Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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subgroup of SO(n)orO(n). Then M carries some

extra geometric structures compatible with g.

Broadly, the smaller Hol(g) is as a subgroup of

O(n), the more special g is, and the more extra

geometric struc tures there are. Therefore, under-

standing and classifying the possible holonomy

groups gives a family of interesting special Rieman-

nian geometri es, such as Ka¨ hler geometry. All of

these speci al geometries have cropped up in physics.

Define the holonomy algebra hol(g) to be the Lie

algebra of Hol(g), regarded as a Lie subalgebra of

o(n), defined up to the adjoint action of O(n).

Define hol

(g) to be the Lie algebra of Hol

(g), as a

Lie subalgebra of o(T

M) ﬃ 



M. The holonomy

algebra hol(g) is intimately connected with the

Riemann curvature tensor R

abcd

= g

bcd

of g.

Theorem 3 The Riemann curvature tensor R

abcd

lies in S

hol

(g) at x, where hol

(g) is regarded as a

subspace of 



M. It also satisfies the first and

second Bianchi identities

abcd

þ R

adbc

þ R

acdb

¼ 0 ½3

abcd

þr

abde

þr

abec

¼ 0 ½4

A related result is the Ambrose–Singer holonomy

theorem, which, roughly speaking, says that hol

(g)

may be reconstructed from R

abcd

for all y 2 M,

moved to x by parallel transport.

If (M, g) and (N, h) are Riemannian manifolds, the

product M  N carries a product metric g  h.Itis

easy to show that Hol(g  h) = Hol(g)  Hol(h). A

Riemannian manifold (M, g) is called reducible if

every point has an open neighborhood isometric to a

Riemannian product and irreducible otherwise.

Theorem 4 Let (M, g) be Riemannian n-manifold.

Then the natural representation of Hol(g) on R

reducible if and only if g is reducible.

There is a class of Riemannian manifolds called

the ‘‘Riemannian symmetric spaces’’ which are

important in the theory of Riemannian holonomy

groups. A Riemannian symmetric space is a

special kind of Riemannian manifold with a

transitive isometry group. The theory of sym-

metric spaces was worked out by E

lie Cartan in

the 1920s, who classified them completely, using

his own classification of Lie groups and their

representations.

A Riemannian metric g is called ‘‘locally sym-

metric’’ if r

abcd

 0, and ‘‘nonsymmetric’’ other-

wise. Every locally symmetric metric is locally

isometric to a Riemannian symmetric space. The

relevance of symmetric spaces to holonomy groups

is that many possible holonomy groups are the

holonomy group of a Riemannian symmetric space,

but are not realized by any nonsymmetric metric.

Therefore, by restricting attention to nonsymmetric

metrics, one considerably reduces the number of

possible Riemannian holono my groups.

A tensor S on M is constant if rS = 0. An

important property of Hol(g) is that it determines

the constant tensors on M.

Theorem 5 Let (M, g) be a Riemannian manifold,

with Levi-Civita connection r. Fix x 2 M, so

that Hol

(g) acts on T

M, and so on the tensor

powers

M 



M. Suppose S 2 C

(

TM 



M) is a constant tensor. Then Sj

is fixed by the action of Hol

(g). Conversely,

if Sj

M 



M is fixed by Hol

(g),

it extends to a unique constant tensor

S 2 C

(

TM 



M).

The main idea in the proof is that if S is a constant

tensor and  : [0, 1] !M is a path from x to y, then



(Sj

) = Sj

, that is, ‘‘constant tensors are invariant

under parallel transport.’’ In particular, they are

invariant under parallel transport around closed

loops based at x, and so under elements of Hol

(g).

Berger’s Classification of Holonomy Groups

Berger classified Riemannian holonomy groups in

1955.

Theorem 6 Let M be a simply connected,

n-dimensional manifold, and g an irreducible, non-

symmetric Riemannian metric on M. Then

(i) Hol(g) = SO(n),

(ii) n = 2m and Hol(g) = SU(m) or U(m),

(iii) n = 4m and Hol(g) = Sp(m) or Sp(m)Sp(1),

(iv) n = 7 and Hol(g) = G

, or

(v) n = 8 and Hol(g) = Spin(7).

To simplify the classification, Berger makes three

assumptions: M is simply connected, g is irreducible,

and g is nonsymmetric. We can make M simply

connected by passing to the ‘‘universal cover.’’ The

holonomy group of a reducible metric is a product

of holonomy groups of irreducible metrics, and the

holonomy groups of locally symmetric metrics

follow from Cartan’s classification of Riemannian

symmetric spaces. Thus, these three assumptions can

easily be removed.

Here is a sketch of Berger’s proof of Theorem 6.

As M is simply connected, Theorem 2 shows Hol(g)

is a closed, connected Lie subgroup of SO(n), and

since g is irreducible, Theorem 4 shows the

representation of Hol(g)onR

is irreducible. So,

suppose that H is a closed, connected subgroup of

442 Riemannian Holonomy Groups and Exceptional Holonomy

SO(n) acting irreducibly on R

, with Lie algebra h.

The classification of all such H follows from the

classification of Lie groups (and is of considerable

complexity). Berger’s method was to take the list of

all such groups H, and to apply two tests to each

possibility to find out if it could be a holonomy

group. The only groups H which passed both tests

are those in the theorem.

Berger’s tests are algebraic and involve the

curvature tensor. Suppose that R

abcd

is the Riemann

curvature of a metric g with Hol(g) = H. Then

Theorem 3 gives R

abcd

2 S

h, and the first Bianchi

identity [3] applies. But if h has large codimension in

o(n), then the vector space R

of elements of S

satisfying [3] will be small, or even zero. However,

the ‘‘Ambrose–Singer holonomy theorem’’ shows that

must be big enough to generate h. For many of the

candidate groups H, this does not hold, and so H

cannot be a holonomy group. This is the first test.

Now r

abcd

lies in (R

)



 R

, and also satisfies

the second Bianchi identity, eqn [4]. Frequently,

these imply that rR = 0, so that g is locally

symmetric. Therefore, we may exclude such H, and

this is Berger’s second test.

Berger’s proof does not show that the groups on

his list actually occur as Riemannian holonomy

groups – only that no others do. It is now known,

though this took another thirty years to find out,

that all possibilities in Theorem 6 do occur.

The Groups on Berger’s List

Here are some brief remarks about each group on

Berger’s list.

(i) SO(n) is the holonomy group of generic

Riemannian metrics.

(ii) Riemannian metrics g with Hol( g)  U(m) are

called ‘‘Ka¨ hler metrics.’’ Ka¨hler metrics are a natural

class of metrics on complex manifolds, and generic

Ka¨ hler metrics on a given complex manifold have

holonomy U(m).

Metrics g with Hol(g) = SU(m) are called Calabi–

Yau metrics. Since SU(m) is a subgroup of U(m), all

Calabi–Yau metrics are Ka¨hler. If g is Ka¨hler and M

is simply connected, then Hol(g)  SU(m) if and

only if g is Ricci-flat. Thus, Calabi–Yau metrics are

locally more or less the same as Ricci-flat Ka¨hler

metrics.

If (M, J) is a compact complex manifold with

trivial canonical bundle admitting Ka¨ hler metrics,

then Yau’s solution of the Calabi conjecture gives a

unique Ricci-flat Ka¨ hler metric in each canonical

class. This gives a way to construct many examples

of Calabi–Yau manifolds, and explains why these

have been named after them.

(iii) Metrics g with Hol(g) = Sp(m)arecalled

‘‘hyper-Ka¨hler.’’ As Sp(m)  SU(2m)  U(2m), hyper-

Ka¨hler metrics are Ricci-flat and Ka¨hler.

Metrics g with holonomy group Sp(m)Sp(1) for

m  2 are called ‘‘quaternionic Ka¨ hler.’’ (Note that

quaternionic Ka¨ hler metrics are not in fact Ka

¨hler.)

They are Einstein, but not Ricci-flat.

(iv), (v) G

and Spin(7) are the exceptional cases,

so they are called the ‘‘exceptional holonomy

groups.’’ Metrics with these holonomy groups are

Ricci-flat.

The groups can be understood in terms of the four

division algebras: the real numbers R, the complex

numbers C, the quaternions H, and the octonions or

Cayley numbers O.



SO(n) is a group of automorphisms of R



U(m)andSU(m) are groups of automorphisms of C



Sp(m) and Sp(m) Sp(1) are automorphism groups

of H



is the automorphism group of Im O ﬃ R

Spin(7) is a group of automorphisms of O ﬃ R

preserving part of the structure on O .

The Exceptional Holonomy Groups

For some time after Berger’s classification, the

exceptional holonomy groups remained a mystery.

In 1987, Bryant used the theory of exterior

differential systems to show that locally there exist

many metrics with these holonomy groups, and gave

some explicit, incomplete examples. Then in 1989,

Bryant and Salamon found explicit, complete

metrics with holonomy G

and Spin(7) on non-

compact manifolds. In 1994–95, the author con-

structed the first examples of metrics with holonomy

and Spin(7) on compact manifolds. For more

information on exceptional holonomy, see Joyce

(2000, 2002).

The Holonomy Group G

Let (x

, ..., x

) be coordinates on R

. Write dx

ij...l

for the exterior form dx

^ dx

^^dx

on R

Define a metric g

, a 3-form ’

, and a 4-form ’

on R

¼ dx

þþdx

’

¼ dx

123

þ dx

145

þ dx

167

þ dx

246

 dx

257

 dx

347

 dx

356

’

¼ dx

4567

þ dx

2367

þ dx

2345

þ dx

1357

 dx

1346

 dx

1256

 dx

1247

½5

The subgroup of GL(7, R) prese rving ’

is the

exceptional Lie group G

. It also preserves g

,  ’

Riemannian Holonomy Groups and Exceptional Holonomy 443

and the orientation on R

. It is a compact,

semisimple, 14-dimensional Lie group, a subgroup

of SO(7).

A G

-structure on a 7-manifold M is a principal

sub-bundle of the frame bundle of M, with

structure group G

.EachG

-structure gives rise

to a 3-form ’ and a metric g on M, such that every

tangent space of M admits an isomorphism with R

identifying ’ and g with ’

and g

, respectively. By

an abuse of notation, (’, g) can be referred to as a

-structure.

Proposition 7 Let M be a 7-manifold and (’, g) a

-structure on M. Then the following are

equivalent:

(i) Hol(g)  G

, and ’ is the induced 3-form;

(ii) r’ = 0 on M, where r is the Levi-Civita

connection of g; and

(iii) d’ = d(’) = 0 on M.

The equations d’ = d(’) = 0 look like linear

partial differential equations on ’. However , it is

better to consider them as nonlinear, for the

following reason. The 3-form ’ determines the

metric g,andg gives the Hodge star  on M.So

’ is a nonlinear function of ’,andd(’) = 0a

nonlinear equation. Thus, constructing and study-

ing G

-manifolds come down to studying solu-

tions of nonlinear elliptic partial differential

equations.

Note that Hol(g)  G

if and only if r’ = 0

follows from Theorem 5 . We call r’ the

‘‘torsion’’ o f the G

-structure (’, g), and when

r’ = 0theG

-structure is ‘‘torsion-free.’’ A triple

(M, ’, g)iscalledaG

-manifold if M is a

7-manifold and (’, g) a torsion-fre e G

-structure

on M.Ifg has holonomy Hol(g)  G

,theng is

Ricci-flat.

Theorem 8 Let M be a compact 7-manifold, and

suppose that (’, g) is a torsion-free G

-structure on M.

Then Hol(g) = G

if and only if 

(M) is finite. In

this case, the moduli space of metrics with holon-

omy G

on M, up to diffeomorphisms isotopic to

the identity, is a smooth manifold of dimension

(M).

The Holonomy Group Spin(7)

Let R

have coordinates (x

, ..., x

). Define a

4-form 

on R



¼dx

1234

þdx

1256

þdx

1278

þdx

1357

dx

1368

dx

1458

dx

1467

 dx

2358

 dx

2367

 dx

2457

þ dx

2468

þ dx

3456

þ dx

3478

þ dx

5678

½6

The subgroup of GL(8, R) preserving 

is the

holonomy group Spin(7). It also preserves the

orientation on R

and the Euclidean metric

= dx

þþdx

. It is a compact, semisimple,

21-dimensional Lie group, a subgroup of SO(8).

A Spin(7)-structure on an 8-manifold M gives rise

to a 4-form  and a metric g on M, such that each

tangent space of M admits an isomorphism with R

identifying  and g with 

and g

, respectively. By

an abuse of notation, the pair (, g) is referred to as

a Spin(7)-structure.

Proposition 9 Let M be an 8-manifold and (, g) a

Spin(7)-structure on M. Then the following are

equivalent:

(i) Hol(g)  Spin(7) and  is the induced 4-form;

(ii) r=0 on M, where r is the Levi-Civita

connection of g; and

(iii) d=0 on M.

We call r the torsion of the Spin(7)-structure

(, g), and ( , g) torsion free if r=0. A triple

(M,, g) is called a Spin(7)-manifold if M is an 8-

manifold and (, g) a torsion-free Spin(7)-structure

on M.Ifg has holonomy Hol(g)  Spin(7), then g is

Ricci-flat.

Here is a result on compact 8-manifolds with

holonomy Spin(7).

Theorem 10 Let (M, 

, g) be a compact Spin(7)-

manifold. Then, Hol(g) = Spin(7) if and only if M is

simply connected, and b

(M) þ b

(M) = b

(M) þ



(M) þ 25. In this case, the moduli space of

metrics with holonomy Spin(7) on M, up to

diffeomorphisms isotopic to the identity, is a smooth

manifold of dimension 1 þ b



(M).

The inclusions between the holonomy groups

SU(m), G

, Spin(7) are

SUð2Þ!SUð3Þ! G

###

SUð2ÞSUð2Þ!SUð4Þ!Spinð7 Þ

½7

The meaning of the above equation is illustrated

by using the inclusion SU(3) ,!G

.AsSU(3)acts

on C

,italsoactsonR  C

ﬃ R

,takingthe

SU(3)-actio n on R to be trivial. Thus, we embed

SU(3) as a subgroup of GL(7,R). It turns out

that SU(3) is contained in the subgroup G

GL(7, R) defined in the section ‘‘The holonomy

group G

.’’

Constructing Compact G

- and Spin(7)-Manifolds

The author’s method of constructing compact

7-manifolds with holonomy G

is based on the

444 Riemannian Holonomy Groups and Exceptional Holonomy

Kummer construction for Calabi–Yau metrics

on the K3 surface and may be divided into four

steps.

Step 1. Let T

be the 7-torus and (’

, g

) a flat

-structure on T

. Choose a finite group  of

isometries of T

preserving (’

, g

). Then the quotient

= is a singular, compact 7-manifold, an orbifold.

Step 2. For certain special groups , there is a

method to resolve the singularities of T

= in a natural

way, using complex geometry. We get a nonsingular,

compact 7-manifold M, together with a map  : M !

=, the resolving map.

Step 3. On M, we explicitly write down a one-

parameter family of G

-structures (’

, g

) depending

on t 2 (0, ). They are not torsion free, but have

small torsion when t is small. As t !0, the

-structure (’

, g

) converges to the singular

-structure 



(’

, g

Step 4. We prove using analysis that for suffi-

ciently small t, the G

-structure (’

, g

)onM, with

small torsion, can be deformed to a G

-structure

(

’

), with zero torsion. Finally, it is shown that

is a metric with holonomy G

on the compact

7-manifold M.

We explain the first two steps in greater detail.

For Step 1, an example of a suitable group  is given

here.

Example 11 Let (x

, ..., x

) be coordinates on

= R

, where x

2 R=Z. Let (’

, g

) be the

flat G

-structure on T

defined by [5]. Let , , and

 be the involutions of T

defined by

 : ðx

; ...; x

7!ðx

; x

; x

Þ½8

 : ðx

; ...; x

7!ðx

; x

; x

;

 x

; x

Þ½9

 : ðx

; ...; x

7!x

; x

; x

; x

;

x

; x

;

 x



½10

By inspection, , , and  preserve (’

, g

because of the careful choice of exactly which signs

to change. Also, 

= 

= 

= 1, and , , and 

commute. Thus, they generate a group

=h, , iﬃZ

of isometries of T

preserving

the flat G

-structure (’

, g

Having chosen a lattice  and finite group , the

quotient T

= is an orbifold, a singular manifold

with only quotient singularities. The singularities of

= come from the fixed points of nonidentity

elements of . We now describe the singularities in

the example.

Lemma 12 In Example 11, , , , and 

have no fixed points on T

. The fixed points of

, , are each 16 copies of T

. The singular set S of

= is a disjoint union of 12 copies of T

,4copies

from each of , , . Each component of S is a

singularity modeled on that of T

 C

={1}.

The most important consideration in choosing 

is that we should be able to resolve the singula-

rities of T

= within holonomy G

,inStep2.We

have no idea how to resolve general orbifold

singularities of G

-manifolds. However, after fifty

years of hard work we understand well how to

resolve orbifold singularities of C alabi–Yau mani-

folds, with holonomy SU( m). This is done by a

combination of algebraic geometry, which pro-

duces the underlying complex manifold by a

crepant resolution, and Calabi–Yau analysis,

which produces the Ricci-flat Ka¨hler metric on

this complex manifold.

Now the holonomy groups SU(2) and SU(3) are

subgroups of G

,asin[7]. Our tactic in Step 2 is to

ensure that all of the singular set S of T

= can

locally be resolved with holonomy SU(2) or SU(3),

and then use Calabi–Yau geometry to do this. In

particular, suppose each connected component of S

is isomorphic to either

1. T

 C

=G, for G a finite subgroup of SU(2); or

2. S

 C

=G, for G a finite subgroup of SU(3)

acting freely on C

n{0}.

One can use complex algebraic geometry to find a

crepant resolution X of C

=G or Y of C

=G. Then

 X or S

 Y gives a local model for how to

resolve the corresponding component of S in T

=.

Thus we construct a nonsingular, compact 7-mani-

fold M by using the patches T

 X or S

 Y to

repair the singularities of T

=. In the case of

Example 11, this means gluing 12 copies of T

 X

into T

=, where X is the blow-up of C

={1} at its

singular point.

By considering different groups  acting on T

and also b y finding topologically distinct resolu-

tions M

, ..., M

of the same orbifold T

=,we

can construct many compact Riemannian 7-mani-

folds with holonomy G

. A good number of

examples are given in Joyce (2000, chap ter 12).

Figure 1 displays the 252 different sets of Betti

numbers of compact, s imply con nected 7-mani-

folds with holonomy G

constructed there

together with 5 more sets from Kovalev. It

seems likely to the author that the Betti numbers

given in Figure 1 are o nly a small propor tion of

Riemannian Holonomy Groups and Exceptional Holonomy 445

the Betti numbers of all compact 7-manifolds with

holonomy G

A different construction of compact 7-manifolds

with holonomy G

was given by Kovalev (2003),

involving gluing together a symptotically cylindrical

Calabi–Yau 3-folds. Compact 8-manifolds with

holonomy Spin(7) were constructed by the author

using two different methods: first, by resolving

singularities of torus orbifolds T

= in a similar way

to the G

case (though the details are different and

more difficult), and second, by resolving Y=hi for Y

a Calabi–Yau 4-orbifold with singularities of a

special kind, and  an antiholomorphic isometric

involution of Y. Details can be found in Joyce (2000).

See also: Calibrated Geometry and Special Lagrangian

Submanifolds.

Further Reading

Bryant RL (1987) Metrics with exceptional holonomy. Annals of

Mathematics 126: 525–576.

Bryant RL and Salamon SM (1989) On the construction of some

complete metrics with exceptional holonomy. Duke Mathe-

matical Journal 58: 829–850.

Gross M, Huybrechts D, and Joyce D (2003) Calabi–Yau

Manifolds and Related Geometries, Universitext Series.

Berlin: Springer.

Joyce DD (2002) Constructing compact manifolds with excep-

tional holonomy, math.DG/0203158, 2002 (a survey paper).

Joyce DD (2004) Constructing compact manifolds with exceptional

holonomy. In: Douglas M, Gauntlett J, and Gross M (eds.)

Strings and Geometry, Clay Mathematics Proceedings 3.

pp. 177–191. Providence, RI: American Mathematical Society.

math.DG/0203158.

Kovalev AG (2003) Twisted connected sums and special

Riemannian holonomy. Journal fu¨ r die Reine und Angewandte

Mathematik 565: 125–160, math.DG/0012189.

0 20 40 60 80 100 120 140 160 180 200

(M )

Figure 1 Betti numbers (b

, b

) of compact G

-manifolds. (From Joyce (2000) and Kovalev (2003).)

446 Riemannian Holonomy Groups and Exceptional Holonomy

Saddle Point Problems

M Schechter, University of California at Irvine,

Irvine, CA, USA

Introduction

Many prob lems arising in science and engineering

call for the solving of the Euler equations of

functionals, that is, equations of the form

ðuÞ¼0 ½ 1

where G(u)isaC

-functional (usually representing

the energy) arising from the given data. As an

illustration, the equation

uðxÞ¼f ðx; uðxÞÞ

is the Euler equation of the functional

GðuÞ¼

kruk



Fðx; uðxÞÞdx

on an appropriate space, where

Fðx; tÞ¼

f ðx; sÞds ½2

and the norm is that of L

. The solving of the Euler

equations is tantamount to finding critical points of

the corresponding functional. The classical approach

was to look for maxima or minima. If one is looking

for a minimum, it is not sufficient to know that the

functional is bounded from below, as is easily

checked. However, one can show that there is a

sequence satisfying

Gðu

Þ!a; G

ðu

Þ!0 ½3

for a = inf G. If the sequence has a converg ent

subsequence, this will produce a minimum.

However, when extrema do not exist, there is no

clear way of obtaining critical points. In particular,

this happens when the functional is not bounded

from either above or below. Until recently, there

was no organized procedure for producing critical

points which are not extre ma. We shall describe an

approach which is very useful in such cases.

To illustrate the technique, we consider the

problem of finding a solution of

u

ðxÞþuðxÞ¼f ðx; uðxÞÞ ½4

x 2 I = [0, 2], under the conditions

uð0Þ¼uð2Þ; u

ð0Þ¼u

ð2Þ½5

We assume that the function f(x, t) is con tinuous in

I  R and is periodic in x with period 2. The

approach begins by asking the question, ‘‘does there

exist a differentiable function G from a space H to

R such that [4], [5] are equivalent to [1]?’’ It is

hoped that one can mimic the methods of calculus to

find critical points and thus solve [1].

Actually, we are asking the following: does there exist

a mapping G from a space H to R such that G has a

critical point u satisfying G

(u) = u

þ u  f (x, u(x))?

In order to solve the problem one has to

1. find G(u) such that

ðG

ðuÞ; vÞ

¼ðu; vÞ

ðf ð; uÞ; vÞ½6

holds for each u, v 2 H,

2. show that there is a function u(x) such that

(u) = 0,

3. show that u

exists in I,

4. show that [1] implies [4].

We used the notation

ðu; vÞ¼

2

uðxÞvðxÞdx

In order to carry out the procedure, we assume

that for each R > 0 there is a constant C

such that

jf ðx; tÞj  C

; x 2 I; t 2 R; jtjR ½7

This assumption is used to carry out step (1). We define

GðuÞ¼

kuk



2

Fðx; uðxÞÞdx ½8

where F(x, t) is given by [2] and we take H to be the

completion of C

(I) with respect to the norm

kuk

¼ðku

þkuk

1=2

½9

where kuk

= (u, u). We have

Theorem 1 If f(x, t) satisfies [7], then G(u) given

by [8] is continuously differe ntiable and satisfies [6].

Once we have reduced the problem to solving [1],

we can search for critical points. The easiest type to

locate are ‘‘saddle points’’ which are local minima in

some directions and local maxima in all others. For

instance, we obtain theorems such as

Theorem 2 Assume that

jf ðx ; tÞj  Cðjtjþ1Þ; x 2 I; t 2 R

2Fðx; tÞ=t

! ðxÞ a.e. as jtj!1

½10

with (x) satisfying

1 þ n

 ðxÞ1 þðn þ 1Þ

1 þ n

6 ðxÞ 6 1 þðn þ 1Þ

½11

and n an integer 0. If G(u) is give n by [8], then

there is a u

2 H such that

ðu

Þ¼0 ½12

In particular, u

is a solution of [4] and [5] in the

usual sense.

In proving this theorem, we shall make use of

Theorem 3 Let M, N be closed subspaces of a

Hilbert space E such that M = N

. Assume that at

least one of these subspaces is finite dimensional.

Let G be a continuously differentiable functional on

E satisfying

¼ sup

v2N

inf

w2M

Gðv þ wÞ 6¼1 ½13

and

¼ inf

w2M

sup

v2N

Gðv þ wÞ 6¼1 ½14

Then there is a sequence {u

}  E such that

Gðu

Þ!c; m

 c  m

; G

ðu

Þ!0 ½15

Theorem 3 allows us to obtain solutions if we can

find subspaces of H such that [13] and [14] hold. We

useittogivetheproofofTheorem 2.

Proof. Note that

kuk

ð1 þ k

Þj

; u 2 H ½16

where the 

are given by



¼ðu;



’

Þ; k ¼ 0; 1; 2; ... ½17

and

’

ðxÞ¼

ﬃﬃﬃﬃﬃﬃ

2

ikx

; k ¼ 0; 1; 2; ... ½18

Let

N ¼fu 2 H : 

¼ 0 for jkj > ng

Thus,

kuk

jkjn

ð1 þ k

Þj

ð1 þ n

Þkuk

; u 2 N ½19

Let

M ¼fu 2 H : 

¼ 0 for jkjng

In this case,

kuk

jkjnþ1

ð1 þ k

Þj

ð1 þðn þ 1Þ

Þkuk

; u 2 M ½ 20

Note that M, N are closed subspaces of H and that

M = N

. Note also that N is finite dimensional. If

we consider the functional [8], it is not difficult to

show that [11] implies

inf

G > 1; sup

G < 1½21

We are now in a position to apply Theorem 3. This

produces a saddle point satisfying [1].

Minimax

Theorem 3 is very useful when extrema do not exist, but

it is not always applicable. One is then forced to search

for other ways of obtaining critical points. Again, one is

faced with the fact that there is no systematic method of

finding them. A useful idea is to try to find sets that

separate the functional. By this we mean the following:

Definition 1 Two sets A, B separate the functional

G(u)if

:¼ sup

G  b

:¼ inf

G ½22

We would like to find sets A and B such that [22]

will imply

9u : GðuÞb

; G

ðuÞ¼0 ½23

This is too much to expect since even semibounded-

ness does not imply the existence of an extremum.

Consequently, we weaken our requirements and

look for sets A, B such that [22] implies

Gðu

Þ!a; G

ðu

Þ!0 ½24

with a  b

. This leads to

Definition 2 We shall say that the set A links the

set B if [22] implies [24] with a  b

for every C

functional G(u).

448 Saddle Point Problems

Of course, [24] is a far cry from [23], but if, for

example, the sequence [24] has a convergent

subsequence, then [24] implies [23]. Whether or

not [24] implies [23] is a property of the functional

G(u). We state this as

Definition 3 We say that G(u) satisfies the Palais–

Smale (PS) condition if [24] always implies [23].

The usual way of verifying this is to show that

every sequence satisfying [24] has a convergent

subsequence (there are other ways).

All of this leads to

Theorem 4 If G satisfies the PS condition and is

separated by a pair of linking sets, then it has a

critical point satisfying [23].

This theorem cannot be applied until one knows if

there are linking sets and functionals that satisfy the

PS condition. Fortunately, they exist. Examples and

sufficient conditions for A to link B are found in the

literature. Obviously, the weaker the conditions, the

more pa irs will qualify. To date, the conditions

described in the next section allow all known

examples.

The Details

Let E be a Banach space, and let  be the set of all

continuous maps =(t) from E  [0, 1] to E such

that

1. (0) = I, the identity map;

2. for each t 2 [0, 1), (t) is a homeomorphism of E

onto E and 

1

(t) 2 C(E  [0, 1), E);

3. (1)E is a single point in E and (t)A converges

uniformly to (1)E as t ! 1 for each bounded

set A  E; and

4. for each t

2 [0, 1) and each bounded set A  E,

sup

0tt

;u2A

fkðtÞukþk

1

ðtÞukg < 1½25

We have the following

Theorem 5 A sufficient condition for A to link B is

(i) A \ B =  and

(ii) for each  2  there is a t 2 (0, 1] such that

ðtÞA \ B 6¼ 

Theorem 6 Let G be a C

-functional on E, and let

A, B be subsets of E such that A, B satisfy [22] and

the hypotheses of Theorem 5. Assume that

a :¼ inf

2

sup

0s1;u2A

GððsÞuÞ½26

is finite. Let (t) be a positive, locally Lipschitz

continuous function on [0, 1) such that

ðrÞdr ¼1 ½27

Then there is a sequence {u

}  E such that

Gðu

Þ!a; G

ðu

Þ= ðku

kÞ ! 0 ½28

If a = b

, then we can also require that

dðu

; BÞ!0 ½29

Corollary 1 Under the hypotheses of Theorem 6

there is a sequence {u

}  E such that

Gðu

Þ!a; ð1 þku

kÞG

ðu

Þ!0 ½30

Proof. We merely take (u) = 1=(1 þkuk)in

Theorem 6.

A useful criterion for finding linking subsets is

Theorem 7 Let F be a continuous map from a

Banach space E to R

, and let Q  E be such that

= Fj

is a homeomorphism of Q onto the closure

of a bounded open subset  of R

. If p 2 , then

1

(@) links F

1

(p).

Some Examples

The following are examples of sets that link.

Example 1 Let M, N be closed subspaces such that

E = M  N (with one finite dimensional). Let

¼fu 2 E : kuk < Rg

and take A = @B

\ N, B = M. Then A links B.

To see this, we identify N with some R

and take

=B

\ N, Q =



. For u 2 E, we write

u ¼ v þ w; v 2 N; w 2 M ½31

and take F to be the projection

Fu ¼ v

Since Fj

= I and M = F

1

(0), we see from Theorem 7

that A links B.

Example 2 We take M, N as in Example 1.Let

6¼ 0 be an element of M, and take

A ¼fv 2 N : kv kRg

[fsw

þ v : v 2 N; s  0; ksw

þ vk¼Rg

B ¼@B



\ M; 0 <<R:

Then A links B. Again we ident ify N with some R

and we may assume kw

k= 1. Let

Q ¼fsw

þ v : v 2 N; s  0; ksw

þ vkRg

Saddle Point Problems 449

Then A = @Q in R

nþ1

.Ifu is given by [31],we

define

Fu ¼ v þkwkw

Then Fj

= I and B = F

1

(w

). We can now apply

Theorem 7 to conclude that A links B.

Example 3 Take M, N as before and let v

6¼ 0be

an element of N. We write N = {v

}  N

. We take

A ¼fv

2 N

: kv

kRg

[fsv

þ v

: v

2 N

; s  0; ksv

þ v

k¼Rg

B ¼fw 2 M : kwkg

[fsv

þ w : w 2 M; s  0; ksv

þ wk¼g

where 0 <<R.ThenA links B.Toseethis,welet

Q ¼fsv

þ v

: v

2 N

; s  0; ksv

þ v

kRg

and reason as before. For simplicity, we assume that

k= 1, E is a Hilber t space and that the splitting

E = N

 {v

}  M is orthogonal. If

u ¼ v

þ w þ sv

; v

2 N

; w 2 M; s 2 R ½32

we define

FðuÞ¼v

þ s þ  

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ



kwk



; kwk

¼ v

þðs þ Þv

; kwk >

Note that Fj

= I while F

1

(v

) is precisely the set

B. Hence we can conclude via Theorem 7 that A

links B.

Example 4 This is the same as Example 3 with A

replaced by A = @B

\ N. The proof is the same

with Q replaced by Q =



\ N.

Example 5 Let M, N be as in Example 1. Take

A = @B



\ N, and let v

be any element in @B

\ N.

Take B to be the set of all u of the form

u ¼ w þ sv

; w 2 M

satisfying any of the following:

(i) kwkR, s = 0,

(ii) kwkR, s = 2R

, and

(iii) kwk= R,0 s  2R

where 0 <<min (R, R

). Then A links B. To see

this, take N = {v

}  N

. Then any u 2 E can be

written in the form [32]. Define

FðuÞ¼v

þ R

 max

kwk; js  R



and Q =





\ N. Again we may ident ify N with

some R

. Then F 2 C(E, N) and Fj

= I. Moreover,

A = F

1

(0). Hence, A links B by Theorem 7.

Example 6 Let M, N be as in Example 1. Let v

be in @B

\ N and write N = {v

}  N

. Let

A = @B



\ N, Q =





\ N, and

B ¼fw 2 M : kwkRg

[fw þ sv

: w 2 M; s  0; kw þ sv

k¼Rg

where 0 <<R. Then A links B. To see this, write

u = w þ v

þ sv

, w 2 M, v

2 N

, s 2 R and take

FðuÞ¼ðcR  maxfckw þ sv

k; jcR  sjgÞv

þ v

where c = =(R  ). Then F is the identity operator

on Q, and F

1

(0) = B. Apply Theorem 7.

Some Applications

Many elliptic semilinear problems can be described

in the following way. Let  be a domain in R

, and

let A be a self-adjoint operator on L

(). We assume

that A  

> 0 and that

ðÞD :¼ DðA

1=2

ÞH

m; 2

ðÞ½33

for some m > 0, wher e C

() denotes the set of test

functions in  (i.e., infinitely differentiable functions

with compact supports in ), and H

m,2

() denotes

the Sobolev space. If m is an integer, the norm in

m,2

() is given by

kuk

m; 2

:¼

jjm



1=2

½34

Here D



represents the generic derivative of order

jj and the norm on the right-hand side of [34] is

that of L

(). We shall not assume that m is an

integer.

Let q be any number satisfying

2  q  2n=ðn  2mÞ; 2m < n

2  q < 1; n  2m

and let f (x, t) be a continuous function on   R.

We make the following assumptions.

Assumption A The function f (x, t) satisfies

jf ðx; tÞj  V

ðxÞ

jtj

q1

þ V

ðxÞW

ðxÞ½35

and

f ðx; tÞ=V

ðxÞ

¼ oðjtj

q1

Þ as jtj!1 ½36

where V

(x) > 0 is a function in L

() such that

 Ckuk

; u 2 D ½37

450 Saddle Point Problems

and W

is a function in L

(). Here

kuk

:¼



juðxÞj



1=q

½38

kuk

:¼kA

1=2

uk½39

and q

= q=(q  1). With the norm [39], D becomes

a Hilbert space. Define G and F by [8] and [2].It

follows that G is a continuously differentiable

functional on the whole of D.

We assume further that

Hðx; tÞ¼2Fðx; tÞtf ðx; tÞ

W

ðxÞ2L

ðÞ; x 2 ; t 2 R ½40

and

Hðx; tÞ!1 a:e: as jtj!1 ½41

Moreover, we assume that there are functions

V(x), W(x) 2 L

() such that multiplication by

V(x) is a compact operator from D to L

() and

Fðx; tÞCðVðxÞ

jtj

þ VðxÞWðxÞjtjÞ ½42

We wish to obtain a solution of

Au ¼ f ðx; uÞ; u 2 D ½43

By a solution of [43] we shall mean a function u 2 D

such that

ðu; vÞ

¼ðf ð; uÞ; vÞ; v 2 D ½44

If f (x, u)isinL

(), then a solution of [44] is in D(A)

and solves [43] in the classical sense. Otherwise we call

it a weak or semistrong solution. We have

Theorem 8 Let A be a self-adjoint operator in

() such that A  

> 0 and [33] holds for some

m > 0. Assume that 

is an eigenvalue of A with

eigenfunction ’

. Assume also

2Fðx; tÞ

; jtj for some >0 ½45

and

2Fðx; tÞ

 W

ðxÞ; t > 0; x 2  ½46

where W

2 L

(). Assume that f ( x, t) satisfies [35],

[36], [40], [41],and[42]. Then [43] has a solution

u 6¼ 0.

Proof. Under the hypotheses of the theorem, it

is know n that the following alternative holds: either

(i) there is an infinite number of y(x) 2 D(A)n{0}

such that

Ay ¼ f ðx; yÞ¼

y ½47

(ii) for each >0 sufficiently small, there is an ">0

such that

GðuÞ"; kuk

¼  ½48

We may assume that option (ii) holds, for otherwise

we are done. By [46] we have

GðR’

ÞR

ðk’

 

k’

Þþ



ðxÞdx



ðxÞdx

By Theorem 6, there is a sequence satisfying [28].

Taking (r) = 1=(r þ 1), we conclude that there is a

sequence {u

}  D such that

Gðu

Þ!c; m

 c  m

;

ð1 þku

ÞG

ðu

Þ!0 ½49

In particular, we have

 2



Fðx; u

Þdx ! c ½50

and

ðf ð; x

Þ; u

Þ!0 ½51

Consequently,



Hðx; u

Þdx !c ½52

These imply



Hðx; u

Þdx  K ½53

If 

= ku

!1, let

= u

=

. Then k

= 1.

Consequently, there is a renamed subsequence such

that

u weakly in D, strongly in L

(), and a.e.

in . We have from [42]

1 ðm

þ Þ=

þ 2C



fVðxÞ

þ VðxÞWðxÞj

j

1

gdx

Consequently,

1  2C



VðxÞ

dx ½54

This shows that

u 6 0. Let 

be the subset of  on

which

u 6¼ 0. Then

ðxÞj ¼ 

ðxÞj ! 1; x 2 

½55

Saddle Point Problems 451