniqueness theorem, in General Relativity and Gravitation,

M. McCallum, ed., Proc. of GR11, Cambridge, 1986, pp. 130–157.

[16] L. Nguyen, On the regularity of singular harmonic maps and axially symmetric
stationary electrovac space-times, preprint.

[17] A. Papapetrou, A static solution of the gravitational field for arbitrary charge
distribution, Proc. R. Irish Acad., A51 (1945), pp. 191–204.

[18] D. Robinson, Uniqueness of the Kerr black hole, Phys. Rev. Lett., 34 (1975),
pp. 905–906.

[19] , A simple proof of the generalization of Israel’s theorem, Gen. Rel. Grav., 8
(1977), pp. 695–698.

[20] P. Smith and E. Stredulinsky, Nonlinear elliptic systems with certain un-
bounded coefficients, Comm. Pure Appl. Math., 37 (1984), pp. 495–510.

[21] R. Wald, General Relativity, The Univ. of Chicago Press, Chicago and London,
1984.

[22] G. Weinstein, On rotating black holes in equilibrium in general relativity, Comm.

Pure Appl. Math., 43 (1990), pp. 903–948.

[23] , The stationary axisymmetric two-body problem in general relativity, Comm.

Pure Appl. Math., 45 (1992), pp. 1183–1203.

[24] , On the force between rotating co-axial black holes, Trans. Amer. Math. Soc.,
343 (1994), pp. 899–906.

[25] , On the Dirichlet problem for harmonic maps with prescribed singularities,
Duke Math. J., 77 (1995), pp. 135–165.

[26] , Harmonic maps with prescribed singularities on unbounded domains, Amer.

J. Math., 118 (1996), pp. 689–700.

[27] , N-black hole stationary and axially symmetric solutions of the Ein-
stein/Maxwell equations, Comm. PDE., 21 (1996), pp. 1389–1430.
 53

Vita

Luc L. Nguyen

2004-2009 Ph.D. in Mathematics, Rutgers University

1998-2002 B.S. in Mathematics from Vietnam National University at Ho Chi Minh

City

2006-2008 Teaching assistant, Department of Mathematics, Rutgers University