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

Подождите немного. Документ загружается.

This preliminary analysis of the singular set can

be refined by doing a so-called tangent cone

analysis.

Tangent Cone Analysis

It is not hard to see that scaling preserves the space

of minimal submanifolds of R

. Namely, if  is

minimal, then so is



y;

¼fy þ 

1

ðx  yÞjx 2  g½56

(To see this, simply note that this scaling multi-

plies the principal curvatures by .) Suppose now

that we fix the point y and take a sequence 

!0.

The monotonicity formula bounds the density of

the rescaled solution, allowing us to extract a

convergent subsequence and limit. This limit,

which is called a ‘‘tangent cone’’ at y,achieves

equality in the monotonicity formula and, hence,

must be homogeneous (i.e., invariant under dila-

tions about y).

The usefulness of tangent cone analysis in

regularity theory is based on two key facts. For

simplicity, we illustrate these when   R

is an

area-minimizing hypersurface. First, if any tangent

cone at y is a hyperplane R

n1

, then  is smooth in a

neighborhood of y. This follows easily from the

Allard regularity theorem since the densi ty at y of

the tangent cone is the same as the density at y of .

The second key fact, known as ‘‘dimension redu-

cing,’’ is due to Almgren and is a refinement of an

argument of Federer. To state this, we first stratify

the singular set S of  into subsets

S

S

n2

½57

where we define S

to be the set of points y 2Sso

that any linear space contained in any tangent cone

at y has dimensi on at most i. (Note that S

n1

= ; by

Allard’s theorem.) The dimension reducing argu-

ment then gives that

dimðS

Þi ½58

where dimension means the Hausdorff dimension.

In particular, the solution of the Bernstein problem

then gives codimension-7 regularity of , that is,

dim (S)  n  8.

Part 2. Constructing Minimal Surfaces

Thus far, we have mainly dealt with regularity and

a priori estimates but have ignored questions of

existence. In this part, we survey some of the most

useful existence resul ts for minimal surfaces. The

following section gives an overview of the classical

Plateau problem. Next, we recall the classical

Weierstrass representation, including a few modern

applications, and the Kapouleas desingularization

method. Then we deal with producing area-mini-

mizing surfa ces and questions of embeddedness.

Finally, we recall the min–max construction for

producing unstable minimal surfaces and, in parti-

cular, doing so while controlling the topology and

guaranteeing embeddedness.

The Plateau Problem

The following fundamental existence problem for

minimal surfaces is known as the Plateau problem:

given a closed curve , find a minimal surface with

boundary . There are various solutions to this

problem depending on the exact definition of a

surface (parametrized disk, integral current, Z

current, or rectifiable varifold). We shall consider

the version of the Plateau problem for parametrized

disks; this was solved independently by J Douglas

and T Rado. The generalization to Riemannian

manifolds is due to C B Morrey.

Theorem 13 Let   R

be a piecewise C

closed

Jordan curve. Then there exists a piecewise C

map

u from D  R

to R

with u(@D)   such that the

image minimizes area among all disks with bound-

ary .

The solution u to the Plateau problem above can

easily be seen to be a branched conformal immer-

sion. R Osserman proved that u does not have true

interior branch points; subsequently, R Gulliver and

W Alt showed that u cannot have false branch

points either.

Furthermore, the solution u is as smooth as the

boundary curve, even up to the bounda ry. A very

general version of this boundary regularity was

proved by S Hildebrandt; for the case of surfaces

in R

, recall the following result of J C C Nitsche:

Theorem 14 If  is a regular Jordan curve of class

k, 

where k  1 and 0 <<1, then a solution u

of the Plateau problem is C

k, 

on all of



The Weierstrass Representation

The classical Weierstrass representation (see Osserman

(1986)) takes holomorphic data (a Riemann surface, a

meromorphic function, and a holomorphic 1-form)

and associates a minimal surface in R

. To be precise,

given a Riemann surface , a meromorphic function g

on , and a holomorphic 1-form  on ,thenwe

Minimal Submanifolds 427

get a (branched) conformal minimal immersion

F :  !R

FðzÞ¼Re

2

ðg

1

ðÞgðÞÞ;



ðg

1

ðÞþgðÞÞ; 1



ðÞ½59

Here, z

2  is a fixed base point and the integra-

tion is along a path 

, z

from z

to z. The choice of

changes F by adding a constant. In general, the

map F may depend on the choice of path (and hence

may not be well defined); this is known as ‘‘the

period problem.’’ However, when g has no zeros or

poles and  is simply connected, then F(z) does not

depend on the choice of path 

, z

Two standard constructions of minimal surfaces

from Weierstrass data are

gðzÞ¼z;ðzÞ¼dz=z;  ¼ Cnf0g

giving a catenoid ½60

gðzÞ¼e

;ðzÞ¼dz;  ¼ C giving a helicoid ½61

The Weierstrass representation is particularly

useful for constructing immersed minimal surfaces.

Typically, it is rather difficult to prove that the

resulting immersion is an embedding (i.e., is 1–1),

although there are some interesting cases where this

can be done. For the first modern example,

D Hoffman and Meeks proved that the surface

constructed by Costa was embedded; this was

the first new complete finite topology properly

embedded minimal surface discovered since the

classical catenoid, helicoid, and plane. This led

to the discovery of many more such surfaces

(see Rosenberg (1992) for more discussion).

Area-Minimizing Surfaces

Perhaps the most natural way to construct minimal

surfaces is to look for ones which minimize area, for

example, with fixed bounda ry, or in a homotopy

class, etc. This has the advantage that often it is

possible to show that the resulting surface is

embedded. We mention a few results along these

lines.

The first embeddedness result, due to Meeks and

Yau, shows that if the boundary curve is embedded

and lies on the boundary of a smooth mean convex

set (and it is null-homotopic in this set), then it

bounds an embedded least area disk.

Theorem 15 (Meeks III and Yau 1982). Let M

a compact Riemannian 3-manifold whose boundary

is mean convex and let  be a simple closed curve in

@M which is null -homotopic in M; then  is

bounded by a least area disk and any such least

area disk is properly embedded.

Note that some restriction on the boundary curve

 is certainly necessary. For instance, if the

boundary curve was knotted (e.g., the trefoil), then

it cou ld not be spanned by any embedded disk

(minimal or otherwise). Prior to the work of Meeks

and Yau, embeddedness was known for extremal

boundary curves in R

with small total curvature by

the work of R Gulliver and J Spruck.

If we instead fix a homotopy class of maps, then

the two fundamental existence results are due to

Sacks–Uhlenbeck and Schoen–Yau (with embed-

dedness proved by Meeks–Yau and Freedman–

Hass–Scott, respectively):

Theorem 16 Given M

, there exist conformal

(stable) minimal immersions u

, ..., u

: S

which generate 

(M) as a Z[

(M)] module.

Furthermore,

(i) if u : S

!Mand[u]



6¼ 0, then Area(u) 

min

Area(u

(ii) each u

is either an embedding or a 2–1 map

onto an embedded two-sided RP

Theorem 17 If 

is a closed surface with genus

g > 0 and i

:  !M

is an embedding which

induces an injective map on 

, then there is a

least area embedding with the same action on 

The Min–Max Construction

of Minimal Surfaces

Variational arguments can also be used to construct

higher index (i.e., nonminimizing) minimal surfaces

using the topology of the space of surfaces. There

are two basic approaches:

1. Applying Morse theory to the energy functional

on the space of maps from a fixed surface  to M.

2. Doing a min–max argument over families of

(topologically nontrivial) sweep-outs of M.

The first approach has the advantage that the

topological type of the minimal surface is easily

fixed; however, the second approach has been more

successful at producing embedded minimal surfaces.

We will highlight a few key results below but refer

to Colding and De Lellis (2003) for a thorough

treatment.

Unfortunately, one cannot directly apply Morse

theory to the energy functional on the space of maps

from a fixed surface because of a lack of compact-

ness (the Palais–Smale condition C does not hold).

428 Minimal Submanifolds

To get around this difficulty, Sacks–Uhlenbeck

introduce a family of perturbed energy functionals

which do satisfy condition C and then obtain

minimal surfaces as limits of critical points for the

perturbed problems:

Theorem 18 If 

(M) 6¼ 0 for some k > 1, then

there exists a branched immer sed minimal 2-sphere

in M (for any metric).

The basic idea of constructing minimal surfaces

via min–max arguments and swe ep-outs goes back

to Birkhoff, who developed it to construct simple

closed geodesics on spheres. In particular, when M is

a topological 2-sphere, we can find a one-parameter

family of curves starting and ending at point curves

so that the induced map F : S

(see Figure 1)

has nonzero degree. The min–max argument pro-

duces a nontrivial closed geodesic of length less than

or equal to the longest curve in the initial one-

parameter family. A curve-shortening argument

gives that the geodesic obtained in this way is

simple.

J Pitts applied a similar argument and geometric

measure theory to get that every closed Riemannian

3-manifold has an embedded minimal surface (his

argument was for dimensions up to seven), but he

did not estimate the genus of the resulting surface.

Finally, F Smith (under the direction of L Simon)

proved (see Colding and De Lellis (2003)):

Theorem 19 Every metric on a topological

3-sphere M admits an embedded minimal 2-sphere.

The main new contribution of Smith was to

control the topological type of the resulting minimal

surface while keeping it embedded.

Part 3. Some Applications of Minimal

Surfaces

In this part, we discuss very briefly a few applica-

tions of minimal surfaces. As mentioned in the

introduction, there are many to choose from and we

have selected just a few.

The Positive-Mass Theorem

The (Riemannian version of the) positive-mass

theorem states that an asymptotically flat

3-manifold M with non-negative scalar curvature

must have positive mass. The Riemannian manifold

M here arises as a maximal spacelike slice in a

(3 þ 1)-dimensional spacetime solution of Einstein’s

equations.

The asymptotic flatness of M arises because the

spacetime models an isolated gravitational system

and hence is a perturbation of the vacuum solution

outside a large compact set. To make this precise,

suppose for simplicity that M has only one end; M

is then said to be asymptoticall y fl at if there is a

compact set   M so that Mn is diffeomorphic

to R

(0) and t he metric on Mn can be

written as

¼ 1 þ

2jxj





þ p

½62

where

jxj

jþjxj

jDp

jþjxj

jC ½63

The consta nt M is the so-called mass of M. Observe

that the met ric g

is a perturbation of the metric on

a constant-time slice in the Schwarzschild spacetime

of mass M; that is to say, the Schwarzschild metric

has p

 0.

A tensor h is said to be O(jxj

p

)ifjxj

jhjþ

jxj

pþ1

jDhjC. For example, an easy calculation

shows that

¼ 1 þ 2 M=j xjðÞ

þ Oðjxj

2

ﬃﬃﬃ



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

det g

¼ 1 þ 3Mjxj

1

þ Oðjxj

2

½64

The positive-mass theorem states that the mass M

of such an M must be non-negative:

Theorem 20 (Schoen an d Yau 1979). With M as

above, M0.

There is a rigidity theorem as well which states that

the mass vanishes only when M is isometric to R

Theorem 21 (Schoen and Yau 1979 ). If jr

O(jxj

5

) and M= 0 in Theorem 20, then M is

isometric to R

We will give a very brief overview of the proof of

Theorem 20, showing in the process where minimal

surfaces appear.

Proof (Sketch). The argument will be by contra-

diction, so suppose that the mass is negative. It is

not hard to prove that the slab between two parallel

Figure 1 A one-parameter family of curves on a 2-sphere

which induces a map F : S

of degree 1. First published in

Surveys in Differential Geometry, volume IX, in 2004, published

by International Press.

Minimal Submanifolds 429

planes is mean convex. That is, we have the

following:

Lemma 3 If M < 0 and M is asymptotically flat,

then there exist R

, h > 0 so that for r > R

the sets

¼fjxj

 r

; h  x

 hg½65

have strictly mean-convex boundary.

Since the compact set C

is mean convex, we can

solve the Plateau problem to get an area-minimizing

(and hence stable) surface 

 C

with boundary

@

¼fjxj

¼ r

; x

¼ hg½66

Using the disk {jxj

 r

, x

= h} as a comparison

surface, we get uniform local area bounds for any

such 

. Combining these local area bounds with the

a priori curvature estimates for minimizing surfaces,

we can take a sequence of r’s going to infini ty and

find a subsequence of 

’s that converge to a

complete area-minimizing surface

 fh  x

 hg½67

Since  is pinched betw een the planes {x

= h}, the

estimates for minimizing surfaces implies that (out-

side a large compac t set)  is a graph over the plane

= 0} and hen ce has quadratic area growth and

finite total curvature. Moreover, using the form of

the metric g

, we see that jruj decays like jxj

1

and



¼ð2s þ Oð1ÞÞðs

1

þ Oðs

2

ÞÞ

¼ 2 þ Oðs

1

Þ½68

where 

= {x

þ x

= s

} \  and k

is the geodesic

curvature of 

(as a curve in ).

To get the contradiction, one combines stability of

 with the positive scalar curvature of M to see that

no such  could have existed. (M was assumed only

to have non-negative scalar curvature. However, a

‘‘rounding off’’ argument shows that the metric on

M can be perturbed to have positive scalar curvature

outside of a compact set and still have negative

mass.) Namely, substituting the Gauss equation into

the stability inequality (this is the stability inequality

in a general 3-manifold; see Colding and Minicozzi II

(1999)) gives



ðjAj

=2 þ Scal

 K



Þ





jrj

½69

Since  has quadratic area growth, we can choose a

sequence of (logarithmic) cutoff functions in [69] to

get

0 <



ðjAj

=2 þ Scal

Þ



< 1½70

since K



may not be positive, we also used that 

has finite total curvature. Moreover, we used that

Scal

is positive outside a compact set to see that

the first integral in [70] was positive. Finally,

substituting [70] into the Gauss–Bonnet formula

gives that



is strictly less than 2 for s large,

contradicting [68].

Black holes

Another way that minimal surfaces enter into

relativity is through black holes. Suppose that we

have a three-dimensional time slice M in a (3 þ 1)-

dimensional spacetime. For simplicity, assume that M

is totally geodesic and hence has non-negative scalar

curvature. A closed surface  in M is said to be

trapped if its mean curvature is everywhere negative

with respect to its outward normal. Physically, this

means that the surface emits an outward shell of light

whose surface area is decreasing everywhere on the

surface. The existence of a closed trapped surface

implies the existence of a black hole in the spacetime.

Given a trapped surface, we can look for the

outermost trapped surface containing it; this outer-

most surface is called an apparent horizon. It is not

hard to see that an apparent horizon must be a

minimal surface and, moreover, a barrier argument

shows that it must be stable. Since M has non-

negative scalar curvature, stability in turn implies

that it must be diffeomorphic to a sphere. See, for

instance, Bray (2002) for references to some results

on black holes, horizons, etc.

Constant Mean Curvature Surfaces

At least since the time of Plateau, minimal surfaces

have been used to model soap films. This is because

the mean curvature of the surface models the surface

tension and this is essentially the only force acting

on a soap film. Soap bubbles, on other hand, enclose

a volume and thus the pressure gives a second

counterbalancing force. It follows easily that these

two forces are in equilibrium when the surface has

constant mean curvature (cmc).

For the same reason, cmc surfaces arise in the

isoperimetric problem. Namely, a surface that mini-

mizes surface area while enclosing a fixed volume must

have cmc. It is not hard to see that such an

isoperimetric surface in R

must be a round sphere.

There are two interesting partial converses to this.

First, by a theorem of Hopf, any cmc 2-sphere in R

must be round. Second, using the maximum principle

(‘‘the method of moving planes’’), Alexandrov showed

that any closed embedded cmc hypersurface in R

must be a round sphere. It turned out, however, that

not every closed immersed cmc surface is round. The

430 Minimal Submanifolds

first examples were immersed cmc tori constructed by

H Wente. Kapouleas constructed many new examples,

including closed higher-genus cmc surfaces.

Many of the techniques developed for studying

minimal surfaces generalize to general cmc surfaces.

Finite Extinction for Ricci Flow

We close this article by indicating how minimal

surfaces can be used to show th at on a hom otopy

3-sphere the Ricci flow becomes extinct in finite

time (see Colding and Minicozzi II (2005) and

Perelman (2003) for details).

Let M

be a smooth closed orientable 3-manifold

and let g(t) be a one-parameter family of metrics on

M evolving by the Ricci flow, so

g ¼2Ric

½71

In an earlier section, we saw that there is a natural

way of constructing minimal surfaces on many

3-manifolds and that comes from the min–max

argument where the minimal of all maximal slices of

sweep-outs is a minimal surface. The idea is then to

look at how the area of this min–max surface changes

under the flow. Geometrically, the area measures a

kind of width of the 3-manifold and as we will see for

certain 3-manifolds (those, like the 3-sphere, whose

prime decomposition contains no aspherical factors),

the area becomes zero in finite time corresponding to

the solution becoming extinct in finite time.

Fix a con tinuous map  : [0, 1] !C

\ L

, M)

where (0) and (1) are constant maps so that  is

in the nontrivial homotopy class [] (such  exists

when M is a homotopy 3-sphere). We define the

width W = W(g,[]) by

WðgÞ¼min

2½

max

s2½0;1

EnergyððsÞÞ ½72

The next theorem gives an upper bound for the

derivative of W(g(t)) under the Ricci flow which forces

the solution g(t) to become extinct in finite time.

Theorem 22 Let M

be a homotopy 3-sphere

equipped with a Riemannian metric g = g(0).

Under the Ricci flow, the width W(g(t)) satisfies

WðgðtÞÞ  4 þ

4ðt þ CÞ

WðgðtÞÞ ½73

in the sense of the limsup of forward difference

quotients. Hence, g(t) must become extinct in finite

time.

The 4 in [73] comes from the Gauss–Bonnet

theorem and the 3/4 comes from the bound on the

minimum of the scalar curvature that the evolution

equation implies. Both of these constants matter

whereas the constant C depends on the initial metric

and the actual value is not important.

To see that [73] implies finite extinction time,

rewrite [73] as

WðgðtÞÞðt þ CÞ

3=4



4ðt þ CÞ

3=4

½74

and integrate to get

ðT þ CÞ

3=4

WðgðTÞÞ C

3=4

Wðgð0ÞÞ

 16 ðT þ CÞ

1=4

 C

1=4

½75

Since W  0 by definition and the right-hand side of

[75] would become negative for T sufficiently large,

we get the claim.

As a corollary of this theorem we get finite

extinction time for the Ricci flow.

Corollary 3 Let M

be a homotopy 3-sphere

equipped with a Riemannian metric g = g(0). Under

the Ricci flow g(t) must become extinct in finite time.

Acknowledgments

The authors were partially supported by NSF Grants

DMS 0104453 and DMS 0405695.

See also: Black Hole Mechanics; Calibrated Geometry

and Special Lagrangian Submanifolds; Geometric Analysis

and General Relativity; Geometric Measure Theory;

Leray–Schauder Theory and Mapping Degree; Ljusternik–

Schnirelman Theory; Singularities of the Ricci Flow.

Further Reading

Bray H (2002) Black holes, geometric flows, and the Penrose

inequality in general relativity. Notices of the American

Mathematical Society 49(11): 1372–1381.

Colding TH and De Lellis C (2003) The Min–Max Construction

of Minimal Surfaces, Surveys in Differential Geometry, vol. 8,

Lectures on Geometry and Topology held in honor of Calabi,

Lawson, Siu, and Uhlenbeck at Harvard University, May 3–5,

The min–max surface

Figure 2 The sweep-out, the min–max surface, and the width

W. First published in the Journal of the American Mathematical

Society in 2005, published by the American Mathematical Society.

Minimal Submanifolds 431

2002, Sponsored by the Journal of Differential Geometry,

pp. 75–107 (math.AP/0303305).

Colding TH and Minicozzi WP II (1999) Minimal Surfaces.

Courant Lecture Notes in Math., vol. 4.

Colding TH and Minicozzi WP II (2003) Disks that are double

spiral staircases. Notices of the American Mathematical

Society 50(3): 327–339.

Colding TH and Minicozzi WP II (2005) Estimates for the

extinction time for the Ricci flow on certain 3-manifolds and a

question of Perelman. Journal of the American Mathematical

Society 18(3): 561–569 (math.AP/0308090).

Colding TH and Minicozzi WP, II (2005) Minimal submanifolds.

Preprint.

Lawson HB (1980) Lectures on Minimal Submanifolds, vol. I.

Berkeley: Publish or Perish.

Meeks W III and Perez J (2004) Conformal properties in classical

minimal surface theory. In: Grigor’yan A and Yau ST (eds.)

Eigenvalues of Laplacians and Other Geometric Operators,

Surveys in Differential Geometry IX, pp. 275–335. Somerville,

MA: International Press.

Meeks WH and Yau ST (1980) Topology of three-dimensional

manifolds and the embedding problems in minimal surface

theory. Annals of Mathematics 112(3): 441–484.

Meeks W III and Yau ST (1982) The classical Plateau problem

and the topology of three dimensional manifolds. Topology

21: 409–442.

Morgan F (1995) Geometric Measure Theory. A Beginner’s

Guide, 2nd edn. San Diego, CA: Academic Press.

Osserman R (1986) A Survey of Minimal Surfaces, 2nd edn.

New York: Dover.

Perelman G Finite extinction time for the solutions to the Ricci

flow on certain three-manifolds, math.DG/0307245.

Perez J (2005) Limits by rescalings of minimal surfaces: minimal

laminations, curvature decay and local pictures. In: Moduli

Spaces of Properly Embedded Minimal Surfaces, Notes for the

Workshop. Palo Alto, CA: American Institute of Mathematics.

(http://www.ugr.es/~jperez/papers/notes.pdf)

Rosenberg H (1992) Some recent developments in the theory of

properly embedded minimal surfaces in R

, Seminare Bour-

baki 1991/92, Asterisque No. 206, pp. 463–535.

Schoen R and Yau ST (1979) On the proof of the positive mass

conjecture in general relativity. Communications in Mathemat-

ical Physics 65(1): 45–76.

Minimax Principle in the Calculus of Variations

A Abbondandolo, Universita

di Pisa, Pisa, Italy

Introduction

When studying a functional f on an infinite-

dimensional function space X, one is ofte n interested

in finding critical points which are not local minima.

A simple yet powerful method to detect those

critical points is the minimax method. The idea

consists in detecting some complexity in the topol-

ogy of X, or in the structure of the sublevels of f,to

find a class  of subsets of X which some how

reveals such a topological complexity, and to show

that the number

c :¼ inf

2

sup

x2

f ðxÞ

is finite (even if the functional may be unbounded

above and below). If the class  is positively

invariant under the action of the negative-gradient

flow of f, and if a suitable compactness assum ption

known as the Pa lais–Smale condition holds, c is

proved to be a critical value of f. Quite remarkably,

the minimax method also works when no topologi-

cal complexity is presen t, but the negative-gradient

flow of f exhibits some kind of rigidity.

In this article we shall describe these ideas,

starting from the simplest minimax result, the

‘‘mountain-pass theorem.’’ We will show how to

apply the minimax method by discussing the

existence question of solutions of a nonlinear elli ptic

boundary value problem, of closed geodesics on

compact manifolds, and of closed characteristics on

compact energy hypersurfaces.

The Mountain-Pass Theorem

Let us start by considering the following familiar

fact. Let f : R

! R be a smooth coercive function

(i.e., its sublevels have compact closure). If a sublevel

{f < a} is not connected – say {f < a} = A [ B,with

A, B disjoint open sets – then f has a critical point x at

level

f ðxÞ¼c :¼ inf

2

max

u2

f ðuÞa

where  is the class of all continuous curves in R

with one end point in A and the other in B. More

figuratively: i f t here are two valleys, then there

must be a mountain pass. Let us examine a possible

proof.

First notice that any curve in the class  will have

to cross the level {f = a}, so c  a. If by contradiction

c is not a critical value of f, by the compactness of the

sublevels there is some >0suchthatjrfj on

{c    f  c þ }. Then the negative-gradient flow

of f, that is, the solution of

ðt; uÞ¼rf ððt; uÞÞ;ð 0; uÞ¼u

432 Minimax Principle in the Calculus of Variations

pulls the sublevel {f  c þ } down into the sublevel

{f  c  } in finite time 2=. Indeed, if ([0, t], u) 

{c    f  c þ }, then the inequalities

2  f ðuÞf ððt; uÞÞ

¼

f ððs ; uÞÞds

jrf ððs ; uÞÞj

ds  

imply that t  2=. By definition of c, we can find a

continuous curve  2  which is contained in {f 

c þ }. But then the curve 

:= (2=, ) still has one

end point in A, the other one in B, and lies in {f 

c  }, contradicting the definition of c.

If we try to generalize this result to functions

defined on an infinite-dimensional real Hilbert space

H, we encounter difficulties due to lack of compact-

ness. Indeed, a continuous function on an infinite-

dimensional Hilbert space can never have compact

sublevels (with respect to the norm topology). If we

look back at the proof, we see that we have used

coercivity to guarantee that if the level set {f = c}

contains no critical points, then rf is bounded away

from zero on the strip {c    f  c þ }, for some

small >0. A natural idea is then to replace the

coercivity assumption by a condition implying the

latter fact.

Definition Let f : H ! R be a continuously differ-

entiable function on a real Hilbert space H.

A sequence (u

)  H is said a Palais–Smale sequence

if f (u

) is bounded and Df (u

) tends to zero. The

function f is said to satisfy the Palais–Smale

condition if every Palais–Smale sequence has a

converging subsequence.

The Palais–Smale condition readily implies the

statement above. Assuming also that f is twice

continuously differentiable, the negative-gradient

flow of f (a well-defined local flow because rf is

continuously differentiable) pulls the sublevel {f 

c þ } down into {f  c  } in finite time. These

observations lead to the following:

Theorem (Mountain pass). Let f be a twice con-

tinuously differentiable function on a real Hilbert

space H, satisfying the Palais–Smale condition.

Assume that a sublevel {f < a} is not connected,

and let A, B be two disjoint open sets such that

A [B = {f < a}. Then f has a critical point x at level

f ðxÞ¼c :¼ inf

2

max

u2

f ðuÞa

where  is the class of all continuous curves in H

with one end point in A and the other one in B.

If we are even more ambitious, and we wish to

consider functions defined on a real Banach space E,

we also encounter the problem of not having a

gradient vector field. Indeed, the differential of f at x,

Df (x), is an element of the dual space E



, but in this

case we have no inner product on E by which we can

represent Df (x) as the product by some vector of E.

This problem can be overcome by the notion of a

pseudogradient vector field. In fact, it can be proved

that if f is continuously differentiable on E, then there

exists a locally Lipschitz vector field V defined on the

complement of the critical points of f,suchthat

kVðuÞk < minfkDf ðuÞk; 1g

Df ðu Þ½VðuÞ >

minfkDf ðuÞk ; 1gkDf ðuÞk

In other words, even if there is no direction of

steepest increase for f, we do have directions along

which the increase of f is steep enough, and these

directions can be selected in a locally Lipschitz way.

Notice that pseudogradients are useful also in the

case of a continuously differentiable function on a

Hilbert space: in this case the gradient of f is just

continuous, so it does not generate a flow. The

Palais–Smale condition, as stated above, makes

perfect sense on the Banach space E (with the only

difference that now Df (u

) tends to zero in the dual

norm of E



), and the mountain-pass theorem holds

for functions of class C

on a Banach space.

Actually, the fact that the domain of f has a vector

structure is not relevant in this statement, and the

mountain-pass theorem holds also for functions

defined on connected infinite-dimensional mani-

folds. Since the essential feature is to dispose of a

pseudogradient vector field, the right level of

generality is to consider a Banach manifold M (i.e., a

manifoldmodeledonaBanachspace)endowedwitha

complete Finsler structure (i.e., a Banach norm on

each tangent space of M, varying in a suitably regular

way, inducing a complete distance on M).

A Nonlinear Elliptic Boundary-Value

Problem

Let us consider a typical application of the mountain-

pass theorem to a semilinear elliptic boundary-value

problem. Let  be a smooth bounded domain in R

and for  2 R, p > 2, consider the problem

u ¼ u þ ujuj

p2

in 

u ¼ 0on@

½1

Let 0 <

<

 

  be the eigenvalues of the

Laplace operator , with domain H

\ H

(), the

Sobolev space of L

-functions on  with weak first

Minimax Principle in the Calculus of Variations 433

two derivatives in L

, vanishing on @. We claim that,

if n = 2, or if n  3and2< p < 2



:= 2n=(n  2), then

problem [1] with <

has a nontrivial solution.

By elliptic regularity, the solutions of [1] are

precisely the critical points of the functional

EðuÞ¼



jruðxÞj

 uðxÞ





juðxÞj

We recall that H

() continuously embeds into

(), for every p < þ1 if n = 2, for every p  2



n  3. So the functional E is well defined, and

actually continuously differentiable, on H

(), a

Hilbert space with the inner product

hu; vi

ðÞ



ruðxÞrvðxÞdx

Since p > 2, near zero the quadratic part of

the functional E dominates over the part with the

-norm. By the Rayleigh characterization of the

first eigenvalue of the Laplacian,



¼ min

u2H

ðÞnf0g



jruðxÞj



uðxÞ

the assumption <

implies that the quadratic

part of E is positive definite. So we can find a small

>0 such that

a :¼ inf

kuk

ðÞ

¼

EðuÞ > 0

On the other hand, the fact that p > 2 implies that

lim

!þ1

EðuÞ¼1

for every u 6¼ 0. Therefore, the sublevel {E < a}

is not connected, and if we can prove the

Palais–Smale condition, the mountain-pass theorem

will imply the existence of a critical point u with

E(u)  a > 0, i.e., a nontrivial solution of [1].

In order to prove the Palais–Smale condition,

notice that the expression for the differential of E,

DEðuÞ½v¼



ruðxÞrvðxÞdx



uðxÞþjuðxÞj

p2

uðxÞ



vðxÞdx

and the compactness of the embedding of H

()

into L

() for p < 2



imply that the gradient of

E has the form

rEðuÞ¼u þ KðuÞ½2

where K : H

() ! H

() is a compact map, that is,

it maps bounded sets into precompact ones. It is

readily seen that when rE has such a form, bounded

Palais–Smale sequences are compact. Thus, it is

enough to show that every Palais–Smale sequence is

bounded. But this follows from the identity

pEðuÞDEðuÞ½u

 1





jruðxÞj

 uðxÞ



together with the fact that the right-hand side term

defines an equivalent norm on H

(), because p > 2

and <

. This concludes the proof.

Actually, using the maximum principle one could

show that under the same assumptions, problem [1]

has a solution which is positive in .

When n  3andp = 2



= 2n=(n  2), the func-

tional f still exhibits a mountain-pass geometry, but

the Palais–Smale condition fails. In fact, the embed-

ding of H

() into L



() is not compact, so the

map K appearing in [2] is not compact, and

bounded Palais–Smale sequences need not have a

converging subsequence. We recall that the non-

compactness of the embedding of H

() into L



()

is due to the fact that the quotient

SðuÞ¼



jruðxÞj



juðxÞj





2=2



is invariant under rescaling u 7!u



(x) = u(x).

When  = 0, the Pohozˇaev identity – an integral

formula obtained by multiplying the equation by

x ru(x)–canbeusedtoprovethatproblem[1]

has no nontrivial solutions, when  is a star-shaped

domain other than the whole R

When  6¼ 0, the presence in the functional of an

-norm – which rescales differently – breaks the

symmetry, and the existence of nontrivial solutions

is again possible. Indeed, Brezis and Nirenberg have

shown that problem [1] with p = 2



has a nontrivial

solution provided that n  4 and 0 <<

,or

n = 3and



<<

, for some 



2 [0, 

] depend-

ing on the domain .

The proof is based on the fact that there is a

certain threshold s > 0, related to the best Sobolev

constant obtained by taking the infimum of S(u)

over all u 2 H

(the domain is irrelevant here),

below which the Palais–Smale condition holds. That

is, every sequence (u

) such that E(u

) converges to

some b less than s,andDE(u

) tends to zero, is

compact. The proof of the mountain-pass theorem

shows that the Palais–Smale condition is needed

only at the minimax level c. In order to conclude, it

is then enough to show that c < s. The value of

c can be estimated by using the fact that the

434 Minimax Principle in the Calculus of Variations

infimum of the quotient S over functions on the

whole R

is attained at the family of functions



ðxÞ¼



nðn  2Þ

ð

þjxj

ðn2Þ=4

which are then solutions of [1] with p = 2



,  = 0,

and =R

Another way to break the symmetry is to keep

 = 0 but to consider domains with a rich topology.

For instance, Bahri and Coron have shown that if 

is a domain with some nonzero singular homology

group H

(; Z

), k  1, then problem [1] with

p = 2



and  = 0 has a positive solution.

Elliptic equations having nonlinearities with the

critical exponent 2



arise naturally in some geo-

metric problems. Consider a manifold M of dimen-

sion n  3, with a metric g having scalar curvature k.

The Yamabe problem calls for finding a metric g

conformally equivalent to g, having constant scalar

curvature. If g

= u

4=(n2)

g, where the positive func-

tion u gives the conformal factor, one finds that

u must solve the equation



4ðn  1Þ

n  2



u ¼ku þ k

ujuj



2

where 

is the Laplace–Beltrami operator associated

with the metric g, and the constant k

is the scalar

curvature of g

. Again, the corresponding functional

satisfies the Palais–Smale condition only below a

certain threshold (actually, the same number s as seen

earlier; this because the lack of compactness is due to

local concentration phenomena, and the metric

structure of the whole ambient becomes irrelevant).

The task is then to show that the minimax level is

below that threshold or, equivalently, that a certain

best Sobolev constant for (M, g)islessthanthe

corresponding constant for R

with the flat metric

(the latter constant is again the infimum of S(u)). This

fact was proved by Aubin in the case n  6or(M, g)

not locally conformally flat. Schoen has then treated

the remaining case, by means of the positive-mass

theorem, a deep result in differential geometry.

A General Minimax Principle

Let us consider again a twice continuously differ-

entiable function f on a real Hilbert space H. The

vector field

VðuÞ¼

rf ðuÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

1 þkrf ðuÞk

has the same nice properties of the gradient vector

field of f, but in addition it is bounded. The

advantage is that the flow of V is globally defined.

When talking about the negative-gradient flow of

f, we will actually refer to such a flow. It will also be

useful to dispose of a negative-gradient flow

truncated below level b. This is the flow of the

vector field V

, where

ðuÞ¼’ðf ðuÞÞVðuÞ

with ’ a smooth function on R which is identically

zero on [1, b], then increases up to reaching the

value 1, and afterwards remains constantly equal to

1. This truncated negative-gradient flow keeps the

points in the sublevel {f  b} fixed, and behaves

as the negative-gradient flow above b (except the

fact that trajectories slow down as the value of

f approaches b).

After these preliminaries, let us consider again the

characterization of the critical level c appearing in the

mountain-pass theorem. This critical level was

obtained as the infimum over a certain class  of

sets  – the curves with end points in different

components of {f < a} – of the maximum of f over .

But if we look back at the proof, we realize that the

fact that these sets were curves was not essential. The

important feature was that the negative-gradient flow

(t,  ) mapped a set of the class  into a set still

belonging to the class ,fort  0. This observation

leads to the following general minimax theorem, due

to Palais:

Theorem (General minimax). Let f be a twice

continuously differentiable function on a real

Hilbert space H, satisfying the Palais–Smale condi-

tion. Let  be a class of subsets of H which is

positively invariant under the action of the negative-

gradient flow  of f (possibly truncated below level

b): that is, if the set  belongs to , then the set (

t , )

belongs to  for all t  0. Then, if the number

c :¼ inf

2

sup

u2

f ðuÞ

is finite (and larger than b), then c is a critical

value of f.

The proof goes along the same lines of the proof of

the mountain-pass theorem: if c is not a critical value

of f, the (possibly truncated) negative-gradient flow

(t

,  ) pulls a sublevel {f  c þ } down into the

sublevel {f  c  } (with c  >b), for some large

, by the Palais–Smale condition. Then we achieve a

contradiction choosing a set  2  on which f does

not exceed c þ , and noticing that (t

, ) is a set

which still belongs to the class , by positive

invariance, and on which f does not exceed c  .

As we shall see in the last section, the possibility

of working with a truncated negative-gradient flow

Minimax Principle in the Calculus of Variations 435

(assuming in this case that c > b) makes the applica-

tion of this theorem easier. Again, an analogous

result holds for continuously differentiable functions

on Banach spaces, or more generally on Banach

manifolds with a complete Finsler structure.

Trivial classes  are the class of all points in H,

and the class consisting of the single set H, yielding

to the infimum and the supremum of f, respectively.

More interesting classes are constructed by fixing a

topological space X and considering the images of

all continuous maps h : X ! H belonging to a

certain relative homotopy class.

Closed Geodesics on Compact Manifolds

A typical application of the general minimax

theorem is Birkhoff proof of the existence of a

closed geodesic on the sphere S

, endowed with an

arbitrary metric g. Closed geodesics are precisely the

critical points of the energy functional

SðxÞ¼

gð

xðtÞ;

xðtÞÞdt

on the Hilbert manifold H

(T, S

) consisting of all

one-periodic loops on S

of Sobolev regularity H

(here T = R=Z denotes the circle parametrized by

[0, 1]). This functional satisfies the Palais–Smale

condition and it is bounded below, but its minima

are just the trivial constant loops, on which S = 0.

Let us use angle coordinates (, ’)onS

, =2 

  =2, 0  ’  2 ( is the latitude, ’ the longi-

tude). A (suitably regular) map h : S

! S

induces a

curve in H

(T, S

) parametrized by : the value of

this curve at  2 [=2, =2] is the loop

t 7!h(,2t). It is a curve that joins two constant

loops. Let  be the set of curves in H

(T, S

) which

are obtained by maps h : S

! S

of topological

degree 1. This class is clearly positively invariant

under the action of the negative-gradient flow of

S (as of every homotopy fixing the constant loops).

If we can show that the minimax level

c :¼ inf

2

sup

u2

SðxÞ

is positive, we will get a positive critical value of S by

the general minimax theorem, hence a nontrivial

closed geodesic. By considering the fact that loops

with small energy also have a small diameter, it

is easy to construct a homotopy on {S < a}, for

some small a > 0, which shrinks every loop to a

point. If h : S

determines a curve  with

max

x2

S(x) < a , composition with this homotopy

yields to a homotopy of h to a map whose image is

acurveinS

. A further homotopy then shows that

the map h is homotopic to a constant, which

is impossible if h has degree 1. This shows that

c a > 0, concluding the proof.

Actually, Ljusternik and Fet have proved that

every compact manifold M has a nontrivial closed

geodesic. Indeed, if M has nonzero fundamental

group, it is enough to minimize S on some nontrivial

homotopy class of loops. Otherwise, the fact that

M is a compact manifold implies that some homo-

topy group 

kþ1

(M), 1  k < dim M, does not van-

ish. A construction similar to the one described

above then allows to associate with every noncon-

tractible map h : S

kþ1

! M a map u :(B

, @B

) !

(T, M ), {S = 0}) which is not homotopically

trivial (here B

denotes the closed unit ball in R

and the notation means that u maps the boundary

of the ball B

into the set of constant loops). Taking

a minimax over the set of images of the maps

u associated with every noncontractible map

h : S

kþ1

! M yields to the desired critical point of

S with positive energy.

It is conjectured that every compact manifold has

infinitely many closed geodesics. Morse theory

allows to prove this fact for the vast majority of

manifolds, but not for the spheres. Bangert and

Franks have established the existence of infinitely

many geodesics on S

by proving that every area-

preserving homeomorphism of the open disk with

two fixed points must have infinitely many periodic

points. Proving the existence of infinitely many

closed geodesics on higher-dimensional spheres is a

challenging open problem.

A Rigidity Property of a Certain

Class of Maps

It is important that the class  in the general

minimax theorem is only required to be invariant

under the action of the negative-gradient flow, and

not, say, under the action of any continuous

homotopy on which the function f is nonincreasing.

Indeed, too many undesirable things can be done on

an infinite-dimensional Hilbert space by arbitrary

continuous maps, whereas the maps arising from

our negative-gradient flow might show some rigid-

ity, forcing them to behave as maps on finite-

dimensional spaces.

Let us clarify this point by considering the follow-

ing example, due to Benci and Rabinowitz. It may

sound a bit artificial at this moment (simpler

examples could be built), but we will find it useful

in the next section. Assume that our Hilbert space is

436 Minimax Principle in the Calculus of Variations