Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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Let
ij
:= R
ikkj
and :=
ii
be the Ri cci tensor and the
scalar curvature, respectively. Then,
E
2
¼
1
4
and E
4
¼
1
32
2
f
2
4jj
2
þjRj
2
g
The Pontrjagin Forms
Since R(x, y) = R(y, x), we can regard R as a
2-form-valued endomorphism of the tangent bundle.
We define the Pontrjagin forms p
i
2 C
1
(
4i
M)by
expanding
det I þ
1
2
R
¼ 1 þ p
1
þ p
2
þ
These differential forms are closed and the corre-
sponding cohomology classes
P
i
¼½p
i
2H
4i
ðM; RÞ
in the de Rham cohomology are independent of
the particular Riemannian metric on M which was
chosen.
The
ˆ
A genus and the Hirzebruch L polynomial
are expressed in terms of these classes using the
splitting principle. Let A be a skew-symmetric
matrix. One sets
pðAÞ :¼ detðI þ AÞ¼1 þ p
1
ðAÞþp
2
ðAÞþ
As A is skew symmetric, it decomposes as the direct
sum of 2 2 blocks of the form
0
i
i
0
We then have
pðAÞ¼
Y
1 þ
2
so
p
i
ðAÞ¼s
i
2
1
;
2
2
; ...
where s
i
is the ith symmetric function;
p
1
¼
X
i
2
i
; p
2
¼
X
i<j
2
i
2
j
and so forth. Let
Lð
~
Þ :¼
Y
tanhð
Þ
¼ 1 þ L
1
ð
~
ÞþL
2
ð
~
Þþ
^
Að
~
Þ :¼
Y
2 sinh
1
2
¼ 1 þ
^
A
1
ð
~
Þþ
^
A
2
ð
~
Þþ
As L
i
and
ˆ
A
i
are even symmetric functions of
~
, one
can write L
i
= L
i
(p
1
(A), ..., p
k
(A)). For example,
L ¼ 1 þ
1
3
p
1
þ
1
45
7p
2
p
2
1
þ
^
A ¼ 1
1
24
p
1
þ
1
5760
ð7p
2
1
4p
2
Þþ
Substituting (1=2)R for A then permits one to
define the Hirzebruch polynomial L(R) and the
ˆ
A
genus
ˆ
A(R).
The Chern Forms
Let V be a k-dimensional complex vector bundle
over M. Let r be a Hermitian connection on V and
let be the associated curvature endomorphism.
The Chern forms c
i
2 C
1
(
2i
M) are defined by
expanding
det I þ
ffiffiffiffiffiffiffi
1
p
2
!
¼ 1 þ c
1
þ c
2
þ
As with the Hirzebruch polynomial and the
ˆ
A genu s,
the Chern character and Todd genus are expressed
in terms of the generating functions:
Tdð
~
Þ¼
Y
1 e
and
chð
~
Þ¼
X
!
One has
Td ¼ 1 þ Td
1
þ Td
2
þ
¼ 1 þ
1
2
c
1
þ
1
12
c
2
1
þ c
2
þ
Ch ¼ ch
0
þ ch
1
þ ch
2
þ
¼ k þ c
1
þ
1
2
c
2
1
2c
2
þ
The Index Theorem
The Gauss–Bonnet Theorem
We return to the de Rham complex. Le t
ðMÞ¼
X
p
ð1Þ
p
dim H
p
ðM; RÞ
be the Euler–Poincare´ characteristic; (M) = 0ifm is
odd. Let M have a simplicial structure with n(k) cells
of degree k; n(0) is the number of vertices, n(1) is the
number of edges, n(2) is the number of triangles, etc.
Then
ðMÞ¼
X
k
ð1Þ
k
nðkÞ
Index Theorems 27
so the Euler–Poincare´ characteristic is a combina-
torial invariant. By the Hodge–de Rham theorem,
indexðd þ Þ¼dim kerð
even
Þdim kerð
odd
Þ
¼ ðMÞ
The Chern–Gauss–Bonnet theorem expresses this
invariant in terms of curvature
ðMÞ¼
Z
M
E
m
dx
where E
m
is the Euler form given above. If one twists
the de Rham complex to take coefficients in an
auxiliary vector bundle V, then no new information
results, since
indexfd þ g
V
¼ ðMÞdimðVÞ
The Hirzebruch Signature Theorem
Let sign (M) be the index of the signature complex
on a manifold of dimension 4k; the index vanishes
in dimensions m 2 mod 4. Let ? be the Hodge
duality operator. As ?
p
?
1
=
mp
, ? preserves the
eigenspaces of the Laplacian. In particular, ? induces
an isomorphism
? : H
p
ðM; RÞ¼kerð
p
Þ
! H
mp
ðM; RÞ¼kerð
mp
Þ
which implements Poincare´ duality. In dimension
2k, ?
2
= Id. Decompose
H
2k
ðM; RÞ¼H
2k;þ
ðM; RÞH
2k;
ðM; RÞ
into the 1 eigenspaces of ?; these may be identified
with ker(
2k,
) acting on C
1
(
2k,
M). As the
contributions to the signature away from the middle
dimension cancel,
signðMÞ¼dim H
2k;þ
ðM; RÞdim H
2k;
ðM; RÞ
As with the de Rham complex, there is a
topological description of this invariant. If and
are closed 2k forms, one sets
h; i :¼
Z
M
^
One can use Stoke’s theorem to see that this
induces a symmetric bilinear form on the de
Rham cohomology groups H
2k
(M; R). Poincare´
duality t hen shows that this symmetric bilinear
form is nondegenerate, so this is a form of type
(p, q); sign(M) is the signature of this quadratic
form:
signðMÞ¼q p
The Hirzebruch signature formula expresses sign
(M) in terms of curvature; if L is the Hirzebr uch
polynomial described above and if m = 4k, then
signðMÞ¼
Z
M
L
k
Let V be an auxiliary coefficient bundle. Taking
coefficients in V then yields the formula
sign
V
ðMÞ¼
X
4iþ2j¼m
2
j
Z
M
L
i
^ ch
j
ðVÞ
The Index of the Yang–Mills Complex
Let YM
V
be the Yang–Mills complex with coeffi-
cients in an auxiliary vector bundle V, then the
index can be evaluated using the formulas given
above as
indexfYM
V
g¼
1
2
fdimðVÞðMÞsignðM; VÞg
¼
1
2
R
M
fdimVE
4
dim VL
1
4ch
2
ðVÞg
The Index of the Dolbeault Complex
If V is a holomorphic bundle over a complex
manifold M, then
indexfð
@ þ
00
Þ
V
g¼
X
2iþ2j¼m
Z
M
Td
i
ðMÞ^ch
j
ðVÞ
The index of the untwisted Dolbeault complex is
called the arithmetic genus and denoted by ag(M).
The Index of the Spin Complex
If M is a spin manifold and if A
V
is the Dirac
operator with coefficients in an auxiliary coefficient
bundle, then
indexfA
V
g¼
X
4iþ2j¼m
Z
M
^
A
i
ðMÞ^ch
j
ðMÞ
The index of the spin complex is called the
ˆ
A genus
and is denoted by
ˆ
A(M). If M is a Spin
c
manifold,
the appropriate formula becomes
indexfA
c
V
g¼
X
4iþ2jþ2k¼m
Z
M
^
A
i
ðMÞ^ch
j
ðMÞ^
k
where =
1
2
c
1
(L), L being the complex line bundle
associated with the Spin
c
structure.
28 Index Theorems
Properties
The classic elliptic complexes defined above are
multiplicative with respect to Cartesian product.
Suppose that M
1
and M
2
are Riemannian manifolds
with the appropriate structures. For the signature
complex, suppose M
1
and M
2
are oriented; for the
Dolbeault complex, suppose M
1
and M
2
are holo-
morphic; for the spin complex, suppose M
1
and M
2
are spin. By taking the twisting coefficient bundle to
be trivial in the interests of simplicity, one has
ðM
1
M
2
Þ¼ðM
1
ÞðM
2
Þ
signðM
1
M
2
Þ¼signðM
1
ÞsignðM
2
Þ
agðM
1
M
2
Þ¼agðM
1
ÞagðM
2
Þ
^
AðM
1
M
2
Þ¼
^
AðM
1
Þ
^
AðM
2
Þ
These complexes behave well under finite coverings.
Let F ! M
2
! M
1
be a finite covering projection
with jF j sheets. Then
ðM
2
Þ¼jFjðM
1
Þ
signðM
2
Þ¼jFjsignð M
1
Þ
agðM
2
Þ¼jFjagðM
1
Þ
^
AðM
2
Þ¼jFj
^
AðM
1
Þ
The connected sum M
1
#M
2
is defined by punching
out small disks about points P
i
in M
i
and then
joining along the spherical boundaries that remain.
It is necessary, of course, to smooth out the resulting
corners. Note that if M
1
and M
2
are complex
manifolds, then M
1
#M
2
is no longer a complex
manifold in general. Since
ðS
m
Þ¼2; signðS
m
Þ¼0; and
^
AðS
m
Þ¼0
the following additivity results follow from the
integral formulas given above:
ðM
1
#M
2
Þ¼ðM
1
ÞþðM
2
Þ2
signðM
1
#M
2
Þ¼signðM
1
ÞþsignðM
2
Þ
^
AðM
1
#M
2
Þ¼
^
AðM
1
Þþ
^
AðM
2
Þ
Examples and Applications
Let S
m
be the standard sphere and let CP
j
be the
complex projective plane. One then has
ðS
4
Þ¼2; signðS
4
Þ¼0
ðS
2
S
2
Þ¼4; signðS
2
S
2
Þ¼0
ðCP
2
Þ¼3; signðCP
2
Þ¼1
In dimension 4, the Riemann–Roch formula yields
agðM
4
Þ¼
1
4
fðMÞþsignðMÞg
This would yield ag(S
4
) =
1
2
; since
1
2
is not an integer,
this shows that S
4
does not admit a complex
structure; a similar argument shows that S
n
does
not admit a complex structure for n 6¼ 2, 6, and it is
not known whether S
6
admits a holomorphic
structure; it does admit an almost-complex
structure.
If we set M = CP
2
#CP
2
, then
agðMÞ¼
1
4
ð3 þ 3 2 þ 1 þ 1Þ¼
3
2
and thus CP
2
#CP
2
does not admit a complex
structure. These examples are typical of the use of
the index theorem to prove the nonexistence of
certain structures.
The General Index Theorem
Let S(T
M) be the sphere bundle of unit cotange nt
vectors and let D(T
M) be the disk bundle of
cotangent vectors of length at most 1. Let
P : C
1
ðV
1
Þ!C
1
ðV
2
Þ
be an elliptic pseudodifferential operator. The
leading symbol p :=
L
(P) induces a smooth map
p : SðT
MÞ!EndðV
1
; V
2
Þ:
We form (M) by gluing two copies of D(M)
together along their common boundary S(M) and
we define a bundle (p, V
1
, V
2
) over (M) by gluing
V
1
to V
2
over S(M) using the clutch ing function p.
The Atiyah– Singer index theorem expresses the
index of P in terms of cohomological data involving
the Chern class of the symbol bundle and the
characteristic classes of the tangent bundle of M.If
(M) is given a suitable orientation, then
indexðPÞ¼
X
2iþ4j¼2m
Z
ðMÞ
ch
i
ððp; V
1
; V
2
ÞÞ^ Td
j
ðMÞ
It specializes to the results given above for
the classical elliptic complexes. Conversely, by
using K -theoretic methods, the index theorem in
full generality can be derived from the special case
of the twisted signature complex.
Manifolds with Boundary
If the boundary of M is nonempty, we must impose
suitable boundary conditions.
Local Boundary Conditions
Choose local coordinates x = (x
1
, ..., x
m
) near the
boundary of M so that x
m
is the geodesic distance to
the boundary. On the boundary, we can decompose
a differential form ! 2 C
1
(M) in the form
! = !
1
þ dx
m
^ !
2
, where !
1
and !
2
are tangential
Index Theorems 29
differential forms. Absolute and relative boundary
conditions are defined by setting
B
a
! :¼ !
2
j
@M
and B
r
! :¼ !
1
j
@M
Let (d þ )
a
and (d þ )
r
be the associated realiza-
tions. These operators preserve the grading of the
exterior algebra M =
even
M
odd
M and define
elliptic complexes
ðd þ Þ
a
: C
1
ð
even
MÞ!C
1
ð
odd
MÞ
ðd þ Þ
r
: C
1
ð
even
MÞ!C
1
ð
odd
MÞ
We consider a collection
J ¼f1 j
1
< < j
p
< mg
of tangential indices and let
dx
J
¼ dx
j
1
^^dx
j
p
The associated absolute boundary conditions for the
Laplacian are defined by
~
B
a
ð
J
dx
J
þ
J
dx
m
^ dx
J
Þ
¼ð
J
j
@M
dx
J
Þ @
x
m
J
j
@M
dx
J
If ? is the Hodge operator, then one sets dually:
~
B
r
ð!Þ¼
~
B
a
ð?!Þ
Let
p
a
and
p
r
be the associated realizations of the
Laplacian with these boundary conditions. The
Hodge–de Rham theore m extends to this setting to
yield isomorphisms
ker
p
a
¼ H
p
ðM; RÞ
and
ker
p
r
¼ H
p
ðM;@M; RÞ
The Hodge ? operator intertwines
p
a
and
mp
r
and implements the Poincare´ duality isomorphism
H
p
(M; R) = H
mp
(M, @M; R). This also shows that
indexðd þ Þ
a
¼
X
p
ð1Þ
p
dim H
p
ðM; RÞ¼ðMÞ
and
indexðd þ Þ
r
¼
X
p
ð1Þ
p
dim H
p
ðM;@M; RÞ
¼ ðM;@MÞ¼ðMÞð@MÞ
Let E
m
be the Euler form if m is even. We set
E
m
= 0ifm is odd. Let L be the second fundam ental
form. Let A = (a
1
, ..., a
m1
)andB = (b
1
, ..., b
m1
)
be collections of distinct indices ranging from 1 to
m 1. Set
L
m1
:¼
X
k
1
k
8
k
k!ðm 1 2kÞ!volðS
m12k
Þ
"
A;B
R
a
1
a
2
b
2
b
1
...R
a
2k1
a
2k
b
2k
b
2k1
L
a
2kþ1
b
2kþ1
...L
a
m1
b
m1
The Chern–Gauss–Bonnet theorem generalizes to
this setting to yield
ðMÞ¼indexðd þ Þ
a
¼
Z
M
E
m
dx þ
Z
@M
L
m1
dy
For example,
ðM
2
Þ¼
1
4
Z
M
2
dx þ 2
Z
@M
2
L
aa
dy
ðM
3
Þ¼
1
8
Z
@M
3
fR
abba
þ L
aa
L
bb
L
ab
L
ab
gdy
ðM
4
Þ¼
1
32
2
Z
M
4
f
2
4jj
2
þjRj
2
gdx
þ
1
24
2
Z
@M
4
f3L
aa
þ 6R
amam
L
bb
þ 6R
acbc
L
ab
þ 2L
aa
L
bb
L
cc
6L
ab
L
ab
L
cc
þ 4L
ab
L
bc
L
ac
gdy
The interior integral vanishes if m is odd. The
boundary integral can be nonzero in any dimen-
sions. Thus, in particular, the index of this elliptic
complex can be nonzero even if m is odd; (D
m
) = 1
for any m . The index of (d þ )
r
is computed
similarly.
Spectral Boundary Conditions
In contrast to the de Rham complex, there do not
exist local boundary conditions for the signature,
spin, and Dolbeault complexes. To simplify the
discussion, we assume that the metric is the product
near the boundary; there are appropriate compen-
sating terms involving the second fundam ental form
in the more general setting. Let A : C
1
(V
1
) !
C
1
(V
2
) denote either the twisted signature or the
twisted spin complexes; there are some additional
difficulties for the Dolbeault complex. Near the
boundary, we can express
A ¼ @
x
m
þ A
T
where A
T
is a self-adjoint tangential operator of
Dirac type on V
1
j
@M
and is a unitary bundle
30 Index Theorems
isomorphism from V
1
j
@M
to V
2
j
@M
. Let {
i
,
i
} be the
discrete spectral resolution of A
T
. One defines
ð A
T
; sÞ¼
X
k
6¼0
sgnð
k
Þj
k
j
s
as a measure of the spectral asymmetry of A
T
. This
is well defined for Re(s) 1 and has a meromorphic
extension to the complex plane C. It turns out that 0
is a regular value and one defines
ðA
T
Þ :¼
1
2
fð A
T
; sÞþdim kerð A
T
Þgj
s¼0
The spectral boundary conditions can now be
imposed. Let
be orthogonal projection in
L
2
(V
1
j
@M
) on the span of the eigensections of A
T
corresponding to non-negative eigenvalues and let
A
be the associated realizati on defined by this
boundary condition.
One can use the Atiyah–Patodi–Singer index
theorem to generalize the relations given above to
this setting. Let f
A
be the local integral given above
that involves the Hirzebruch L polynomial for the
signature complex or the
ˆ
A genus for the spin
complex. One then has
indexðA
Þ¼ðA
T
Þþ
Z
M
f
A
There are suitable correction formulas involving
integrals of polynomials in the second fundamental
form and in the curvature tensor if the structures are
not product near the boundary.
Equivariant Problems
The Classical Lefschetz Formula
Let M be a compact Riemannian manifold without
boundary. Let T be a smooth map from M to M.Then
pullback T
induces an action on C
1
(
p
M)which
commutes with the exterior derivative d and hence an
action on the de Rham cohomology groups H
p
(M; R).
The Lefschetz number of T is then given by
LðTÞ¼
X
p
ð1Þ
p
trfT
on H
p
ðM; RÞg
To illustrate the Lefschetz number, let M = T
2
be
the two-dimensional torus. Let e
1
:= dx
1
, let
e
2
:= dx
2
, and let e
12
:= dx
1
^ dx
2
. Then,
H
0
ðT
2
; RÞ¼1 R
H
1
ðT
2
; RÞ¼e
1
R þ e
2
R
H
2
ðT
2
; RÞ¼e
12
R
Let T(x
1
, x
2
) = (n
11
x
1
þ n
12
x
2
, n
21
x
1
þ n
22
x
2
). Then,
T
ð1Þ¼1
T
ðe
1
Þ¼n
11
e
1
þ n
12
e
2
T
ðe
2
Þ¼n
21
e
1
þ n
22
e
2
T
ðe
12
Þ¼ðn
11
n
22
n
12
n
21
Þe
12
and, consequently, the Lefschetz number becomes
LðTÞ¼detðI T
Þ
¼ 1 ðn
11
þ n
22
Þþðn
11
n
22
n
12
n
21
Þ
The classical Lefschetz fixed-point formula expresses
L in terms of data for the fixed-point set F(T)andisan
example of the equivariant index theorem. One
assumes that the fixed-point set of T consists of smooth
submanifolds N
1
, ..., N
k
and that the induced map
dT
on the normal bundles of these manifolds is
nondegenerate. This means that det (I dT
) 6¼ 0, that
is, that there are no infinitesimal normal directions
which are left fixed. One then has
LðTÞ¼
X
i
signðdetðI dT
ÞÞðN
i
Þ
The Lefschetz Formula for the Other Classical
Elliptic Complexes
Let T be an orientation-preserving isometry of M.
When dealing with the spin complex, suppose that T
preserves the spin structure; when dealing with the
Dolbeault complex, suppose that T preserves the
holomorphic structure. If
A : C
1
ðV
1
Þ!C
1
ðV
2
Þ
is one of the classical elliptic complexes, then by
assumption T
commutes with A and hence pre-
serves the eigenspaces of the associated Laplac ians.
The Lefschetz number is defined by setting
L
A
ðTÞ :¼trðT
on kerðA
AÞÞ
trðT
on kerðAA
ÞÞ
Setting T = Id, one recovers the standard index.
To simplify the discussion, we assume henceforth
that T is an orientation-preserving isometry of M
with only isolated fixed points. Let {
1
, ...,
m=2
}be
the rotation angles of dT at a fixed point x of T. Set
j
:¼ cosð
j
Þþ
ffiffiffiffiffiffiffi
1
p
sinð
j
Þ
We take the sum over the isolated fixed points x and
then the product over the rotation angle s 1 j
m=2 to express
Index Theorems 31
L
sign
ðTÞ¼
X
x
Y
j
ffiffiffiffiffiffiffi
1
p
cot
j
2
L
spin
ðTÞ¼
X
x
Y
j
1
2
ffiffiffiffiffiffiffi
1
p
csc
j
2
L
Dolb
ðTÞ¼
X
x
Y
j
ð1
j
Þ
1
In considering the spin complex, we assume T
preserves the spin structure. This permits us to lift dT
from SO(m)toSpin(m) and defines liftings of the
rotation angles
i
from [0, 2] to [0, 4] in such a way
that the formula given above for the spin complex is
well defined. In considering the Dolbeault complex,
we assume that T preserves a complex structure, so the
formula given above for the Dolbeault complex
involving the complex eigenvalues
j
is well defined.
Acknowledgements
Research of P Gilkey was partially supported by
the MPI (Leipzig, Germany). Research of R Ivanova
was partially supported by the UHH Seed Money
Grant. Research of K Kirsten was partially sup-
ported by the Baylor University Summer Sabbatical
Program and by the MPI (Leipzig, Germany).
Research of J H Park was supported by the Korea
Research Foundation Grant funded by the Korean
Government (MOEHRD) (KRF-2005-204-C00007).
See also: Anomalies; Clifford Algebras and their
Representations; Cohomology Theories; Dirac Operator
and Dirac Field; Gerbes in Quantum Field Theory;
Intersection Theory; Instantons: Topological Aspects;
K-Theory; Path-Integrals in Non Commutative Geometry;
Quillen Determinant; Riemann Surfaces; Spinors and
Spin Coefficients.
Further Reading
Atiyah MF and Segal GB (1968) The index of elliptic operators II.
Annals of Mathematics 87: 531–545.
Atiyah MF and Singer IM (1968) The index of elliptic operators I,
III, IV, V. Annals of Mathematics 87: 484–530, 546–604.
Atiyah MF and Singer IM (1971) The index of elliptic operators I,
III, IV, V. Annals of Mathematics 93: 119–138, 139–149.
Atiyah MF, Patodi VK, and Singer IM (1975) Spectral asymmetry and
Riemannian geometry I. Mathematical Proceedings of the Cam-
bridge Philosophical Society 77: 43–69; 78: 405–432.
Atiyah MF, Patodi VK, and Singer IM (1976) Spectral asymmetry
and Riemannian geometry I. Mathematical Proceedings of the
Cambridge Philosophical Society 79: 71–79.
Berline N, Getzler E, and Vergne M (1992) Heat Kernels and
Dirac Operators, Grundlehren der Mathematishen Wis-
senschaften, vol. 298. Berlin: Springer.
Bordag M, Mohideen U, and Mostepanenko VM (2001) New
developments in the Casimir effect. Physics Reports 353: 1–205.
Eguchi T, Gilkey PB, and Hanson AJ (1980) Gravitation, gauge
theories and differential geometry. Physics Reports 66: 213–393.
Elizalde E, Odintsov SD, Romeo A, Bytsenko AA, and Zerbini S
(1994) Zeta Regularization Techniques with Applications.
Singapore: World Scientific.
Esposito G (1998) Dirac Operators and Spectral Geometry.
Cambridge: Cambridge University Press.
Gilkey P (1995) Invariance Theory, the Heat Equation, and the
Atiyah–Singer Index Theorem, 2nd edn, Studies in Advanced
Mathematics. Boca Raton, FL: CRC Press.
Grubb G (1996) The Functional Calculus of Pseudo-Differential
Boundary Problems, 2nd edn., Progress in Mathematics, vol.
65. Boston, MA: Birkha¨ user Boston.
Hirzebruch F and Zagier DB (1974) The Atiyah–Singer Index
Theorem and Elementary Number Theory. Wilmington:
Publish or Perish.
Kirsten K (2001) Spectral Functions in Mathematics and Physics.
Boca Raton, FL: Chapman and Hall/CRC Press.
Melrose R (1993) The Atiyah–Patodi–Singer Index Theorem,
Research Notes in Mathematics, vol. 4. Wellesley, MA:
AKPeters,Ltd.
Palais RS et al. (1965) Seminar on the Atiyah–Singer index
theorem. Annals of Mathematical Studies, 57. Princeton:
Princeton University Press.
Vassilevich DV (2003) Heat kernel expansion: user’s manual.
Physics Reports 388: 279–360.
Inequalities in Sobolev Spaces
M Vaugon, Universite
´
P.-M. Curie, Paris VI, Paris,
France
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
Given 1 p < n, it was shown by Sobolev that there
exists a constant K > 0 such that, for any u 2
C
1
0
(R
n
), the space of smooth functions with
compact support in R
n
,
Z
R
n
juj
p
?
dx
1=p
?
K
Z
R
n
jruj
p
dx
1=p
½1
where ru is the gradient of u and p
?
= np=(n p). It
is easily seen that p
?
in [1] is critical in the following
sense. Let kk
p
stand for the L
p
-norm. For u 2
C
1
0
(R
n
), and >0, let also u
be the function given
by u
(x) = u(x). For p and q two real numbers,
kru
k
p
¼
1ðn=pÞ
kruk
p
ku
k
q
¼
n=q
kuk
q
Letting ! 0 and !þ1, it follows that an
inequality like kuk
q
Kkruk
p
holds true for all u
(in particular for the u
’s) only when q = p
?
.To
32 Inequalities in Sobolev Spaces
prove [1], the approach of Sobolev was based on the
straightforward represent ation formula
uðxÞ¼
ðn=2Þ
2
n=2
Z
R
n
X
n
k¼1
x
k
y
k
jx yj
n
@
k
uðyÞdy
where is the Gamma function, and on an
n-dimensional versio n of a theorem of Hardy –
Littlewood concerning fractional integrals that we
apply to the right-hand side of the above representa-
tion formula. More direct arguments were later
discovered in independent works by Gagliardo and
Nirenberg. In particular, the explicit inequality
Z
R
n
juj
n=ðn1Þ
dx
ðn1Þ=n
1
2
Y
n
k¼1
Z
R
n
jD
k
ujdx
1=n
1
2
Z
R
n
jrujdx ½2
was proved to hold, where D
k
is the partial
derivative D
k
= @=@x
k
. Inequality [2] is of the form
[1] when p = 1, since 1
?
= n=(n 1). By geometric
measure theory, and the coarea formula, it can be
expressed as an isoperimetric type inequality.
There have been several symbols and several
definitions for Sobolev spaces. Before they became
generally associated with the name of Sobolev, they
were someti mes referred to by other names, for
instance, as ‘‘Beppo Levi spaces.’’ We often find two
definitions and two notation s in the literature. For
a domain in R
n
, p 1 real, an d u of class C
m
in ,
we let
kuk
m;p
¼
X
0jjm
kD
uk
p
p
0
@
1
A
1=p
½3
when the right-hand side makes sense, where kk
p
is
the L
p
-norm, = (
1
, ...,
n
) is a multi-index,
jj=
P
i
i
, and D
= D
1
1
D
n
n
. We define
H
m; p
ðÞ¼the completion of
fu 2 C
m
ðÞ s.t. kuk
m;p
< þ1g
with respect to the norm kk
m; p
W
m; p
ðÞ¼ u 2 L
p
ðÞ s.t. D
u 2 L
p
ðÞ
f
for all 0 j jm
g
where D
is the weak (or distributional) partial
derivative of u with respect to the multi-index .Both
H
m, p
()andW
m, p
() are Banach spaces (and even
Hilbert when p = 2). It is easily seen that H
m, p
()
W
m, p
(), but we had to wait for the work of Meyers
andSerrintorealizethatH
m, p
() = W
m, p
(). The
spaces H
m, p
(), also denoted W
m, p
(), are referred to
as Sobolev spaces. The spaces H
m, p
0
(), also denoted
W
m, p
0
(), are defined as the closure of C
1
0
()in
H
m, p
(), where C
1
0
() is the space of smooth
functions with compact support in .
Inequality [1] states that the Sobolev space
H
1, p
0
(R
n
) is naturally embedded in the Lebesgue
space L
p
?
(R
n
), a particular case of what we now
refer to as Sobolev embeddings.
Sobolev Inequalities and the Sobolev
Embedding Theorem in Its First Part
Let m be an integer and let p 1 be real. The
Sobolev space H
m, p
(R
n
), also denoted by W
m, p
(R
n
),
is defined by in one of the two equivalent ways:
H
m; p
ðR
n
Þ¼the completion of
fu 2 C
m
ðR
n
Þ s.t: kuk
m;p
< þ1g
with respect to the norm kk
m;p
or
H
m; p
ðR
n
Þ¼ u 2 L
p
ðR
n
Þs.t: D
u 2 L
p
ðR
n
Þ
f
for all 0 j jm
g
where D
is the weak (or distributional) partial
derivative of u with respect to the multi-index ,and
kk
m, p
is as in [3]. The Sobolev space (H
m, p
(R
n
),
kk
m, p
) is a Banach space, and even a Hilbert space
when p = 2. The space is reflexive when p > 1, and
we also have that H
m, p
(R
n
) = H
m, p
0
(R
n
), where
H
m, p
0
(R
n
) is defined as the closure of C
1
0
(R
n
)in
H
m, p
(R
n
). What we usually refer to as the first part
of Sobolev inequalities can be expressed as follows.
Sobolev embeddings (Part I). For p, q two real
numbers with 1 q < p,andk, m two integers with
0 m < k,if1=p = 1=q (k m)=n, then H
k, q
H
m, p
, and there exists K > 0 such that kuk
m, p
Kkuk
k, q
for all u 2 H
k, q
.
The Sobolev theorem in its first part states that
the above Sobolev embe ddings (resp. inequalities)
hold true for the Euclidean space. A particular case
of interest is when k = 1. In this case, we get, as in
the introduction, that for any 1 p < n, H
1, p
(R
n
)
L
p
?
(R
n
) where p
?
= np=(n p). The embedding for
the Euclidean space reduces to the Sobolev inequal-
ity [1]. An important remark is that there is a
hierarchy for Sobolev embeddings. In particular,
that if H
1, 1
L
n=(n1)
,1
?
= n=(n 1), then all the
other embeddings H
k, q
H
m, p
hold true. Thanks to
this remark, the Sobolev embedding theorem for
Euclidean space easily follows from an inequality
like [2]. The hierarchy for Sobolev embeddings is an
easy consequence of Ho¨ lder’s inequalities when
k = 1, and of Ho¨ lder’s inequalities together with
Kato’s inequality when k > 1.
Inequalities in Sobolev Spaces 33
There are several extensions of Sobolev inequal-
ities in the literature. Famous extensions were
discovered by Gagliardo and Nirenber g. The Nash
inequality, which reads as
Z
R
n
u
2
dx
ðnþ2Þ=n
K
Z
R
n
jujdx
4=n
Z
R
n
jruj
2
dx ½4
for all u 2 H
1, 2
(R
n
), is one of the Gagli ardo–
Nirenberg’s inequalities. The Nash inequality easily
follows from [1] when p = 2 and Ho¨ lder’s inequal-
ity. There are also extensions of Sobolev spaces, for
instance, spaces of BV-functions or Orlicz–Sobolev
spaces.
The Sobolev Embedding Theorem in Its
Second Part
For m inte ger, let C
m
B
(R
n
) be the space of functions
of class C
m
in R
n
for which the norm
kuk
C
m
¼
X
0jjm
sup
x2R
n
jD
uðxÞj
is finite. What we usually refer to as the second part
of Sobolev inequalities can be expressed as follows.
Sobolev embeddings (Part II). For q 1 a real
number, and k, m two integers with 0 m < k,if
1=q (k m)=n < 0, then H
k, q
C
m
B
, and there
exists K > 0 such that kuk
C
m
Kkuk
k, q
for all
u 2 H
k, q
.
The Sobolev theorem in its second part states that
the above Sobolev embe ddings (resp. inequalities)
hold true for the Euclidean space. Refinements were
then obtained by Morrey with embeddings in
Ho¨ lder spaces. Let, for instance, C
0,
(R
n
) be the
Ho¨ lder space of continuous functions in R
n
for
which the norm
kuk
C
0;
¼ sup
x2R
n
juðxÞj þsup
x6¼y
juðyÞuðxÞj
jy xj
is finite. For k = 1, m = 0, and q 1 such that 1=q
1=n < 0, the embedding H
1, q
(R
n
) C
0
B
(R
n
) can be
refined into an embedding like H
1, q
(R
n
) C
0,
(R
n
),
where 2 (0, 1) is such that 1=q (1 )=n < 0.
The Case of Domains and the Kondrakov
Theorem
The Sobolev embeddings in their first and second
parts extend to regular domains . A typical
condition is that satisfies a cone property. When
is bounded, and thus of finite v olume, an
embedding like H
1, p
() L
p
?
() implies that we
also have that H
1, p
() L
q
() for all 1 q p
?
.
The Kondrakov theorem states that such embed-
dings are all compact, unless q = p
?
, in the sense that
bounded sequences of functions in H
1, p
possess
converging subsequences in L
q
.
For p 1 real, the Sobolev embedding theorem in
its first part provides embeddings of H
1, p
into
Lebesgue spaces when p < n, while the Sobo lev
embedding theorem in its second part provides
embeddings of H
1, p
into Ho¨ lder spaces when p > n.
For p = n, it is false that H
1, n
can be embedded
into L
1
. However, when is bounded, we can
prove that exp (u) 2 L
1
() when u 2 H
1, n
0
(), and
that
Z
expðuÞdx K expðkuk
n
1;n
Þ
where , K > 0 are independent of u. We also have
that
Z
expðjuj
n=ðn1Þ
Þdx K
for all u 2 H
1, n
0
()suchthatkruk
n
1, where ,
K > 0 are independent of u. Such inequalities are often
referred to as Moser–Tru¨ dinger type inequalities.
The Case of Riemannian Manifolds
Riemannian manifolds are natural extensions of
Euclidean space. For ( M, g) a Riemannian manifold,
m integer, and p 1 real, we define the Sobolev
space H
m, p
(M)by
H
m;p
ðMÞ¼the completion of
fu 2 C
m
ðMÞ s.t. kuk
m;p
< þ1g
with respect to the norm kk
m;p
where kuk
m, p
=
P
m
i = 0
kr
i
uk
p
, r
i
u is the ith covari-
ant derivative of u,andkk
p
is the L
p
-norm
in (M, g). A notation like kr
i
uk
p
stands for the
L
p
-norm of the pointwise norm jr
i
uj of r
i
u. Sobolev
spaces on manifolds are Banach spaces, even Hilbert
when p = 2, and they are reflexive when p > 1. They
do not depend on the metric when M is compact.
For compact Riemannian manifolds, everything
works as for bounded domains. The Sobolev
embeddings in their first and second parts remain
valid. The Kondrakov theorem also remains valid.
However, since constant functions are in Sobolev
spaces when the manifold is compact, the L
p
-norm
of u in the H
1, p
-norm of u should be added to the
right-hand side in inequalities like [1]. More
precisely, if (M, g) is a compact Riemannian
34 Inequalities in Sobolev Spaces
manifold of dimension n,and1 p < n, then the
inequality for the embedding H
1, p
(M) L
p
?
(M)
reads as: there exists K > 0 such that for any
u 2 H
1, p
(M),
Z
M
juj
p
?
dv
g
p=p
?
K
Z
M
jruj
p
dv
g
þ
Z
M
juj
p
dv
g
½5
where dv
g
is the Riemannian volume element with
respect to g. When (M, g) is no longer compact, the
Sobolev embedding theorem might become false. A
nontrivial key observation is that a Sobolev inequal-
ity lik e [5] on a complete manifold (M, g ) implies the
existence of a uniform (with respect to the center)
lower bound for the volume of balls of radius 1. It
follows that for any n 2, there exist complete
Riemannian n-manifolds (M, g) for which, for any
p 2 [1, n), H
1, p
(M) 6 L
p
?
(M). Possible examples are
warped products of the real line R and the
(n 1)-sphere S
n1
. When the Ricci curvature is
bounded from below, the condition that there is a
uniform (with respect to the center) lower bound for
the volume of balls of radius 1 is necessary and
sufficient in order to get that the Sobolev embed-
dings are valid.
Isoperimetric and Euclidean
Type Inequalities
Let (M, g) be a complete Riemannian n-manifold.
Euclidean type inequalities are said to hold on (M, g)
if there exists K > 0 such that for any 1 p < n,
and any u 2 H
1, p
(M),
Z
M
juj
p
?
dv
g
1=p
?
K
Z
M
jruj
p
dv
g
1=p
½6
where p
?
= np=n p. As for the Euclidean space, if
the above inequality holds for some p
0
, then it
holds, with distinct K, for all p
0
p < n.In
particular, if the inequality holds for p = 1, it holds
for all p’s. The inequality when p = 1 was shown to
be true by Hoffman and Spruck when the manifold
is simply connected of nonpositive sectional curva-
ture. Such manifolds are referred to as Cartan–
Hadamard manifolds. The inequality when p = 2is
related to the nonparabolicity of the manifold,
namely the existence of a minimal Green’s function,
and to the behavior of the minimal Green’s function.
By geometric measure theory and the coarea
formula, [6] when p = 1 is equivalent to the
isoperimetric inequality
Area
g
ð@Þ
1
C
Vol
g
ðÞ
ðn1Þ=n
½7
where C > 0, is a smooth bounded dom ain in
M,Area
g
(@) is the volume of @ for the metric
induced by g,andVol
g
() is the volume of with
respect t o g. Moreover, the constants C and K
(for p = 1) are the same in the sense that if [6] for
p = 1 holds with K,then[7] holds with C = K,and
if [7] holds with C,then[6] for p = 1 holds with
K = C.
The sharp constant for the isoperimetric inequal-
ity [7] in Euclidean space is known. When n = 2 its
value is C(2) = 1=(4) and the sharp isoperimetric
inequality is the well-known inequality L
2
4A,
where A is the volume of a smooth bounded domain
in R
2
, and L is the length of its boundary. For
arbitrary n, the sharp constant C(n) for the isoperi-
metric inequality is given by
CðnÞ¼
1
n
n
!
n1
1=n
½8
where !
n1
is the volume of the unit (n 1)-sphere.
Moreover, still for the Euclidean space, equality
holds in the sharp isoperimetric inequality if and
only if is a ball. A famous conjecture concerning
sharp isoperimetric inequalities, often referred to as
the Cartan–Hadamard conjecture, is that the sharp
isoperimetric inequality holds on Cartan–Hadamard
manifolds. Thanks to works by Croke, Kleiner, and
Weil, the conjecture is known to be true in
dimensions 2, 3, and 4. From the Bishop–Gromov
comparison theorem, we also get that the only
complete manifold of non-negative Ricci curvature
for which the sharp isoperimetric inequality holds is
the Euclidean space itself.
The sharp constants K = K(n, p)for[6] when p > 1
have been computed in Euclidean space by Aubin,
Rodemich, and Talenti. The extremal functions were
also computed, where, by definition, an extremal
function is a function which realizes the case of
equality in the inequality. We get that
Kðn; pÞ¼
1
n
nðp 1Þ
n p
ðp1Þ=p
ðn þ 1Þ
ðn=pÞðn þ 1 n= pÞ!
n1
1=n
½9
where, as above, is the gamma function. More-
over, u is an extremal function for the sharp
inequality in Euclidean space if and only if, up to a
scale factor,
uxðÞ¼
2
þ
jx aj
p=ðp1Þ
nðn 2Þ
0
B
B
B
@
1
C
C
C
A
ðnpÞ=p
½10
Inequalities in Sobolev Spaces 35
for some >0, and a 2 R
n
.Whenp = 2, the
functions u in [10] are both the only extremal
functions for the sharp Sobolev inequality in Euclidean
space, and the only positive solutions of the equation
u = u
2
?
1
in R
n
,where=
P
i
D
2
i
is the Laplace–
Beltrami operator (the usual Laplacian with a minus
sign in front of it). Sharp constants are also known for
several of the Gagliardo–Nirenberg inequalities in
Euclidean space. The sharp constant for the Nash
inequality in Euclidean space was computed by Carlen
and Loss. If the sharp isoperimetric inequality holds on
a complete Riemannian n-manifold, then the sharp
inequalities [6] hold for all 1 p < n.
Sharp Inequalities on Compact
Riemannian Manifolds
The study of sharp Sobolev inequalities on compact
manifolds if often referred to as the AB program for
Sobolev inequalities. For (M, g ) a compact Rieman-
nian n-manifold, and 1 p < n, [5] can be rewritten
in two different forms:
Z
M
juj
p
?
dv
g
1=p
?
A
Z
M
jruj
p
dv
g
1=p
þ B
Z
M
juj
p
dv
g
1=p
½11
and
Z
M
juj
p
?
dv
g
p=p
?
A
0
Z
M
jruj
p
dv
g
þ B
0
Z
M
juj
p
dv
g
½12
where A, B, A
0
, B
0
are positive constants independent
of u. An easy remark is that if [12] holds with
constants A
0
and B
0
, then [11] holds with A = (A
0
)
1=p
and B = (B
0
)
1=p
. The sharp first (resp. second)
constants in [11] and [12] are defined as the lowest
possible values for A and A
0
(resp. for B and B
0
)in
[11] and [12]. The sharp first constants are
independent of the manifold and are given by
A
0
= A
p
= K(n, p)
p
, where K(n, p)isasin[9]. The
sharp second constants depend on the manifold
and are given by B
0
= B
p
= V
p=n
g
, where V
g
is the
volume of (M, g). A typical question in the AB
program is to know whether or not we can take A
or B to be the sharp constants in [11] and, similarly,
whether or not we can take A
0
or B
0
to be the sharp
constants in [12]. Another typical question in the AB
program is whether or not there are nonzero
extremal functions for the saturated form of the
sharp inequalities when they are valid. Concerning
the B-part of the program, the sharp inequality [11]
with B = V
1=n
g
is true on any manifold, and constant
functions are extremal functions. On the other hand,
it can be proved that the stronger [12] with
B
0
= V
p=n
g
is always false when p > 2, whatever
the manifold. Concerning the A-part of the
AB-program, Hebey and Vaugon proved that the
sharp inequality [12] with A
0
= K( n,2)
2
is true on
any manifold. In other words, for any compact
Riemannian manifold (M, g) of dimension n 3,
there exists B
0
> 0 such that, for any u 2 H
1, 2
(M),
Z
M
juj
2
?
dv
g
2=2
?
Kðn; 2Þ
2
Z
M
jruj
2
dv
g
þ B
0
Z
M
juj
2
dv
g
½13
We then get the saturated form of [13] by taking
B
0
= B
0
(g) to be the lowest possible B
0
in [13].In
general, when p 6¼ 2, we can prove that the sharp
inequality [11] with A = K(n, p)istrueonany
manifold, and that there are nonzero extremal
functions for the saturated form of the sharp inequal-
ity. On the other hand, the stronger [12] with
A
0
= K(n, p)
p
when p > 2 is false when the curvature
is positive, but true when the curvature is negative.
The p = 2caseintheA-part of the AB program is of
importance for its connection with the Yamabe
problem. The p = 1caseintheA-part of the AB
program is of importance for its connection with the
isoperimetric inequality. The AB program has also
been considered for Gagliardo–Nirenberg inequal-
ities, including the Nash inequality, and Sobolev–
Poincare´ inequalities on compact manifolds.
Further Reading
Adams RA (1978) Sobolev Spaces. San Diego: Academic Press.
Aubin T (1998) Some Nonlinear Problems in Riemannian
Geometry. Springer Monographs in Mathematics. Berlin:
Springer.
Druet O and Hebey E (2002) The AB Program in Geometric
Analysis: Sharp Sobolev Inequalities and Related Problems.
Memoirs of the American Mathematical Society, 160, 761.
American Mathematical Society.
Evans LC and Gariepy RF (1992) Measure Theory and Fine
Properties of Functions. CRC Press.
Hebey E (2000) Nonlinear Analysis on Manifolds: Sobolev Spaces
and Inequalities. Courant Lecture Notes in Mathematics, vol.
5. American Mathematical Society.
Maz’ja VG (1985) Sobolev Spaces, Springer Series in Soviet
Mathematics. Berlin: Springer.
36 Inequalities in Sobolev Spaces