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London Mathematical Society Lecture Note Series: 394
Variational Problems in
Differential Geometry
University of Leeds 2009
Edited by
R. BIELAWSKI
K. HOUSTON
J.M. SPEIGHT
University of Leeds

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Variational problems in differential geometry : University of Leeds, 2009 /
edited by R. Bielawski, K. Houston, J.M. Speight.
p. cm. – (London Mathematical Society lecture note series ; 394)
Includes bibliographical references.
ISBN 978-0-521-28274-1 (pbk.)
1. Geometry, Differential – Congresses. I. Bielawski, R. II. Houston, Kevin, 1968–
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Contents
List of contributors pageviii
Preface xi
1 The supremum of ﬁrst eigenvalues of conformally
covariant operators in a conformal class 1
Bernd Ammann and Pierre Jammes
1.1 Introduction 1
1.2 Preliminaries 4
1.3 Asymptotically cylindrical blowups 11
1.4 Proof of the main theorem 14
Appendix A Analysis on (M
∞,g∞)19
References 22
2K-Destabilizing test conﬁgurations with smooth
central ﬁber 24
Claudio Arezzo, Alberto Della Vedova, and Gabriele La Nave
2.1 Introduction 24
2.2 The case of normal singularities 29
2.3 Proof of Theorem 2.1.8 and examples 32
References 34
3 Explicit constructions of Ricci solitons 37
Paul Baird
3.1 Introduction 37
3.2 Solitons from a dynamical system 40
3.3 Reduction of the equations to a 2-dimensional system 44
3.4 Higher dimensional Ricci solitons via projection 48
3.5 The 4-dimensional geometry Nil
4 50
References 55
v

vi Contents
4 Open Iwasawa cells and applications to surface theory 56
Josef F. Dorfmeister
4.1 Introduction 56
4.2 Basic notation and the Birkhoff decomposition 58
4.3 Iwasawa decomposition 59
4.4 Iwasawa decomposition via Birkhoff decomposition 60
4.5 A function deﬁning the open Iwasawa cells 62
4.6 Applications to surface theory 64
References 66
5 Multiplier ideal sheaves and geometric problems 68
Akito Futaki and Yuji Sano
5.1 Introduction 68
5.2 An overview of multiplier ideal sheaves 72
5.3 Direct relationships between multiplier ideal sheaves and
the obstructionF 83
References 90
6 Multisymplectic formalism and the covariant phase space 94
Fr´ed´eric H´elein
6.1 The multisymplectic formalism 95
6.2 The covariant phase space 110
6.3 Geometric quantization 117
References 123
7 Nonnegative curvature on disk bundles 127
Lorenz J. Schwachh¨ofer
7.1 Introduction 127
7.2 Normal homogeneous metrics and Cheeger deformations 128
7.3 Homogeneous metrics of nonnegative curvature 130
7.4 Collar metrics of nonnegative curvature 131
7.5 Bundles with normal homogeneous collar 132
7.6 Cohomogeneity one manifolds 139
References 140
8 Morse theory and stable pairs 142
Richard A. Wentworth and Graeme Wilkin
8.1 Introduction 142
8.2 Stable pairs 146
8.3 Morse theory 154
8.4 Cohomology of moduli spaces 174
References 180

Contents vii
9 Manifolds withk-positive Ricci curvature 182
Jon Wolfson
9.1 Introduction 182
9.2 Manifolds withk-positive Ricci curvature 183
9.3 Fill radius and an approach to Conjecture 1 192
9.4 The fundamental group and ﬁll radius bounds 198
References 200

Contributors
Bernd Ammann
Facult¨at f¨ur Mathematik, Universit¨at Regensburg, 93040 Regensburg,
Germany
Pierre Jammes
Laboratoire J.-A. Dieudonn´e, Universit´e Nice – Sophia Antipolis, Parc
Valrose, F-06108 NICE Cedex 02, France
Claudio Arezzo
Abdus Salam International Center for Theoretical Physics, Strada Costiera
11, Trieste (Italy) and Dipartimento di Matematica, Universit`adiParma,
Parco Area delle Scienze 53/A, Parma, Italy
Alberto Della Vedova
Fine Hall, Princeton University, Princeton, NJ 08544 and Dipartimento di
Matematica, Universit`a di Parma, Parco Area delle Scienze 53/A, Parma, Italy
Gabriele La Nave
Department of Mathematics, Yeshiva University, 500 West 185 Street,
New York, NY, USA
Paul Baird
D´epartement de Math´ematiques, Universit´e de Bretagne Occidentale,
6 Avenue Le Gorgeu – CS 93837, 29238 Brest, France
Josef F. Dorfmeister
Fakult¨at f¨ur Mathematik, Technische Universit¨at M¨unchen, Boltzmannstr. 3,
D-85747 Garching, Germany
viii

List of contributors ix
Akito Futaki
Department of Mathematics, Tokyo Institute of Technology, 2-12-1,
O-okayama, Meguro, Tokyo 152-8551, Japan
Yuji Sano
Department of Mathematics, Kyushu University, 6-10-1, Hakozaki,
Higashiku, Fukuoka-city, Fukuoka 812-8581 Japan
Fr´ed´eric H´elein
Institut de Math´ematiques de Jussieu, UMR CNRS 7586, Universit´e Denis
Diderot Paris 7, 175 rue du Chevaleret, 75013 Paris, France
Lorenz J. Schwachh¨ofer
Fakult¨at f¨ur Mathematik, Technische Universit¨at Dortmund, Vogelpothsweg
87, 44221 Dortmund, Germany
Richard A. Wentworth
Department of Mathematics, University of Maryland, College Park, MD
20742, USA
Graeme Wilkin
Department of Mathematics, University of Colorado, Boulder, CO 80309,
USA
Jon Wolfson
Department of Mathematics, Michigan State University, East Lansing,
MI 48824, USA

Preface
The workshopVariational Problems in Differential Geometrywas held at the
University of Leeds from March 30 to April 2nd, 2009.
The aim of the meeting was to bring together researchers working on
disparate geometric problems, all of which admit a variational formulation.
Among the topics discussed were recent developments in harmonic maps and
morphisms, minimal and CMC surfaces, extremal K¨ahler metrics, the Yam-
abe functional, Hamiltonian variational problems, and topics related to gauge
theory and to the Ricci ﬂow.
The meeting incorporated a special session in honour of John C. Wood, on
the occasion of his 60th birthday, to celebrate his seminal contributions to the
theory of harmonic maps and morphisms.
The following mathematicians gave one-hour talks: Bernd Ammann, Clau-
dio Arezzo, Paul Baird, Olivier Biquard, Christoph Boehm, Francis Burstall,
Josef Dorfmeister, Akito Futaki, Mark Haskins, Frederic Helein, Nicolaos
Kapouleas, Mario Micallef, Frank Pacard, Simon Salamon, Lorenz Schwach-
hoefer, Peter Topping, Richard Wentworth, and Jon Wolfson.
There were about 50 participants from the UK, US, Japan and several Euro-
pean countries. The schedule allowed plenty of opportunities for discussion
and interaction between ofﬁcial talks and made for a successful and stimulat-
ing meeting.
The workshop was ﬁnancially supported by the London Mathematical Soci-
ety, the Engineering and Physical Sciences Research Council of Great Britain
and the School of Mathematics, University of Leeds.
The articles presented in this volume represent the whole spectrum of the
subject.
The supremum of ﬁrst eigenvalues of conformally covariant operators in a
conformal classby Ammann and Jammes is concerned with the ﬁrst eigenvalues
of the Yamabe operator, the Dirac operator, and more general conformally
xi

xii Preface
covariant elliptic operators on compact Riemannian manifolds. It is well known
that the inﬁmum of the ﬁrst eigenvalue in a given conformal class reﬂects a rich
geometric structure. In this article, the authors study the supremum of the ﬁrst
eigenvalue and show that, for a very general class of operators, this supremum
is inﬁnite.
The article,K-Destabilizing test conﬁgurations with smooth central ﬁber
by Arezzo, Della Vedova, and La Nave is concerned with the famous Tian-
Yau-Donaldson conjecture about existence of constant scalar curvature K¨ahler
metrics. They construct many new families ofK-unstable manifolds, and,
consequently, many new examples of manifolds which do not admit K¨ahler
constant scalar curvature metrics in some cohomology classes.
As has been now understood, a very natural extension of Einstein metrics
are the Ricci solitons. These are the subject of Paul Baird’s articleExplicit
constructions of Ricci solitons, in which he does precisely that: he constructs
many explicit examples, including some in the more exotic geometries Sol
3,
Nil
3, and Nil4.
Josef Dorfmeister is concerned with a more classical topic: that of constant
mean curvature and Willmore surfaces. In recent years, many new examples of
such surfaces were constructed using loop groups. The method relies on ﬁnding
“Iwasawa-like” decompositions of loop groups and the articleOpen Iwasawa
cells in twisted loop groups and some applications to harmonic mapsdiscusses
such decompositions and their singularities.
The currently extremely important notions ofK-stability andK-
polystability are the topic of the paper by Futaki and SanoMultiplier ideal
sheaves and geometric problems. This is an expository article giving state-of-
the-art presentation of the powerful method of multiplier ideal sheaves and
their applications to K¨ahler-Einstein and Sasaki-Einstein geometries.
Multisymplectic formalism and the covariant phase spaceby Fr´ed´eric H´elein
takes us outside Riemannian geometry. The author presents an alternative (in
fact, two of them) to the Feynman integral as a foundation of quantum ﬁeld
theory.
Lorenz Schwachh¨ofer’sNonnegative curvature on disk bundlesis a survey of
the glueing method used to construct Riemannian manifolds with nonnegative
sectional curvature - one of the classical problems in geometry.
Morse theory and stable pairsby Wentworth and Wilkin introduces new
techniques to compute equivariant cohomology of certain natural moduli
spaces. The main ingredient is a version of Morse-Atiyah-Bott theory adapted
to singular inﬁnite dimensional spaces.
The ﬁnal article,Manifolds withk-positive Ricci curvature,byJonWolf-
son, is a survey of results and conjectures about Riemanniann-manifolds with

Preface xiii
k-positive Ricci curvature. These interpolate between positive scalar curva-
ture (n-positive Ricci curvature) and positive Ricci curvature (1-positive Ricci
curvature), and the author shows how the results aboutk-positive Ricci curva-
ture, 1<k<n, also interpolate, or should do, between what is known about
manifolds satisfying those two classical notions of positivity.
We would like to extend our thanks to our colleague John Wood for his help
and assistance in preparing these proceedings.
R. Bielawski
K. Houston
J.M. Speight
Leeds, UK

1
The supremum of ﬁrst eigenvalues of
conformally covariant operators
in a conformal class
bernd ammann and pierre jammes
Abstract
Let (M,g) be a compact Riemannian manifold of dimension≥3. We show that
there is a metric˜gconformal togand of volume 1 such that the ﬁrst positive
eigenvalue of the conformal Laplacian with respect to˜gis arbitrarily large.
A similar statement is proven for the ﬁrst positive eigenvalue of the Dirac
operator on a spin manifold of dimension≥2.
1.1 Introduction
The goal of this article is to prove the following theorems.
Theorem 1.1.1Let(M,g
0,χ)be compact Riemannian spin manifold of
dimensionn≥2. For any metricgin the conformal class[g
0], we denote
the ﬁrst positive eigenvalue of the Dirac operator on(M,g,χ)byλ
+
1
(Dg).
Then
sup
g∈[g 0]
λ
+
1
(Dg)Vol(M,g)
1/n
=∞.
Theorem 1.1.2Let(M,g
0,χ)be compact Riemannian manifold of dimension
n≥3. For any metricgin the conformal class[g
0], we denote the ﬁrst positive
eigenvalue of the conformal LaplacianL
g:=∞ g+
n−2
4(n−1)
Scalg(also called
Yamabe operator) on(M,g,χ)byλ
+
1
(Lg). Then
sup
g∈[g 0]
λ
+
1
(Lg)Vol(M,g)
2/n
=∞.
The Dirac operator and the conformal Laplacian belong to a large fam-
ily of operators, deﬁned in details in subsection1.2.3. These operators are
1

2 B. Ammann and P. Jammes
called conformally covariant elliptic operators of orderkand of bidegree
((n−k)/2,(n+k)/2), acting on manifolds (M,g)ofdimensionn>k.In
our deﬁnition we also claim formal self-adjointness.
All such conformally covariant elliptic operators of orderkand of bidegree
((n−k)/2,(n+k)/2) share several analytical properties, in particular they are
associated to the non-compact embeddingH
k/2
→L
2n/(n−k)
. Often they have
interpretations in conformal geometry. To give an example, we deﬁne for a
compact Riemannian manifold (M,g
0)
Y(M,[g
0]) :=inf
g∈[g 0]
λ1(Lg)Vol(M,g)
2/n
,
whereλ
1(Lg) is the lowest eigenvalue ofL g.IfY(M,[g 0])>0, then the
solution of the Yamabe problem [29] tells us that the inﬁmum is attained and
the minimizer is a metric of constant scalar curvature. This famous problem
was ﬁnally solved by Schoen and Yau using the positive mass theorem.
In a similar way, forn=2 the Dirac operator is associated to constant-mean-
curvature conformal immersions of the universal covering intoR
3
. If a Dirac-
operator-analogue of the positive mass theorem holds for a given manifold
(M,g
0), then the inﬁmum
inf
g∈[g 0]
λ
+
1
(Dg)Vol(M,g)
1/n
is attained [3]. However, it is still unclear whether such a Dirac-operator-
analogue of the positive mass theorem holds in general.
The Yamabe problem and its Dirac operator analogue, as well as the
analogues for other conformally covariant operators are typically solved by
minimizing an associated variational problem. As the Sobolev embedding
H
k/2
→L
2n/(n−k)
is non-compact, the direct method of the calculus of variation
fails, but perturbation techniques and conformal blow-up techniques typically
work. Hence all these operators share many properties.
However, only few statements can be proven simultaneously for all confor-
mally covariant elliptic operators of orderkand of bidegree ((n−k)/2,(n+
k)/2). Some of the operators are bounded from below (e.g. the Yamabe and
the Paneitz operator), whereas others are not (e.g. the Dirac operator). Some
of them admit a maximum principle, others do not. Some of them act on func-
tions, others on sections of vector bundles. The associated Sobolev spaceH
k/2
has non-integer order ifkis odd, hence it is not the natural domain of a dif-
ferential operator. For Dirac operators, the spin structure has to be considered
in order to derive a statement as Theorem1.1.1forn=2. Because of these
differences, most analytical properties have to be proven for each operator
separately.

The supremum of ﬁrst eigenvalues 3
We consider it hence as remarkable that the proof of our Theorems1.1.1
and1.1.2can be extended to all such operators. Our proof only uses some few
properties of the operators, deﬁned axiomatically in1.2.3. More exactly we
prove the following.
Theorem 1.1.3LetP
gbe a conformally covariant elliptic operator of order
k,ofbidegree((n−k)/2,(n+k)/2)acting on manifolds of dimensionn>k.
We also assume thatP
gis invertible onS
n−1
×R(see Deﬁnition1.2.4). Let
(M,g
0)be compact Riemannian manifold. In the case thatP gdepends on
the spin structure, we assume thatMis oriented and is equipped with a spin
structure. For any metricgin the conformal class[g
0], we denote the ﬁrst
positive eigenvalue ofP
gbyλ
+
1
(Pg). Then
sup
g∈[g 0]
λ
+
1
(Pg)Vol(M,g)
k/n
=∞.
The interest in this result is motivated by three questions. At ﬁrst, as already
mentioned above the inﬁmum
inf
g∈[g 0]
λ
+
1
(Dg)Vol(M,g)
1/n
reﬂects a rich geometrical structure [3], [4], [5], [7], [8], similarly for the
conformal Laplacian. It seems natural to study the supremum as well.
The second motivation comes from comparing Theorem1.1.3to results
about some other differential operators. For the Hodge Laplacian∞
g
p
acting
onp-forms, we have sup
g∈[g 0]
λ1(∞
g
p
)Vol(M,g)
2/n
=+∞forn≥4 and 2≤
p≤n−2([19]). On the other hand, for the Laplacian∞
g
acting on functions,
we have
sup
g∈[g 0]
λk(∞
g
)Vol(M,g)
2/n
<+∞
(the casek=1isprovenin[20] and the general case in [27]). See [25]fora
synthetic presentation of this subject.
The essential idea in our proof is to construct metrics with longer and longer
cylindrical parts. We will call this anasymptotically cylindrical blowup. Such
metrics are also calledPinocchio metricsin [2,6]. In [2,6] the behavior of Dirac
eigenvalues on such metrics has already been studied partially, but the present
article has much stronger results. To extend these existing results provides the
third motivation.
AcknowledgmentsWe thank B. Colbois, M. Dahl, and E. Humbert for
many related discussions. We thank R. Gover for some helpful comments on
conformally covariant operators, and for several references. The ﬁrst author

4 B. Ammann and P. Jammes
wants to thank the Albert Einstein institute at Potsdam-Golm for its very kind
hospitality which enabled to write the article.
1.2 Preliminaries
1.2.1 Notations
In this articleB y(r) denotes the ball of radiusraroundy,S y(r)=∂B y(r)
its boundary. The standard sphereS
0(1)⊂R
n
inR
n
is denoted byS
n−1
, its
volume isω
n−1. For the volume element of (M,g) we use the notationdv
g
.In
our article,→(V) (resp.→
c(V)) always denotes the set of all smooth sections
(resp. all compactly supported smooth sections) of the vector bundleV→M.
For sectionsuofV→Mover a Riemannian manifold (M,g) the Sobolev
normsL
2
andH
s
,s∈N, are deﬁned as
u
2
L
2
(M,g)
:=
χ
M
|u|
2
dv
g
u
2
H
s
(M,g)
:=u
2
L
2
(M,g)
+∇u
2
L
2
(M,g)
+···+∇
s
u
2
L
2
(M,g)
.
The vector bundleVwill be suppressed in the notation. IfMandg
are clear from the context, we write justL
2
andH
s
. The completion of
{u∈→(V)|u
H
s
(M,g)<∞}with respect to theH
s
(M,g)-norm is denoted
by→
H
s
(M,g)(V), or if (M,g)orVis clear from the context, we alternatively
write→
H
s(V)orH
s
(M,g)for→ H
s
(M,g)(V). The same deﬁnitions are used for
L
2
instead ofH
s
. And similarly→ C
k
(M,g)(V)=→ C
k(V)=C
k
(M,g)istheset
of allC
k
-sections,k∈N∪{∞}.
1.2.2 Removal of singularities
In the proof we will use the following removal of singularities lemma.
Lemma 1.2.1(Removal of singularities lemma)Let≤be a bounded open
subset ofR
n
containing0. LetPbe an elliptic differential operator of orderk
on≤,f∈C
∞
(≤), and letu∈C
∞
(≤\{0})be a solution of
Pu=f (1.1)
on≤\{0}with
lim
ε→0
χ
B0(2ε)−B 0(ε)
|u|r
−k
=0andlim
ε→0
χ
B0(ε)
|u|=0 (1.2)

The supremum of ﬁrst eigenvalues 5
whereris the distance to0. Thenuis a (strong) solution of(1.1)on≤. The
same result holds for sections of vector bundles over relatively compact open
subset of Riemannian manifolds.
ProofWe show thatuis a weak solution of (1.1) in the distributional sense, and
then it follows from standard regularity theory, that it is also a strong solution.
This means that we have to show that for any given compactly supported smooth
test functionψ:≤→Rwe have
χ
≤
uP
∗
ψ=
χ
≤
fψ.
Letη:≤→[0,1] be a test function that is identically 1 onB
0(ε), has
support inB
0(2ε), and with|∇
m
η|≤C m/ε
m
. It follows that
sup|P
∗
(ηψ)|≤C(P,≤,)ε
−k
,
onB
0(2ε)\B 0(ε) and sup|P
∗
(ηψ)|≤C(P,≤,)onB 0(ε) and hence
λ
λ
λ
λ
χ
≤
uP
∗
(ηψ)
λ
λ
λ
λ
≤Cε
−k
χ
B0(2ε)\B 0(ε)
|u|+C
χ
B0(ε)
|u|
≤C
χ
B0(2ε)\B 0(ε)
|u|r
−k
+C
χ
B0(ε)
|u|→0.
(1.3)
We conclude
χ
≤
uP
∗
ψ=
χ
≤
uP
∗
(ηψ)+
χ
≤
uP
∗
((1−η)ψ)
=
χ
≤
uP
∗
(ηψ)
∞
≥∈→
→0
+
χ
≤
(Pu)(1−η)ψ
∞
≥∈ →
→
≤
≤
fψ
(1.4)
forε→0. Hence the lemma follows. χ
Condition (1.2) is obviously satisﬁed if
≤
≤
|u|r
−k
<∞. It is also satisﬁed if
χ
≤
|u|
2
r
−k
<∞andk≤n, (1.5)
as in this case
χ
B0(2ε)\B 0(ε)
|u|r
−k

2
≤
χ
≤
|u|
2
r
−k
χ
B0(2ε)\B 0(ε)
r
−k
∞
≥∈ →
≤C
.

6 B. Ammann and P. Jammes
1.2.3 Conformally covariant elliptic operators
In this subsection we present a class of certain conformally covariant elliptic
operators. Many important geometric operators are in this class, in particular
the conformal Laplacian, the Paneitz operator, the Dirac operator, see also
[18,21,22] for more examples. Readers who are only interested in the Dirac
operator, the Conformal Laplacian or the Paneitz operator, can skip this part
and continue with section1.3.
Such a conformally covariant operator is not just one single differential oper-
ator, but a procedure how to associate to ann-dimensional Riemannian manifold
(M,g) (potentially with some additional structure) a differential operatorP
g
of orderkacting on a vector bundle. The important fact is that ifg 2=f
2
g1,
then one claims
P
g2
=f
−
n+k
2Pg1
f
n−k
2. (1.6)
One also expresses this by saying thatPhas bidegree ((n−k)/2,(n+k)/2).
The sense of this equation is apparent ifP
gis an operator fromC
∞
(M)
toC
∞
(M). IfP gacts on a vector bundle or if some additional structure (as
e.g. spin structure) is used for deﬁning it, then a rigorous and careful deﬁni-
tion needs more attention. The language of categories provides a good formal
framework [30]. The concept of conformally covariant elliptic operators is
already used by many authors, but we do not know of a reference where a
formal deﬁnition is carried out that ﬁts to our context. (See [26] for a similar
categorial approach that includes some of the operators presented here.) Often
an intuitive deﬁnition is used. The intuitive deﬁnition is obviously sufﬁcient if
one deals with operators acting on functions, such as the conformal Laplacian
or the Paneitz operator. However to properly state Theorem1.1.3we need the
following deﬁnition.
LetRiem
n
(resp.Riemspin
n
) be the categoryn-dimensional Riemannian
manifolds (resp.n-dimensional Riemannian manifolds with orientation and
spin structure). Morphisms from (M
1,g1)to(M 2,g2) are conformal embed-
dings (M
1,g1)→(M 2,g2) (resp. conformal embeddings preserving orienta-
tion and spin structure).
LetLaplace
n
k
(resp.Dirac
n
k
) be the category whose objects are
{(M,g),V
g,Pg}
where (M,g) in an object ofRiem
n
(resp.Riemspin
n
), whereV gis a vector
bundle with a scalar product on the ﬁbers, whereP
g:→(V g)→→(V g)isan
elliptic formally self-adjoint differential operator of orderk.

The supremum of ﬁrst eigenvalues 7
A morphism (ι, κ) from{(M
1,g1),Vg1
,Pg1
}to{(M 2,g2),Vg2
,Pg2
}consists
of a conformal embeddingι:(M
1,g1)→(M 2,g2) (preserving orientation
and spin structure in the case ofDirac
n
k
) together with a ﬁber isomorphism
κ:ι
∗
Vg2
→V g1
preserving ﬁberwise length, such thatP g1
andP g2
sat-
isfy the conformal covariance property (1.6). For stating this property pre-
cisely, letf>0 be deﬁned byι
∗
g2=f
2
g1, and letκ ∗:→(V g2
)→→(V g1
),
κ
∗(ϕ)=κ◦ϕ◦ι. Then the conformal covariance property is
κ
∗Pg2
=f
−
n+k
2Pg1
f
n−k
2κ∗. (1.7)
In the following the mapsκandιwill often be evident from the context
and then will be omitted. The transformation formula (1.7) then simpliﬁes
to (1.6).
Deﬁnition 1.2.2Aconformally covariant elliptic operator of orderkand of
bidegree((n−k)/2,(n+k)/2) is a contravariant functor fromRiem
n
(resp.
Riemspin
n
)toLaplace
n
k
(resp.Dirac
n
k
), mapping (M,g)to(M,g,V g,Pg)in
such a way that the coefﬁcients are continuous in theC
k
-topology of metrics
(see below). To shorten notation, we just writeP
gorPfor this functor.
It remains to explain theC
k
-continuity of the coefﬁcients.
For Riemannian metricsg,g
1,g2deﬁned on a compact setK⊂Mwe set
d
g
C
k
(K)
(g1,g2):=max
t=0,...,k
(∇g)
t
(g1−g2)C
0
(K).
For a ﬁxed background metricg, the relationd
g
C
k
(K)
(·,·) deﬁnes a distance
function on the space of metrics onK. The topology induced byd
g
is inde-
pendent of this background metric and it is called theC
k
-topology of metrics
onK.
Deﬁnition 1.2.3We say that the coefﬁcients ofParecontinuous in theC
k
-
topology of metricsif for any metricgon a manifoldM, and for any compact
subsetK⊂Mthere is a neighborhoodUofg|
Kin theC
k
-topology of met-
rics onK, such that for all metrics˜g,˜g|
K∈U, there is an isomorphism of
vector bundles ˆκ:V
g|K→V ˜g|Kover the identity ofKwith induced map
ˆκ
∗:→(V g|K)→→(V ˜g|K) with the property that the coefﬁcients of the differ-
ential operator
P
g−(ˆκ∗)
−1
P˜gˆκ∗
depend continuously on˜g(with respect to theC
k
-topology of metrics).

8 B. Ammann and P. Jammes
1.2.4 Invertibility onS
n−1
×R
LetPbe a conformally covariant elliptic operator of orderkand of bide-
gree ((n−k)/2,(n+k)/2). For (M,g)=S
n−1
×R, the operatorP gis a
self-adjoint operatorH
k
⊂L
2
→L
2
(see Lemma1.3.1and the comments
thereafter).
Deﬁnition 1.2.4We say thatPis invertible onS
n−1
×RifP gis an invertible
operatorH
k
→L
2
wheregis the standard product metric onS
n−1
×R.In
order words there is a constantσ>0 such that the spectrum ofP
g:→H
k(Vg)→
→
L
2(Vg) is contained in (−∞,−σ]∪[σ,∞) for anyg∈U. In the following,
the largest suchσwill be calledσ
P.
We conjecture that any conformally covariant elliptic operator of orderk
and of bidegree ((n−k)/2,(n+k)/2) withk<nis invertible onS
n−1
×R.
1.2.5 Examples
Example 1:The Conformal Laplacian
Let
L
g:=∞ g+
n−2
4(n−1)
Scal
g,
be the conformal Laplacian. It acts on functions on a Riemannian manifold
(M,g), i.e.V
gis the trivial real line bundleR
.Letι:(M 1,g1)→(M 2,g2)
be a conformal embedding. Then we can chooseκ:=Id :ι
∗
Vg2
→V g1
and
formula (1.7) holds fork=2 (see e.g. [15, Section 1.J]). All coefﬁcients of
L
gdepend continuously ongin theC
2
-topology. HenceLis a conformally
covariant elliptic operator of order 2 and of bidegree ((n−2)/2,(n+2)/2).
The scalar curvature ofS
n−1
×Ris (n−1)(n−2). The spectrum ofL gon
S
n−1
×RofL gcoincides with the essential spectrum ofL gand is [σ L,∞) with
σ
L:=(n−2)
2
/4. HenceLis invertible onS
n−1
×Rif (and only if)n>2.
Example 2:The Paneitz operator
Let (M,g) be a smooth, compact Riemannian manifold of dimensionn≥5.
The Paneitz operatorP
gis given by
P
gu=(∞ g)
2
u−div g(Agdu)+
n−4
2
Q
gu
where
A
g:=
(n−2)
2
+4
2(n−1)(n−2)
Scal
gg−
4
n−2
Ric
g,
Q
g=
1
2(n−1)
∞
gScalg+
n
3
−4n
2
+16n−16
8(n−1)
2
(n−2)
2
Scal
2
g
−
2
(n−2)
2
|Ricg|
2
.

The supremum of ﬁrst eigenvalues 9
This operator was deﬁned by Paneitz [32] in the casen=4, and it was general-
ized by Branson in [17] to arbitrary dimensions≥4. We also refer to Theorem
1.21 of the overview article [16]. The explicit formula presented above can
be found e.g. in [23]. The coefﬁcients ofP
gdepend continuously ongin the
C
4
-topology
As in the previous example we can choose forκthe identity, and then the
Paneitz operatorP
gis a conformally covariant elliptic operator of order 4 and
of bidegree ((n−4)/2,(n+4)/2).
OnS
n−1
×Rone calculates
A
g=
(n−4)n
2
Id+4π
R>0
whereπ
Ris the projection to vectors parallel toR.
Q
g=
(n−4)n
2
8
.
We conclude
σ
P=Q=
(n−4)n
2
8
andPis invertible onS
n−1
×Rif (and only if)n>4.
Examples 3:The Dirac operator.
Let˜g=f
2
g.Let gMresp. ˜gMbe the spinor bundle of (M,g) resp.
(M,˜g). Then there is a ﬁberwise isomorphismβ
g
˜g
:gM→ ˜gM, preserving
the norm such that
D
˜g◦β
g
˜g
(ϕ)=f
−
n+1
2β
g
˜g
◦Dg

f
n−1
2ϕ

,
see [24,14] for details. Furthermore, the cocycle conditions
β
g
˜g
◦β
˜g
g
=Id and β
ˆg
g
◦β
˜g
ˆg
◦β
g
˜g
=Id
hold for conformal metricsg,˜gandˆg. We will hence use the mapβ
g
˜g
to identify

gMwith ˜gM. Hence we simply get
D
˜gϕ=f
−
n+1
2◦Dg

f
n−1
2ϕ

. (1.8)
All coefﬁcients ofD
gdepend continuously ongin theC
1
-topology. Hence
Dis a conformally covariant elliptic operator of order 1 and of bidegree
((n−1)/2,(n+1)/2).
The Dirac operator onS
n−1
×Rcan be decomposed in a partD vertderiving
alongS
n−1
and a partD horderiving alongR,D g=Dvert+Dhor,see[1]or[2].

10 B. Ammann and P. Jammes
Locally
D
vert=
n−1

i=1
ei·∇ei
for a local frame (e 1,...,en−1)ofS
n−1
.Here·denotes the Clifford multi-
plicationTM⊗
gM→ gM. FurthermoreD hor=∂t·∇∂t
, wheret∈Ris
the standard coordinate ofR. The operatorsD
vertandD horanticommute. For
n≥3, the spectrum ofD
vertcoincides with the spectrum of the Dirac operator
onS
n−1
, we cite [12] and obtain
specD
vert=
◦
±

n−1
2
+k

|k∈N
0
⊗
.
The operator (D
hor)
2
is the ordinary Laplacian onRand hence has spectrum
[0,∞). Together this implies that the spectrum of the Dirac operator onS
n−1
×
Ris the set (−∞,−σ
D]∪[σ D,∞) withσ D=
n−1
2
.
In the casen=2 these statements are only correct if the circleS
n−1
=S
1
carries the spin structure induced from the ball. Only this spin structure extends
to the conformal compactiﬁcation that is given by adding one point at inﬁnity
for each end. For this reason, we will understand in the whole article that all
circlesS
1
should be equipped with this bounding spin structure. The exten-
sion of the spin structure is essential in order to have a spinor bundle on the
compactiﬁcation. The methods used in our proof use this extension implicitly.
HenceDis invertible onS
n−1
×Rif (and only if)n>1.
Most techniques used in the literature on estimating eigenvalues of the
Dirac operators do not use the spin structure and hence these techniques cannot
provide a proof in the casen=2.
Example 4:The Rarita-Schwinger operator and many other Fegan type
operators are conformally covariant elliptic operators of order 1 and of bide-
gree ((n−1)/2,(n+1)/2). See [21] and in the work of T. Branson for more
information.
Example 5:Assume that (M,g) is a Riemannian spin manifold that carries
a vector bundleW→Mwith metric and metric connection. Then there is a
natural ﬁrst order operator→(
gM⊗W)→→( gM⊗W), theDirac opera-
tor twisted byW. This operator has similar properties as conformally covariant
elliptic operators of order 1 and of bidegree ((n−1)/2,(n+1)/2). The meth-
ods of our article can be easily adapted in order to show that Theorem1.1.3
is also true for this twisted Dirac operator. However, twisted Dirac operators
are not “conformally covariant elliptic operators” in the above sense. They
could have been included in this class by replacing the categoryRiemspin
n
by

The supremum of ﬁrst eigenvalues 11
Figure 1.1Asymptotically cylindrical metricsg L(alias Pinocchio metrics) with
growing nose lengthL.
a category of Riemannian spin manifolds with twisting bundles. In order not to
overload the formalism we chose not to present these larger categories.
The same discussion applies to the spin
c
-Dirac operator of a spin
c
-manifold.
1.3 Asymptotically cylindrical blowups
1.3.1 Convention
From now on we suppose thatP gis a conformally covariant elliptic operator of
orderk,ofbidegree((n−k)/2,(n+k)/2), acting on manifolds of dimension
nand invertible onS
n−1
×R.
1.3.2 Deﬁnition of the metrics
Letg 0be a Riemannian metric on a compact manifoldM. We can suppose
that the injectivity radius in a ﬁxed pointy∈Mis larger than 1. The geodesic
distance fromytoxis denoted byd(x,y).
We choose a smooth functionF
∞:M\{y}→[1,∞) such such that
F
∞(x)=1ifd(x,y)≥1,F ∞(x)≤2ifd(x,y)≥1/2 and such thatF ∞(x)=
d(x,y)
−1
ifd(x,y)∈(0,1/2]. Then forL≥1 we deﬁneF Lto be a smooth
positive function onM, depending only ond(x,y), such thatF
L(x)=F ∞(x)
ifd(x,y)≥e
−L
andF L(x)≤d(x,y)
−1
=F∞(x)ifd(x,y)≤e
−L
.
For anyL≥1orL=∞setg
L:=F
2
L
g0. The metricg ∞is a complete
metric onM
∞.
The family of metrics (g
L) is called anasymptotically cylindrical blowup,
in the literature it is denoted as a family ofPinocchio metrics[6], see also
Figure1.1.
1.3.3 Eigenvalues and basic properties on(M,g L)
For theP-operator associated to (M,g L),L∈{0}∪[1,∞) (or more exactly
its self-adjoint extension) we simply writeP
Linstead ofP gL
.AsMis compact
the spectrum ofP
Lis discrete.

12 B. Ammann and P. Jammes
We will denote the spectrum ofP
Lin the following way
...≤λ
−
1
(PL)<0=0...=0<λ
+
1
(PL)≤λ
+
2
(PL)≤...,
where each eigenvalue appears with the multiplicity corresponding to the
dimension of the eigenspace. The zeros might appear on this list or not, depend-
ing on whetherP
Lis invertible or not. The spectrum might be entirely positive
(for example the conformal LaplacianY
gon the sphere) in which caseλ
−
1
(PL)
is not deﬁned. Similarly,λ
+
1
(PL) is not deﬁned if the spectrum of (P L)is
negative.
1.3.4 Analytical facts about(M ∞,g∞)
The analysis of non-compact manifolds as (M ∞,g∞) is more complicated than
in the compact case. Nevertheless (M
∞,g∞) is an asymptotically cylindrical
manifold, and for such manifolds an extensive literature is available. One pos-
sible approach would be Melrose’s b-calculus [31]: our cylindrical manifold is
such ab-manifold, but for simplicity and self-containedness we avoid this the-
ory. We will need some few properties that we will summarize in the following
proposition.
We assume in the whole section thatPis a conformally covariant elliptic
operator that is invertible onS
n−1
×R, and we writeP ∞:=P g∞
for the operator
acting on sections of the bundleVover (M
∞,g∞).
Proposition 1.3.1P
∞extends to a bounded operator from
→
H
k
(M∞,g∞)(V)→→ L
2
(M∞,g∞)(V)
and it satisﬁes the following regularity estimate
(∇
∞
)
s
uL
2
(M∞,g∞)≤C(u L
2
(M∞,g∞)+P ∞uL
2
(M∞,g∞)) (1.9)
for allu∈→
H
k
(M∞,g∞)(V)and alls∈{0,1,...,k}. The operator
P
∞:→H
k
(M∞,g∞)(V)→→ L
2
(M∞,g∞)(V)
is self-adjoint in the sense of an operator in→
L
2
(M∞,g∞)(V).
The proof of the proposition will be sketched in the appendix.
Proposition 1.3.2The essential spectrum ofP
∞coincides with the essen-
tial spectrum of theP-operator on the standard cylinderS
n−1
×R. Thus the
essential spectrum ofP
∞is contained in(−∞,−σ P]∪[σ P,∞).

The supremum of ﬁrst eigenvalues 13
This proposition follows from the characterization of the essential spectrum
in terms of Weyl sequences, a well-known technique which is for example
carried out and well explained in [13].
The second proposition states that the spectrum ofP
∞in the interval
(−σ
P,σP) is discrete as well. Eigenvalues ofP ∞in this interval will be called
small eigenvalues ofP
∞. Similarly to above we use the notationλ
±
j
(P∞)for
the small eigenvalues ofP
∞.
1.3.5 The kernel
Having recalled these well-known facts we will now study the kernel of con-
formally covariant operators.
Ifgand˜g=f
2
are conformal metrics on a compact manifoldM, then
ϕσ →f
−
n−k
2ϕ
obviously deﬁnes an isomorphism from kerP
gto kerP ˜g. It is less obvious that
a similar statement holds if we compareg
0andg ∞deﬁned before:
Proposition 1.3.3The map
kerP
0→kerP ∞
ϕ0σ →ϕ ∞=F
−
n−k
2
∞ϕ0
is an isomorphism of vector spaces.
ProofSupposeϕ
0∈kerP 0. Using standard regularity results it is clear that
sup|ϕ
0|<∞. Then
χ
M∞
|ϕ∞|
2
dv
g∞
≤
χ
M\B y(1/2)
|ϕ∞|
2
dv
g∞
+sup|ϕ 0|
2
χ
By(1/2)
F
−(n−k)
∞
dv
g∞
≤2
k
χ
M\B y(1/2)
|ϕ0|
2
dv
g0
+sup|ϕ 0|
2
ωn−1
χ
1/2
0
r
n−1
r
k
dr<∞.
(1.10)
Here we used that up to lower order termsdv
g∞
coincides with the product
measure of the standard measure on the sphere with the measured(logr)=
1
r
dr. Furthermore, formula (1.6) impliesP ∞ϕ∞=0. Hence the map is well-
deﬁned. In order to show that it is an isomorphism we show that the obvious
inverseϕ
∞σ →ϕ 0:=F
n−k
2
∞ϕ∞is well deﬁned. To see this we start with an
L
2
-section in the kernel ofP ∞.

14 B. Ammann and P. Jammes
We calculate
χ
M
F
k
∞
|ϕ0|
2
dv
g0
=
χ
M∞
|ϕ∞|
2
dv
g∞
.
Using again (1.6) we see that this section satisﬁesP
0ϕ0=0onM\{y}. Hence
condition (1.5) is satisﬁed, and together with the removal of singularity lemma
(Lemma1.2.1) one obtains that the inverse map is well deﬁned. The proposition
follows. χ
1.4 Proof of the main theorem
1.4.1 Stronger version of the main theorem
We will now show the following theorem.
Theorem 1.4.1LetPbe a conformally covariant elliptic operator of order
k,ofbidegree((n−k)/2,(n+k)/2), on manifolds of dimensionn>k.We
assume thatPis invertible onS
n−1
×R.
Iflim inf
L→∞|λ
±
j
(PL)|<σ P, then
λ
±
j
(PL)→λ
±
j
(P∞)∈(−σ P,σP)forL→∞.
In the case Spec(P
g0
)⊂(0,∞) the theorem only makes a statement about
λ
+
j
, and conversely in the case that Spec(P g0
)⊂(−∞,0) it only makes a
statement aboutλ
−
j
.
Obviously this theorem implies Theorem1.1.3.
1.4.2 The supremum part of the proof of Theorem1.4.1
At ﬁrst we prove that
lim sup
L→∞
(λ
+
j
(PL))≤λ
+
j
(P∞). (1.11)
Letϕ
1,...,ϕjbe sequence ofL
2
-orthonormal eigenvectors ofP ∞to
eigenvaluesλ
+
1
(P∞),...,λ
+
j
(P∞)∈[−ˉλ,ˉλ],ˉλ<σ P. We choose a cut-off
functionχ:M→[0,1] withχ(x)=1for−log(d(x,y))≤T,χ(y)=0
for−log(d(x,y))≥2T, and|(∇
∞
)
s
χ|g∞
≤Cs/T
s
for alls∈{0,...,k}.
Letϕbe a linear combination of the eigenvectorsϕ
1,...,ϕj. From Propo-
sition1.3.1we see that
(∇
∞
)
s
ϕL
2
(M∞,g∞)≤Cϕ L
2
(M∞,g∞)

The supremum of ﬁrst eigenvalues 15
whereConly depends on (M
∞,g∞). Hence for sufﬁciently largeT
P
∞(χϕ)−χP ∞ϕL
2
(M∞,g∞)≤kC/Tϕ L
2
(M∞,g∞)≤2kC/Tχϕ L
2
(M∞,g∞)
asχϕ L
2
(M∞,g∞)→ϕ L
2
(M∞,g∞)forT→∞. The sectionχϕcan be inter-
preted as a section on (M,g
L)ifL>2T, and on the support ofχϕ
we haveg
L=g∞andP ∞(χϕ)=P L(χϕ). Hence standard Rayleigh quo-
tient arguments imply that ifP
∞hasmeigenvalues (counted with mul-
tiplicity) in the interval [a,b] thenP
Lhasmeigenvalues in the interval
[a−2kC/T, b+2kC/T]. Taking the limitT→∞we obtain (1.11).
By exchanging some obvious signs we obtain similarly
lim sup
L→∞
(−λ
−
j
(PL))≤−λ
−
j
(P∞). (1.12)
1.4.3 The inﬁmum part of the proof of Theorem1.4.1
We now prove
lim inf
L→∞
(±λ
±
j
(PL))≥±λ
±
j
(P∞). (1.13)
We assume that we have a sequenceL
i→∞, and that for eachiwe have a
system of orthogonal eigenvectorsϕ
i,1,...,ϕ i,mofPLi
, i.e.P Li
ϕi,ł=λi,łϕi,ł
forł∈{1,...,m}. Furthermore we suppose thatλ i,ł→ˉλ ł∈(−σ P,σP)for
ł∈{1,...,m}.
Then
ψ
i,ł:=

F
Li
F∞
n−k
2
ϕi,ł
satisﬁes
P
∞ψi,ł=hi,łψi,łwithh i,ł:=

F
Li
F∞

k
λi,ł.
Furthermore
ψ
i,ł
2
L
2
(M∞,g∞)
=
χ
M

F
Li
F∞

−k
|ϕi,ł|
2
dv
gL
i
≤sup
M
|ϕi,ł|
2
χ
M

F
Li
F∞

−k
dv
gL
i
Because of
χ
M

F
Li
F∞

−k
dv
gL
≤C
χ
r
n−1−k
dr <∞

16 B. Ammann and P. Jammes
(forn>k) the normψ
i,łL
2
(M∞,g∞)is ﬁnite as well, and we can renormalize
such that
ψ
i,łL
2
(M∞,g∞)=1.
Lemma 1.4.2For anyδ>0and anył∈{0,...,m}the sequence
σ
ψ
i,łC
k+1
(M\B y(δ),g∞)
π
i
is bounded.
Proof of the lemma.After removing ﬁnitely manyi, we can assume thatλ
i≤
2ˉλande
−Li
<δ/2. HenceF L=F∞andh i=λionM\B y(δ/2). Because of
χ
M\B y(δ/2)
|(P∞)
s
ψi|
2
dv
g∞
≤(2ˉλ)
2s
χ
M\B y(δ/2)
|ψi|
2
dv
g∞
≤(2ˉλ)
2s
we obtain boundedness ofψ iin the Sobolev spaceH
sk
(M\B y(3δ/4),g ∞),
and hence, for sufﬁciently largesboundedness inC
k+1
(M\B y(δ),g∞). The
lemma is proved. χ
Hence after passing to a subsequenceψ
i,łconverges inC
k,α
(M\B y(δ),g∞)
to a solutionˉψ
łof
P
∞
ˉψ
ł=ˉλł
ˉψ
ł.
By taking a diagonal sequence, one can obtain convergence inC
k,α
loc
(M∞)of
ψ
i,łtoˉψł. It remains to prove thatˉψ 1,...,ˉψ mare linearly independent, in
particular that anyˉψ
łπ=0. For this we use the following lemma.
Lemma 1.4.3For anyε>0there isδ
0andi 0such that
ψ
i,łL
2
(By(δ0),g∞)≤εψ i,łL
2
(M∞,g∞)
for alli≥i 0and allł∈{0,...,m}. In particular,
ψ
i,łL
2
(M\B y(δ0),g∞)≥(1−ε)ψ i,łL
2
(M∞,g∞).
Proof of the lemma.Because of Proposition1.3.1and
P
∞ψi,łL
2
(M∞,g∞)≤|ˉλł|ψi,łL
2
(M∞,g∞)=|ˉλł|

The supremum of ﬁrst eigenvalues 17
we get
(∇
∞
)
s
ψi,łL
2
(M∞,g∞)≤C
for alls∈{0,...,k}.Letχbe a cut-off function as in Subsection1.4.2with
T=−logδ. Hence
P
∞
σ
(1−χ)ψ
i,ł
π
−(1−χ)P
∞(ψi,ł)L
2
(M∞,g∞)≤
C
T
=
C
−logδ
.(1.14)
On the other hand (B
y(δ)\{y},g ∞) converges for suitable choices of base
points forδ→0toS
n−1
×(0,∞)intheC
∞
-topology of Riemannian man-
ifolds with base points. Hence there is a functionτ(δ) converging to 0 such
that
P
∞
σ
(1−χ)ψ
i,ł
π

L
2
(M∞,g∞)≥(σp−τ(δ))(1−χ)ψ i,łL
2
(M∞,g∞).(1.15)
Using the obvious relation
(1−χ)P
∞(ψi,ł)L
2
(M∞,g∞)≤|λi,ł|(1−χ)ψ i,łL
2
(M∞,g∞)
we obtain with (1.14) and (1.15)
ψ
i,łL
2
(By(δ
2
),g∞)≤(1−χ)ψ i,łL
2
(M∞,g∞)
≤
C
|logδ|(σ P−τ(δ)−|λ i,ł|)
.
The right hand side is smaller thanεforisufﬁciently large andδsufﬁ-
ciently small. The main statement of the lemma then follows forδ
0:=δ
2
.
The Minkowski inequality yields.
ψ
i,łL
2
(M\B y(δ
2
),g∞)≥1−ψ i,łL
2
(By(δ
2
),g∞)≥1−ε. χ
The convergence inC
1
(M\B y(δ0)) implies strong convergence inL
2
(M\
B
y(δ0),g∞)ofψ i,łtoˉψł. Hence
ˉψ
łL
2
(M\B y(δ0),g∞)≥1−ε,
and thusˉψ
łL
2
(M∞,g∞)=1. The orthogonality of these sections is pro-
vided by the following lemma, and the inequality (1.13) then follows
immediately.
Lemma 1.4.4The sectionsˉψ
1,...,ˉψ mare orthogonal.

18 B. Ammann and P. Jammes
Proof of the lemma.The sectionsϕ
i,1,...,ϕi,łare orthogonal. For any ﬁxed
δ
0(given by the previous lemma), it follows for sufﬁciently largeithat
λ
λ
λ
χ
M\B y(δ0)
ψi,ł,ψ
i,˜łŁdv
g∞
λ
λ
λ=
λ
λ
λ
χ
M\B y(δ0)
ϕi,ł,ϕ
i,˜łŁdv
gL
i
λ
λ
λ
=
λ
λ
λ
χ
By(δ0)
ϕi,ł,ϕ
i,˜łŁdv
gL
i
λ
λ
λ
=
λ
λ
λ
χ
By(δ0)

F
Li
F∞

k
∞≥∈→
≤1
ψi,ł,ψ
i,˜łŁdv
g∞
λ
λ
λ
≤ε
2
(1.16)
Because of strongL
2
convergence onM\B y(δ0)thisimplies
λ
λ
λ
χ
M\B y(δ0)
ˉψł,ˉψ˜łŁdv
g∞
λ
λ
λ≤ε
2
(1.17)
for˜łπ =ł, and hence in the limitε→0 (andδ
0→0) we get the orthogonality
ofˉψ
1,...,ˉψ m. χ

Appendix A Analysis on (M ∞,g∞)
The aim of this appendix is to sketch how to prove Proposition1.3.1.All
properties in this appendix are well-known to experts, but explicit references
are not evident to ﬁnd. Thus this summary might be helpful to the reader.
The geometry of (M
∞,g∞) is asymptotically cylindrical. The metricg ∞
is even ab-metric in the sense of Melrose [31], but to keep the presentation
simple, we avoid theb-calculus.
If (r, γ)∈R
+
×S
n−1
denote polar normal coordinates with respect to the
metricg
0, and if we sett:=−logr, then (t,γ) deﬁnes a diffeomorphismα:
B
(M,g0)
y
(1/2)\{y}→[log 2,∞)×S
n−1
such that (α
−1
)
∗
g∞=dt
2
+htfor a
family of metrics such that (α
−1
)
∗
g∞, all of its derivatives, its curvature, and all
derivatives of the curvature tend to the standard metric on the cylinder, and the
speed of the convergence is majorised by a multiple ofe
t
. Thus the continuity
of the coefﬁcients property implies, thatP
∞extends to a bounded operator
from→
H
k
(M∞,g∞)(V)→→ L
2
(M∞,g∞)(V).
The formal self-adjointness ofP
∞implies that
χ
M∞
ψ, P∞ϕŁ=
χ
M∞
P∞ψ, ϕŁ (A.18)
holds forϕ,ψ∈→
c(V) and as→ c(V) is dense inH
k
, property (A.18) follows
allH
k
-sectionsϕ,ψ.
To show Proposition1.3.1it remains to prove the regularity estimate and
then to verify that the adjoint ofP
∞:→H
k
(M∞,g∞)(V)→→ L
2
(M∞,g∞)(V) has
domain→
H
k
(M∞,g∞)(V).
For proving the regularity estimate we need the following local estimate.
Lemma A.1LetKbe a compact subset of a Riemannian manifold(U,g).
LetPbe an elliptic differential operator onUof orderk≥1. Then there is a
19

20 B. Ammann and P. Jammes
constantC=C(U,K,P,g)such that
u
H
k
(K,g)≤C
σ
u L
2
(U,g)+Pu L
2
(U,g)
π
. (A.19)
Here theH
k
(K,g)-norm is deﬁned via the Levi-Civita connection forg.
This estimate holds uniformly in anε-neighborhood ofPandgin the
following sense. Assume that˜Pis another differential operator, and that the
C
0
-norm of the coeffcients of˜P−Pis at mostε, whereεis small. Also
assume that˜gisε-close togin theC
k
-topology. Then the estimate(A.19)
holds for˜Pinstead ofPand for˜ginstead ofgand again for a constant
C=C(U,K,P,g,ε).
Proof of the lemma.We cover the compact setKby a ﬁnite number of
coordinate neighborhoodsU
1,...,Um. We choose open setsV i⊂Uisuch that
the closure ofV
iis compact inU iand such thatK⊂V 1∪...∪V m. One can
choose compact setsK
i⊂Visuch thatK=K 1∪...∪K m. To prove (A.19)
it is sufﬁcient to proveu
H
k
(Ki,g)≤C(u L
2
(Vi,g)+Pu L
2
(Vi,g)) for anyi.
We write this inequality in coordinates. As the closure ofV
iis a compactum
inU
i, the transition to coordinates changes the above inequality only by a
constant. The operatorP, written in a coordinate chart is again elliptic.
We have thus reduced the prove of (A.19) to the prove of the special case
thatUandKare open subsets of ﬂatR
n
.
The proof of this special case is explained in detail for example in in [33,
Corollary III 1.5]. The idea is to construct a parametrix forP, i.e. a pseudodif-
ferential operator of order−ksuch thatS
1:=QP−Id andS 2:=PQ−Id are
inﬁnitely smoothing operators. ThusQis bounded fromL
2
(U) to the Sobolev
spaceH
k
(U), in particularQ(P(u)) H
k≤CP(u) L
2. Smoothing operators
map the Sobolev spaceL
2
continuously toH
k
. We obtain
u
H
k
(K)≤u H
k
(U)≤Q(P(u)) H
k
(U)+S 1(u) H
k
(U)
≤C
λ
σ
P(u)
L
2
(U)+u L
2
(U)
π
.
See also [28, III §3] for a good presentation on how to construct and work with
such a parametrix.
To see the uniformicity, one veriﬁes that
λ
λ
λ
λ
u
H
k
(K,˜g)
u H
k
(K,g)
−1
λ
λ
λ
λ
≤C˜g−g C
k≤Cε
and
λ
λ
λ
λ
˜P(u)
L
2
(U)
P(u) L
2
(U)
−1
λ
λ
λ
λ
≤Cεu H
k
(U).
The unformicity statement thus follows. χ

The supremum of ﬁrst eigenvalues 21
Proof of the regularity estimate in Proposition1.3.1.We writeM
∞asM B∪
([0,∞)×S
n−1
), such that the metricg ∞is asymptotic (in theC
∞
-sense) to the
standard cylindrical metric. The metricg
∞restricted to [R−1,R+2]×S
n−1
then converges in theC
k
-topology to the cylindrical metricdt
2
+σ
n−1
on
[0,3]×S
n−1
forR→∞. As the coefﬁcients ofP gdepend continuously on
the metric, theP-operators on [R−1,R+2]×S
n−1
is in anε-neighborhood
ofP,forR≥R
0=R0(ε). Applying the preceding lemma forK=[R,R+
1]×S
n−1
andU=(R−1,R+2)×S
n−1
we obtain
∇
s
uL
2
([R,R+1]×S
n−1
,g∞)≤C
σ
u L
2
((R−1,R+2)×S
n−1
,g∞)
+P ∞uL
2
((R−1,R+2)×S
n−1
,g∞)
π
. (A.20)
Similarly, applying the lemma toK=M
B∪([0,R 0]×S
n−1
) andU=
M
B∪([0,R 0+1)×S
n−1
)gives
∇
s
uL
2
(MB∪([0,R 0]×S
n−1
),g∞)≤C
σ
u L
2
(MB∪([0,R 0+1)×S
n−1
),g∞)
+P ∞uL
2
(MB∪([0,R 0+1)×S
n−1
),g∞)
π
.(A.21)
Taking the sum of estimate (A.21), of estimate (A.20)forR=R
0,again
estimate (A.20)butforR=R
0+1, and so for allR∈{R 0+2,R 0+3,...}
we obtain (1.9), with a larger constantC. χ
Now we study the domainDof the adjoint of
P
∞:→H
k
(M∞,g∞)(V)→→ L
2
(M∞,g∞)(V).
By deﬁnition a sectionϕ:→
L
2
(M∞,g∞)(V)isinDif and only if
→
H
k
(M∞,g∞)(V)łuσ →
χ
M∞
P∞u, ϕŁ (A.22)
is bounded as a map fromL
2
toR.Forϕ∈→ H
k
(M∞,g∞)(V) we know that
P
∞ϕisL
2
and thus property (A.18) directly implies this boundedness. Thus
→
H
k
(M∞,g∞)(V)⊂D.
Conversely assume the boundedness of (A.22). Then there is av∈
→
L
2
(M∞,g∞)(V) such that
≤
M∞
u, vŁ=
≤
M∞
P∞u, ϕŁ, or in other wordsP ∞ϕ=
vholds weakly. Standard regularity theory implies
ϕ∈→
H
k
(M∞,g∞)(V).
We obtain→
H
k
(M∞,g∞)(V)=D, and thus the self-adjointness ofP ∞follows.
Proposition1.3.1is thus shown.

22 B. Ammann and P. Jammes
References
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G´eom Inst. Fourier Grenoble16(1998), 33–42.
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dimensional tori, J. Geom. Phys.56(2006), 623–642.
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Japan48(1996), 69–83.
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nigfaltigkeiten, Teubner Verlag, 1981.
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1031–1042.
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metric conformal deformations, Proc. of Am. Math. Soc.134(2006), 715–721.
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et volume conforme, Math. Ann.275(1986), 257–267.
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Math. Oxford, II. series27(1976), 371–378.
[22] R. Gover and L. J. Peterson,Conformally invariant powers of the Laplacian,
Q-curvature, and tractor calculus, Comm. Math. Phys.235(2003), 339–378.

The supremum of ﬁrst eigenvalues 23
[23] E. Hebey and F. Robert,Coercivity and Struwe’s compactness for Paneitz type
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(2001), 491–517.
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Authors’ addresses:
Bernd Ammann
Facult¨at f¨ur Mathematik
Universit¨at Regensburg
93040 Regensburg
Germany
[email protected]
Pierre Jammes
Laboratoire J.-A. Diendonn´e, Universit´e Nice-Sophia Antipolis,
Parc Valrose, F-06108 Nice Cedex02, France
[email protected]

2
K-Destabilizing test conﬁgurations with
smooth central ﬁber
claudio arezzo, alberto della vedova, and
gabriele la nave
Abstract
In this note we point out a simple application of a result by the authors in
[2]. We show how to construct many families of strictlyK-unstable polarized
manifolds, destabilized by test conﬁgurations with smooth central ﬁber. The
effect of resolving singularities of the central ﬁber of a given test conﬁguration
is studied, providing many new examples of manifolds which do not admit
K¨ahler constant scalar curvature metrics in some classes.
2.1 Introduction
In this note we want to speculate about the following Conjecture due to Tian-
Yau-Donaldson ([23], [24], [25], [7]):
Conjecture 2.1.1A polarized manifold(M,A)admits a K¨ahler metric of
constant scalar curvature in the classc
1(A)if and only if it isK-polystable.
The notion ofK-stability will be recalled below. For the moment it sufﬁces to
say, loosely speaking, that a polarized manifold, or more generally a polarized
variety (V,A), isK-stable if and only if anyspecialdegeneration ortest
conﬁgurationof (V,A) has an associatednon positiveweight, called Futaki
invariant and that this is zero only for the product conﬁguration, i.e. the trivial
degeneration.
We do not even attempt to give a survey of results about Conjecture2.1.1,but
as far as the results of this note are concerned, it is important to recall the reader
that Tian [24], Donaldson [7], Stoppa [22], using the results in [3] and [4], and
Mabuchi [17] have proved the sufﬁciency part of the Conjecture. Destabilizing
a polarizing manifold then implies non existence results of K¨ahler constant
scalar curvature metrics in the corresponding classes.
24

K-Destabilizing test conﬁgurations 25
One of the main problems in this subject is that under a special degeneration
a smooth manifold often becomes very singular, in fact just a polarized scheme
in general. This makes all the analytic tool available at present very difﬁcult to
use.
Hence one naturally asks which type of singularities must be introduced to
make the least effort to destabilize a smooth manifold without cscK metrics.
The aim of this note is to provide a large class of examples of special
degenerations with positive Futaki invariant andsmoothlimit. In fact we want to
provide a “machine” which associates to any special degeneration of a polarized
normalvariety (V,A) with positive Futaki invariant a special degenerationfor
a polarized manifold(˜M,˜A) with smooth central ﬁber and still positive Futaki
invariant.
To the best of our knowledge, before this work the only known examples of
special degeneration with non negative Futaki invariant and smooth central ﬁber
are the celebrated example of Mukai-Umemura’s Fano threefold ([18]) used
by Tian in [24] to exhibit the ﬁrst examples of Fano manifolds with discrete
automorphism group and no K¨ahler-Einstein metrics (other Fano manifolds
with these properties have been then produced in [1]). In this case there exist
non trivial special degenerations with smooth limit andzeroFutaki invariant
(hence violating the deﬁnition ofK-stability). It then falls in the borderline
case, making this example extremely interesting and delicate. We stress that
our “machine” does not work in this borderline case, because a priori the Futaki
invariant of the new test conﬁguration is certainly small (by [2]) but we cannot
control its sign.
To state our result more precisely we now recall the relevant deﬁnitions:
Deﬁnition 2.1.2Let (V,A)bean-dimensional polarized variety or scheme.
Given a one-parameter subgroupρ:C
∗
→Aut(V) with a linearization onA
and denoted byw(V,A) the weight of theC
∗
-action induced on

top
H
0
(V,A),
we have the following asymptotic expansions ask−0:
h
0
(V,A
k
)=a 0k
n
+a1k
n−1
+O(k
n−2
) (2.1)
w(V,A
k
)=b 0k
n+1
+b1k
n
+O(k
n−1
) (2.2)
The (normalized)Futaki invariantof the action is the rational number
F(V,A,ρ)=
b
1
a0
−
b
0a1
a
2
0
.
Deﬁnition 2.1.3Atest conﬁguration(X, L) for a polarized variety (V,A)
consists of a schemeXendowed with aC
∗
-action that linearizes on a line
bundleLoverX, and a ﬂatC
∗
-equivariant mapf:X→C(whereChas the

26 C. Arezzo, A. Della Vedova, and Gabriele La Nave
usual weight oneC
∗
-action) such thatL| f
−1
(0)is ample onf
−1
(0) and we have
(f
−1
(1),L| f
−1
(1))∕(V,A
r
)forsomer>0.
When (V,A) has aC
∗
-actionρ:C
∗
→Aut(V), a test conﬁguration where
X=V×CandC
∗
acts onXdiagonally throughρis calledproduct conﬁgu-
ration.
Given a test conﬁguration (X, L) we will denote byF(X, L) the Futaki
invariant of theC
∗
-action induced on the central ﬁber (f
−1
(0),L| f
−1
(0)).
If (X, L) is a product conﬁguration as above, clearly we haveF(X, L)=
F(V,A,ρ).
Deﬁnition 2.1.4The polarized manifold (M,A)isK-stableif for each test
conﬁguration for (M,A) the Futaki invariant of the induced action on the central
ﬁber (f
−1
(0),L| f
−1
(0)) is less than or equal to zero, with equality if and only if
we have a product conﬁguration.
A test conﬁguration (X, L) is calleddestabilizingif the Futaki invariant of
the induced action on (f
−1
(0),L| f
−1
(0)) is greater than zero.
Test conﬁgurations for an embedded varietyV⊂P
N
endowed with the hyper-
plane polarizationAcan be constructed as follows. Given a one-parameter
subgroupρ:C
∗
→GL(N+1), which induces an obvious diagonalC
∗
-action
onP
N
×C, it clear that the subscheme
X=
Ł
(z, t)∈P
N
×C|tπ =0,(ρ(t
−1
)z, t)∈V
ł
⊂P
N
×C,
is invariant and projects equivariantly onC. Thus considering the relatively
ample polarizationLinduced by the hyperplane bundle gives test conﬁguration
for (V,A). On the other hand, given a test conﬁguration (X, L) for a polarized
variety (V,A), the relative projective embedding given byL
r
, withrsufﬁciently
large, realizesXas above (see details in [21]).
We can now describe our “machine”: consider a test conﬁguration (X, L)
for a polarized normal variety (V,A) withF(X, L)>0. Up to raiseLto a
suitable power – which does not affect the Futaki invariant – we can suppose
being in the situation above withX⊂P
N
×Cinvariantly, andLinduced by the
hyperplane bundle ofP
N
. At this point we consider the central ﬁberX 0⊂P
N
,
which is invariant with respect toρ, and we apply the (equivariant) resolution of
singularities [14, Corollary 3.22 and Proposition 3.9.1]. Thus there is a smooth
manifold˜Pacted on byC
∗
and an equivariant map
β:˜P→P
N
which factorizes through a sequence of blow-ups, such that the strict transform
˜X
0ofX0is invariant and smooth. The key observation is that the strict transform
˜X
1of the ﬁberX 1⊂Xdegenerate toX 0under the givenC
∗
action on˜P, thus it

K-Destabilizing test conﬁgurations 27
must be smooth. This gives an invariant family˜X⊂P×Cand an equivariant
birational morphism
π:˜X→X.
Some comments are in order:
1allthe ﬁbers of˜Xare smooth, butπis never a resolution of singularities of
X(except the trivial case when the central ﬁber ofXwas already smooth)
since it fails to be an isomorphism on the smooth locus ofX;
2˜L=π
∗
Lis not a relatively ample line bundle any more, but just a big and nef
one. It is not then even clear what it means to compute its Futaki invariant;
3 the ﬁber over the generic point ofCof the new (big and nef) test conﬁguration
(˜X,˜L) is different fromV;
4 the family˜Xis not unique since the resolutionβit is not.
The issue raised at point (2) was addressed in [2] and it was proved that the
following natural (topological) deﬁnition makes the Futaki invariant a continu-
ous function around big and nef points in the K¨ahler cone. We will give simple
self-contained proofs in the cases of smooth manifolds and varieties with just
normal singularities in Section 2.
Deﬁnition 2.1.5LetVbe a projective variety or scheme endowed with aC
∗
-
action and letBbe a big and nef line bundle onV. Choosing a linearization of
the action onBgives aC
∗
-representation on
−
dimV
j=0
H
j
(V,B
k
)
(−1)
j
(here the
E
−1
denotes the dual ofE). We setw(V,B
k
)=trA k, whereA kis the generator
of that representation. Ask→+∞we have the following expansion
w(V,B
k
)
χ(V,B
k
)
=F
0k+F 1+O(k
−1
),
and we deﬁne
F(V,B)=F
1
to be theDonaldson–Futaki invariantof the chosen action on (V,B)
The existence of the expansion involved in deﬁnition above follows from the
standard fact thatχ(V,B
k
) is a polynomial of degree dimV, whose proof (see
for example [11]) can be easily adapted to show thatw(V,B
k
) is a polynomial
of degree at most dimV+1.

28 C. Arezzo, A. Della Vedova, and Gabriele La Nave
The key technical Theorem proved in [2] is then the following:
Theorem 2.1.6LetB,Abe linearized line bunldes on a schemeVacted on
byC
∗
. Suppose thatBis big and nef andAample. We have
F(V,B
r
⊗A)=F(V,B)+O

1
r

,asr→∞.
Having established a good continuity property of the Futaki invariant up to these
boundary point, we need to address the question of the effect of a resolution
of singularities of the central ﬁber. This is a particular case of the following
non trivial extension of previous analysis by Ross and Thomas [21] which was
proved in [2] where the general case of birational morphisms has been studied:
Theorem 2.1.7Given a test conﬁgurationf:(X, L)→Cas above, letf
λ
:
(X
λ
,L
λ
)→Cbe another ﬂat equivariant family withX
λ
normal and letβ:
(X
λ
,L
λ
)→(X, L)be aC
∗
-equivariant birational morphism such thatf
λ
=
f◦βandL
λ
=β
∗
L. Then we have
F(X
λ
,L
λ
)≥F(X, L),
with strict inequality if and only if the support ofβ
∗(OX
λ)/OXhas codimension
one.
The proof of these results uses some heavy algebraic machinery, yet their proof
when (V,A) or the central ﬁber of (X, L) have only normal singularities (a
case largely studied) is quite simple and we give it in Section 2.
The Corollary of Theorem2.1.6and Theorem2.1.7we want to point out in
this note is then the following:
Theorem 2.1.8Let(X, L)be a test conﬁguration for the polarized normal
variety(V,A)with positive Futaki invariant. Let moreover(˜X,˜L)be a (big
and nef) test conﬁguration obtained from(X, L)as above and let(˜M,˜B)be
the smooth (big and nef) ﬁber over the point1∈C. LetRbe any relatively
ample line bundle over˜X.
Then(˜X,˜L
r
⊗R)is a test conﬁguration for(˜M,˜B
r
⊗R| ˜M)with following
properties:
1 smooth central ﬁber;
2 positive Futaki invariant forrsufﬁciently large.
In particular˜Mdoes not admit a constant scalar curvature K¨ahler metric in
any class of the formc
1(˜B
r
⊗R| ˜M), withrlarge enough.

K-Destabilizing test conﬁgurations 29
While this Theorem clearly follows from Theorems2.1.6and Theorem2.1.7,
but for the speciﬁc case of central ﬁber with normal singularities it follows
from the much simpler Proposition2.2.1and Theorem2.2.3.
The range of applicability of the above theorem is very large. We go through
the steps of the resolution of singularities in an explicit example by Ding-
Tian [6] of a complex orbifold of dimension 2. In this simple example explicit
calculations are easy to perform, yet we point out that the ﬁnal example is
somehow trivial since it ends on a product test conﬁguration. On the other
hand abundance of similar examples even in dimension 2 can be obtained by
the reader as an exercise using the results of Jeffres [12] and Nagakawa [19],
in which cases we loose an explicit description of the resulting destabilized
manifold, but we get new nontrivial examples. In fact in higher dimensions one
can use the approach described in this note to test also the Arezzo-Pacard blow
up theorems [3][4], when the resolution of singularities requires a blow up of
a scheme of positive dimension.
2.2 The case of normal singularities
In this section we give simple proofs of the continuity of the Futaki invariant
at boundary points for smooth manifolds or varieties with normal singularities.
More general results of this type have been proved in [2] but we want to stress
that under these assumptions proofs become much easier.
The fundamental continuity property we will need, and proved in Corollary
2.1.6, can be stated in the following form for smooth bases:
Proposition 2.2.1LetA,Lbe respectively an ample and a big and nef line
bundle on a smooth projective manifoldM. For everyC
∗
-action onMthat
linearizes toAandL,asr→+∞we have
F(M,L
r
⊗A)=F(M,L)+O

1
r

.
ProofThe result is a simple application of the equivariant Riemann-Roch
Theorem. We present here the details of the calculations involved, since we
could not ﬁnd precise references for them.
Fix an hermitian metrics onAthat is invariant with respect to the action of
S
1
⊂C
∗
and suppose that the curvatureωis a K¨ahler metric. SinceLis nef,
for eachr>0 we can choose an invariant metric onLwhose curvatureη
r
satisfyrη r+ω>0. In other wordsrη r+ωis a K¨ahler form which coincides
with the curvature of the induced hermitian metric on the line bundleL
r
⊗A.

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Foundation, Inc.; 20Nov70; MU8249.
ONE BIG OCEAN. Reinald Werrenrath, Jr. Made by Journal Films.
10 min., sd., color, 16 mm. © Reinald Werrenrath, Jr.; 1Apr70;
MP20824.
ONE DAY'S GROWTH. See
[REMINGTON ELECTRIC SHAVER TELEVISION COMMERCIALS]
ONE SMALL STEP FOR MAN. See
THE BOLD ONES.
ONLY PEOPLE PRODUCE PROFITS. Caltex Petroleum Corp. Made
by Visualscope. 10 min., sd., color, Super 8 mm. Loop film. ©
Caltex Petroleum Corp.; 13Nov70; MU825O.
OPEN WIDE. Terrytoons. 5 min., sd., b&w, 16 mm. (Deputy Dawg)
© Terrytoons, a division of CBS Films, Inc.; 19Aug64 (in notice:

1963); LP38381.
OPENING A CHECKING ACCOUNT. See
[HANDLING FINANCES]
OPENING A SAVINGS ACCOUNT. See
[HANDLING FINANCES]
OPENING NEW DOORS (Filmstrip) Troll Associates. 36 fr., color, 35
mm. (Developing good work and study habits) Author, Brian
James; illustrator, Gordon Hart. © Troll Associates; 6Feb70;
A166387.
OPERA WITH HENRY BUTLER. Learning Co. of America. 26 min.,
sd., color, 16 mm. © Learning Co. of America, division of
Columbia Pictures Industries, Inc.; 2Nov70; LP38518.
OPERATION OF THE pH METER, BECKMAN MODEL 72. John Wiley
& Sons. 4 min., si, color, Super 8 mm. Loop film. Appl. author:
Wendell H. Slabaugh. © John Wiley & Sons, Inc.; 18Jul69;
MP20554.
OPPORTUNITY AND THE GOOD LIFE. Aerojet-General Corp. 7
min., sd., color, 16 mm. © Aerojet-General Corp.; 29Jul69;
LP38118.
OPTOMETRIC ASSISTANT. See
[CAREER DEVELOPMENT: SELF-EVALUATION]
ORAL MEDICATIONS. See
ADMINISTRATION OF MEDICATIONS.
ORAL REPORTS: HOW TO TALK OUT LOUD (Filmstrip) Troll
Associates. 33 fr., color, 35 mm. (Developing good work and
study habits) Author, Anne Brent; illustrator, Herbert Leopold. ©
Troll Associates; 6Feb70; A166391.

ORBIT A LITTLE BIT. Terrytoons. 5 min., sd., b&w, 16 mm.
(Deputy Dawg) © Terrytoons, a division of CBS Films, Inc.;
19Aug64 (in notice: 1963); LP38339.
ORBITAL SHAPES & PATHS. Teaching Films. 12 min., sd., color.
(Space technology series) © Teaching Films, Inc., subsidiary of
A-V Corp.; 21Sep70; MU8237.
ORDEAL OF MAJOR GRIGSBY. See
THE LAST GRENADE.
ORGANIZING FOR SALES (Filmstrip) American Training Academy.
58 fr., color, 35 mm. Original material, research & direction by
James L. Miller. Appl. states prev. pub. 26Apr67, JP10254. NM:
additions & revisions. © James L. Miller; 11Feb70; JU12643.
THE ORIGIN OF THE ALPHABET. See
OUR LITERARY HERITAGE.
THE ORIGIN OF WRITING. See
OUR LITERARY HERITAGE.
OSCAR'S BIRTHDAY PRESENT. Terrytoons. 5 min., sd., b&w, 16
mm. (Astronut) © Terrytoons, a division of CBS Films, Inc.;
21Jul65; LP38427.
OSCAR'S THINKING CAP. Terrytoons. 5 min., sd., b&w, 16 mm.
(Astronut) © Terrytoons, a division of CBS Films, Inc.; 21Jul65;
LP38424.
OSMOSIS (Filmstrip) No. 1565. Popular Science Audio-Visuals. 42
fr., color, 35 mm. (A Biology release) With Filmstrip guide, 5 p. ©
Popular Science Audio-Visuals, Inc.; 6Oct69; A196087.
OTHER SIDE OF THE CHART. See
MARCUS WELBY, M.D.

OTHER WOMAN. See
MY THREE SONS.
OTROS TRABAJADORES DE LA COMUNIDAD. See
[SPANISH PROGRAM]
OUR CHILDREN'S HERITAGE (Filmstrip) Cooper Films & Records. 6
filmstrips, color, 35 mm. With Teacher's guide, 31 p. Contents:
Peter and the wolf. 32 fr.--Toads and diamonds. 29 fr.--The ugly
duckling. 42 fr.--The town mouse and the country mouse. 26 fr.--
Pinocchio. 31 fr.--Silly Joe. 28 fr. Illustrated by Carroll E. Spinney.
© Cooper Films & Records, Inc.; 17Aug70 (1969 in notices on
filmstrips); A182384.
OUR COMMUNITY. See
WHAT IS A COMMUNITY.
OUR FRIEND THE ROBIN. Troll Associates. 4 min., si., color, Super
8 mm. Loop film. © Troll Associates; 16Jan70; MP20680.
OUR HERITAGE FROM ANCIENT ROME (Filmstrip) Harcourt, Brace
& World. 2 filmstrips (pt. 1, 92 fr.; pt. 2, 117 fr.), color, 35 mm.
(Adventure in literature) Appl. author: Guidance Associates of
Pleasantville. © Harcourt, Brace & World, Inc.; 19Dec69;
JP12421.
OUR HERITAGE FROM THE PAST. See
ARAB WORLD AND ISLAM.
THE BYZANTINE EMPIRE.
LIFE IN ANCIENT EGYPT.
LIFE IN THE DARK AGES.
LIFE IN THE MIDDLE AGES.

MESOPOTAMIA, CRADLE OF CIVILIZATION.
THE RENAISSANCE.
SPLENDOR OF ANCIENT GREECE.
WORLD OF ANCIENT ROME.
OUR LITERARY HERITAGE (Filmstrip) Educational Filmstrips. 6
filmstrips, color, 35 mm. With filmstrip manual, 24 p. Contents:
Pt. 1-6: The origin of writing. 68 fr.--The origin of the alphabet.
69 fr.--Classical books and libraries. 60 fr.--Monasticism and the
book. 63 fr.--The book during the Islamic naissance and
European Renaissance. 65 fr.--The development and
dissemination of printing. 71 fr. By George & Suzanne Russell. ©
Educational Filmstrips; 1Sep70; JP12627.
OUR TEACHING TASK (Filmstrip) Broadman Films. 50 fr., color, 35
mm. With manual, 1 v. © Broadman Films; 1Jul70; A173130.
OUTPUT ADMITTANCE--HYBRID. T. M. Adams. 4 min., si., color,
Super 8 mm. Loop film. © T. M. Adams; 1Sep70; MP20929.
OVER THE HILL TO THE MOON. Teaching Films. 9 min., sd., color.
(Space technology series) © Teaching Films, Inc., subsidiary of
A-V Corp.; 21Sep70; MU8240.
OVEREDGING. Pt. 1, job 1: Closing shoulders (Filmstrip) National
Knitted Outerwear Assn. 149 fr., color, 35 mm. © National
Knitted Outerwear Assn.; 8Aug69; JP12590.
THE OWL AND THE PUSSYCAT. Raster Productions. Released by
Columbia Pictures. 95 min., sd., color, 35 mm. Panavision. Based
on the play by Bill Manhoff. © Raster Productions, Inc.; 1Nov70;
LP38197.

P
PAINT WITH BRUSHES. See
PHONO-VIEWER PROGRAM, ART SERIES 2/DISCOVERING
TECHNIQUES.
PAINT YOUR WAGON. Alan Jay Lerner Productions. 164 min., sd.,
color, 35 mm. Panavision. Produced with Malpaso Co. Based
upon the musical play. © Paramount Pictures Corp. & Alan Jay
Lerner Productions, Inc.; 6Oct69; LP38469. (See also Paint your
wagon); 9Oct69; LP38115.
PAINT YOUR WAGON. Alan Jay Lerner Productions. Released by
Paramount Pictures Corp. 137 min., sd., color, 35 mm.
Panavision. Based upon the musical play. © Paramount Pictures
Corp. & Alan Jay Lerner Productions, Inc.; 9Oct69; LP38115.
(See also Paint your wagon); 6Oct69; LP38469.
PAINTING (Filmstrip) Mt. San Jacinto College. 46 fr., color, 35 mm.
(Automotive technology, N-5) Through the cooperation of
California Department of Education. Planned & written by
Sheldon Abbott & Paul Wells. © Mt. San Jacinto College a.a.d.o.
Mt. San Jacinto Junior College District (in notice; Mt. San Jacinto
College); 1Jan69; JP12527.
PAINTING PROBLEMS (Filmstrip) Mt. San Jacinto College. 36 fr.,
color, 35 mm. (Automotive technology, unit N-6) Planned &
written by Paul Wells; audio-visual production, A. H. Waterman;
art, Pam Weaver. © Mt. San Jacinto College a.a.d.o. Mt. San
Jacinto Junior College District (in notice: Mt. San Jacinto
College); 1Jan68; JP12524.
THE PANTHERS. American Broadcasting Companies. 28 min, sd.,
color, 16 mm. (Now) A presentation of ABC News. © American
Broadcasting Companies, Inc.; 7Apr70; MP20924.

PAPER SHAPES. See
PHONO-VIEWER PROGRAM, ART SERIES 2/DISCOVERING
TECHNIQUES.
PARADE. See
[FOCUS ON SELF-DEVELOPMENT, STAGE ONE: AWARENESS]
PARAGRAPHS: HENRY LEARNS SOMETHING NEW (Filmstrip) Troll
Associates. 35 fr., color, 35 mm. (New adventures in language)
Author, Edward McCullough; illustrator, Dodie O'Keefe. © Troll
Associates; 10Jan70; A166443.
PARTS OF SPEECH: UP AND AWAY IN A FLYING BOAT (Filmstrip)
Troll Associates. 30 fr., color, 35 mm. (New adventures in
language) Author, Edward McCullough; illustrator, Karen Tureck.
© Troll Associates; 14Jan70; A166432.
PARTY PLANNING. Dart Industries. 13 min., sd., color, Super 8
mm. (Stop and go learning, session 4) © Dart Industries, Inc.;
1Oct69; MP20619.
THE PASSION OF ANNA. A. B. Svensk Filmindustri. Released by
United Artists Corp. 99 min., sd., color, 35 mm. © A. B. Svensk
Filmindustri; 28Mar70 (in notice: 1969); LP38433.
EL PATITO FEO. See
[SPANISH PROGRAM]
EL PATITO VALIENTE Y EL VIENTO FRIO DEL NORTE (Filmstrip) S-
132. Educational Projections Corp. 34 fr., color, 35 mm. Traducido
del ingles por Carlos Rivera. © Educational Projections Corp.;
23Dec68; JP12383.
PATRIOTIC POETRY BY AMERICAN WRITERS. See
A VISUAL ANTHOLOGY OF POETRY.

PATTON. Twentieth Century-Fox Film Corp. 171 min., sd., color, 70
mm. Dimension 150. Based on factual material from Patton:
Ordeal and triumph, by Ladislas Farago, & A soldier's story, by
Omar N. Bradley. © Twentieth Century-Fox Film Corp.; 30Dec69;
LP38179.
THE PATTY DUKE SHOW. Chrislaw Productions. Canada. Released
by United Artists Television. Approx. 27 min. each, sd., b&w, 16
mm. © United Artists Television, Inc.
The actress. © 26Nov63; LP38237.
Are mothers people. © 18Feb64; LP38233.
Auld lang syne. © 31Dec63; LP38241.
Christmas present. © 24Dec63; LP38240.
The con artists. © 25Feb64; LP38234.
The elopement. © 1Oct63; LP38236.
Horoscope. © 7Jan64; LP38232.
The perfect teenager. © 3Mar64; LP38235.
The princess Cathy. © 17Dec63; LP38239.
The song writers. © 10Dec63; LP38238.
PAUL BUNYAN. See
AMERICAN FOLKLORE.
PAUL BUNYAN AND HIS BLUE OX. See
TELL ME A STORY.
PAUL BUNYAN AND HIS GREAT BLUE OX (Filmstrip) Troll
Associates. 41 fr., color, 35 mm. (American folk heroes and tall
tales) Illustrator, Gloria Fletcher. © Troll Associates; 8Jan70;
A166481.

PAUL CEZANNE (Filmstrip) Films & Slides. 15 fr., color, 35 mm. ©
Films & Slides; 1Dec61; JP12700.
PEACE. Todd N. Tuckey. 6 min., si., color, 16 mm. © Todd N.
Tuckey; 31Aug70; MU8231.
PEACH PLUCKIN' KANGAROO. Terrytoons. 5 min., sd., b&w, 16
mm. (Deputy Dawg) © Terrytoons, a division of CBS Films, Inc.;
19Aug64 (in notice: 1963); LP38345.
PECOS BILL. See
AMERICAN FOLKLORE.
PECOS BILL AND LIGHTNING (Filmstrip) Troll Associates. 43 fr.,
color, 35 mm. (American folk heroes and tall tales) Illustrator,
Ettie de Laczay. © Troll Associates; 8Jan70; A166483.
PECOS BILL AND THE LONG LASSO, See
TELL ME A STORY.
PEEK-A-BOO. See
[CHAS. PFIZER & CO. TELEVISION COMMERCIALS]
PEGASUS THE WINGED HORSE (Filmstrip) Troll Associates. 44 fr.,
color, 35 mm. (Myths and legends of ancient Greece) Illustrator,
Regina Fisher. © Troll Associates; 6Feb70; A166490.
PEOPLE AGAINST ORTEGA. See
THE BOLD ONES.
PEOPLE PLEASERS (Filmstrip) Chrysler Corp. Made by Ross Roy,
Inc. 59 fr., color, 35 mm. © Chrysler Corp.; 17Sep70; JP12538.
PEOPLE SOUP. Pangloss Productions. Released by Columbia
Pictures Corp. 11 min., sd., color, 35 mm. © Pangloss
Productions, Inc.; 1Apr70 (in notice: 1969); MP20551.

PERCENT (Filmstrip) No. 769. Educational Projections Corp. 39 fr.,
color, 35 mm. (Mathematics, level 6) By Donovan R. Lichtenberg
& Charles W. Engel. © Educational Projections Corp.; 27Feb70;
JP12680.
PERCEPTION. Appleton-Century-Crofts. 1 reel, sd., color, 16 mm.
(Analysis of behavior) Appl. authors: Robert Johnson & Michael
Ball. © Meredith Corp.; 5Nov70; MP20952.
PERCHING BIRDS, LARGEST FAMILY OF BIRDS. Troll Associates. 4
min., si., color, Super 8 mm. Loop film. © Troll Associates;
16Jan70; MP20682.
PEREGRINE FALCON. See
THE WONDERFUL WORLD OF DISNEY .
PERFECT TEENAGER. See
THE PATTY DUKE SHOW.
PERFORMA PANTY HOSE. See
[PRO-TEL PRODUCTS TELEVISION COMMERCIALS]
PERFORMANCE OF DOWNCOMERS IN DISTILLATION COLUMNS.
Fractionation Research. 15 min., sd., b&w, 16 mm. ©
Fractionation Research, Inc.; 18Feb70; MP20595.
PERIODONTAL DISEASE. Teaching Films. 9 min., sd., color, 16 mm.
(Prevention & control of dental disease) © Teaching Films, Inc.,
division of A-V Corp.; 26Oct70; MU8258.
PERSEUS AND MEDUSA (Filmstrip) Troll Associates. 43 fr., color, 35
mm. (Myths and legends of ancient Greece) Illustrator, Regina
Fisher. © Troll Associates; 6Feb70; A166488.
PERSONAL DEVELOPMENT. See
GETTING LOST.

GOING TO SCHOOL.
LEARNING TO DO THINGS FOR YOURSELF.
LEARNING TO HELP OTHERS.
LEARNING TO LISTEN CAREFULLY.
WHAT TO DO WHEN YOU VISIT.
PERSONALITY IN BUSINESS (Filmstrip) No. 422. Popular Science
Audio-Visuals. 41 fr., color, 35 mm. (A Guidance release) With
Filmstrip guide, 5 p. © Popular Science Audio-Visuals, Inc.;
5Jan70; A196144.
PERU: INCA HERITAGE. Hartley Productions. 18 min., sd., color, 16
mm. Appl. author: Elda Hartley. © Hartley Productions, Inc.;
5Nov70; MP2O983.
PETER AND THE WOLF. See
OUR CHILDREN'S HERITAGE.
PETER PAN. See
FAVORITE CHILDREN'S BOOKS.
PETS CAN READ. Dade County School B Board. 6 min., sd., color,
16 mm. © Dade County School Board; 25Aug70; MP20874.
THE PHARMACIST AND CANCER. American Cancer Society. Made
by Campus Film Productions. 22 min., sd., color, 16 mm. ©
American Cancer Society, Inc.; 16Apr69; MP20790.
PHONO-VIEWER PROGRAM, ART SERIES 1/EXPLORING
MATERIALS (Filmstrip) General Learning Corp., Early Learning
Division. 5 filmstrips (15 fr. each), color, 16 mm. With Kit.
Contents: This is finger paint.--This is paint.--These are crayons.--
This is paper.--This is clay. Prepared in cooperation with Binney &
Smith, Inc.; art consultant: Margaret Johnson; photographer:
John Naso; designed by Sara Stein; written by Carol Murdock.

Appl. author: General Learning Corp., employer for hire. ©
General Learning Corp.; 31Dec69; A189588-189592.
PHONO-VIEWER PROGRAM, ART SERIES 2/DISCOVERING
TECHNIQUES (Filmstrip) General Learning Corp., Early Learning
Division. 5 filmstrips (15 fr. each), color, 16 mm. With kit.
Contents: Paper shapes.--Crayon over, crayon under.--Paint with
brushes.--Print with paint.--Putting together. Prepared in
cooperation with Binney & Smith, Inc.; art consultant: Margaret
Johnson; photographer: John Naso; designed by Sara Stein;
written by Carol Murdock. Appl. author: General Learning Corp.,
employer for hire. © General Learning Corp.; 31Dec69;
A189583-189587.
PHOTORECEPTION AND FLOWERING. Regents of University of
Colorado. 4 min., si., color, Super 8 mm. (BSCS single topic
inquiry films) Loop film. Appl. author: Biological Sciences
Curriculum Study. © Regents of University of Colorado; 1Jul69
(in notice: 1968); MP20755.
PHOTOSYNTHESIS: THE BIOCHEMICAL PROCESS. Coronet
Instructional Films. 17 min., sd., b&w, 16 mm. © Coronet
Instructional Films, a division of Esquire, Inc.; 1Apr70; MP20847.
THE PHYLA: WHO'S WHO IN THE ANIMAL KINGDOM. Reela
Educational Films, a division of Wometco Enterprises. Released
by Sterling Movies, Educational Films Division. 17 min., sd., color,
16 mm. Produced in cooperation with University of Miami School
of Marine & Atmospheric Sciences, Dade County Public Schools &
Editors of International Oceanographic Foundation Publications.
© Reela Educational Films a.a.d.o. Reela Films, a division of
Reela Films Laboratories, Inc.; 21Jul70; MP20809.
PHYSICAL FITNESS: SLOW DOWN, I CAN'T KEEP UP (Filmstrip)
McGraw-Hill Book Co. Made by McGraw-Hill Films. 49 fr., color, 35
mm. (Learning to learn series) With guide. © McGraw-Hill, Inc.;
30Dec69 (in notice: 1968); JP12496.
PHYSIOLOGY FILM SERIES. See

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Variational Problems in Differential Geometry 1st Edition Professor Roger Bielawski

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