Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
949
cinomraH spaM
Proceedings of the N.S.E-C.B.M.S. Regional Conference,
Held at Tulane University, New Orleans
December 51 - ,91 1980
Edited by Knill, R.J. .M Kalka, and H.CJ. Sealey
galreV-regnirpS
Berlin Heidelberg New York 1982
Editors
Ronald J. Knill
Morris Kalka
Department of Mathematics, Tulane University
New Orleans, LA 70118, USA
Howard C.J. Sealey
Department of Mathematics, University of Utah
Salt Lake City, UT 84112, USA
AMS Subject Classifications (1980): 53-06, 53 C 05
ISBN 3-540-11595-1 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-11595-1 Springer-Verlag New York Heidelberg Berlin
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PREFACE
These proceedings report the substance of talks and papers contributed by
participants In the N.S.F.-C.B.M.S. Reglonal Conference on HarmonlcMaps at Tulane
University Dec. 15-19, 1980. The prlnclpal lecturer at that conference was
James Eells of the University of Warwick. His lectures, as ls customary with such
conferences, will be published separately in the blue series of SMBC Regional
conference reports. That report was co-authored by Luc Lemeire.
The format of the conference was ten lectures by Eells, ten lectures by
selected participants. These latter lectures along wlth two contributed papers
occur here. Thus the Eells-Lemalre report and these lectures are to be regarded as
companion volumes. The Eells-Lemelre work carefully lays down the foundation for
the formalism, in the contest of differential geometry, necessary for the
development of the theory of harmonic maps, and systematically applies that
formalism to selected topics.
These proceedings concern related results In the area of harmonic maps. The
two volumes together are not exhaustive of the current state of the theory, however
they represent the recent efforts of soma of the leading contributors to its
development.
The editorial committee would like to acknowledge first of all James Eells for
the selfless hard work and preparation that went into his lectures, for his
leadership, and his overall good nature which contributed to the pleasant and
stimulating exchanges at the conference.
We would also llke to use this opportunity to thank Ms. Jackle Bollng whose
administrative expertise kept the conference running smoothly,
Mrs. Heater Paternostro who ably handled all correspondence for the conference, and
Mrs. Phuong .Q maL for her efficient help in editing and retyping several of the
papers occuring here.
The editorial coe~aittee was chaired by Ronald J Kntll who would like to
personally acknowledge the contribution of eM Kalka, H. C. J. Sealey, A. L. Vitter
and P.-W. Wong for their expert advice and support throughout.
eW would all like to acknowledge the financial support of the National Science
Foundation for the conference. In addition the editors wish to acknowledge Tulane
University for substantial financial and staff support without which these
proceedings would not have been produced.
TABLE OF CONTENTS
Milnor Number and Classification of Isolated Singularities
of Holomorphic Maps, Bruce Bennett & Stephen S.-T. Yau . . . . . . . . . . . 1
Harmonic Curvature for Gravitational and Yang-Mills Fields,
Jean-Pierre Boum~xignon . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Harmonic Maps from ~pl to ~pn , D. Burns . . . . . . . . . . . . . . . . . 48
Vector Cross Products, Harmonic Maps and the Cauchy Riemann
Equations, Alfred Gray . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
Harmonic Maps in Kahler Geometry and Deformation Theory,
M. Kalka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Harmonic Foliations, Franz W. l~er & Phillippe Tondeur . . . . . . . . . . . 87
On the Stability of Harmonic Maps, Pui-Fai Leung . . . . . . . . . . . . . . . 221
Stability of Harmonic MaRs Between Symmetric Spaces,
T. Nagano . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
On A Class of Harmonic Maps, J. H. Sc~son . . . . . . . . . . . . . . . . . . 138
Harmonic Diffeomorphisms of Surfaces, H. C. J. Sealey . . . . . . . . . . . . 140
Equivariant Harmonic Maps into Spheres, Karen K. Uhlenbeck . . . . . . . . . . 146
MILNOR NUMBER AND CLASSIFICATION OF ISOLATED SINGULARITIES
OF HOLOMORPHIC MAPS
by
Bruce Bennett & Stephen S.-T. Yau
§ .O Introduction
A map :f M ~- N between Riemannian manifolds is said to be harmonic
if it is a critical map for the energy functional E(f) (cf. § .)1 In ]4[
Eells-Sampson proved the following fundamental theorem. If M and N are
both cempact and N has nonpositive sectional curvature, then every continuous
map from M to N is homotopic to a harmonic map. Hartman ]7[ proved that
the harmonic map is unique in each hemotopy class if N has strictly negative
curvature.
For more detail theory of harmonic maps, the reader should consult the
excellent survey article ]3[ by Eells and Lemaire. Perhaps one of the
interesting and difficult problems is the classification of singularities of
harmonic maps between complex K~ihler manifolds. In [26, 27] .J .C Wood gave
a complete classification of singularities of harmonic maps between surfaces
(real dimension). It is well known that a holcmorphic or conjugate holemorphic
map between Eahlerm anifolds is always harmonic. The problem of determining
under what conditions the converse holds is an important and difficult problem.
Recently Siu [24] has defined a notion of strongly negative curvature tensor
and proved the following important theorem.
Theorem (Siu) Suppose M and N are compact ~hler manifolds and the
curvature tensor of N is strongly negative. Suppose :f M ~ N is a harmonic
map and the rank over ql of the differential df of f is at least 4 at
some point of M . Then f is either holcmorphic or conjugate holcmorphic.
In view of these facts, we shall only discuss singularities of holomorphic
maps in this paper. At least this is the first step towards understanding
singularities of harmonic maps. Actually only isolated singularities of
holemorphic maps will be considered. Under these assumptions, the techniques
from algebraic geometry and complex analysis can be brought in.
Let :f UC ~3 ~ be a holomorphic function on an open neighborhood U
of 0 in ~3 . A point x in U is called a singular point of a map f
if the complex gradient ~f is zero at x . We assize that 0 e U is an
isolated singularity of f and f(O) = 0 . In 2, 13, and 30, the theory
of classification of singularities was quite well developed by using the
geometric genus of the singularity. In this paper, we shall classify singu-
larities according to Mi_uor number. Although the later classification turns
out not as natural as the former one, in practice the later one may be more
useful because the Milnor number is easier to compute than the geometric genus.
The later approach was due to Mather 15, Siersma 21, 22 and Arnold i.
In § 1 , we recall some of their results as well as Milnor's results on
topology of hypersurface singularities. In § 2 we give a formula which
relates the Milnor number and the invariants of any resolution of the singu-
larity (cf. Theorem A and Theorem B). These results were obtained some
years ago and have been informally circulated to some extent. It seems to us
that there is still interest in this article. In § 3 we develop certain
results related to sheaves of p-forms at isolated hypersurface singularities.
We would like to thank .P Griffiths, .H Hironaka, M. Kuga, J. Milnor,
D. Mtm~ford, .H Laufer and Y.-T. Siu for discussions related to this work.
Isolated sin6ularities of holemorphic maps
Suppose :f M ~- N is a map between two Riemannianmanifolds with
Riemannian metrics
ds 2 = gijdxidxJ
* 2
(Here the summation convention is used. ) We can define trace 2 = f dSN
ds N
at p e M as the s~n of the critical values of
* 2
f ds N
on the set of all nonzero tangent vectors of M at p , with each critical
value counted as many times as the dimension of its associated critical set.
It is clear that
trace 2 f * ds N 2 = gmJ(ho~.f) •
d~ ~x ~ ~x j
We define the energy density e(f) of f by
1 * 2
e(f) = ~ trace 2 f ds N
ds M
and the energy E(f) of f by
E(f) = f e(f)
M
Definition: A map :f N ~ M between Riemannian manifolds is said to be
harmonic if it is a critical map for the energy functional E(f).
The Euler-Lagrange equation for the energy functional E(f) is
f2~fji ~ ..~M ~f3 N~ ~6 ~
)*(
g i j=o
where MFk iJ and N FGG are respectively the Christoffel symbols of M and N.
Frc~ now on we assume that M , N are K~hler manifold with ~ahler metrics
gi.-dwldwJ~ and ho~-dz~dz 6 ~ respectively. For any smooth map :f M ~ N let
~: ~.f~
l 3w I
Then equation (*) becomes
[~
=0
w~i j G6F
Hence a holomorphic or conjugate bolomorphic map between K'ahler manifolds is
always harmonic.
Let :f M ~- N be a hol~norphic map. We shall always assume n = dim M >
dim N = k . A point x in M is said to be a singular point of the map f
if the Jacobian of f at x is not of maximal rank. In this paper, we shall
asst~ne that f has only isolated singularities. Clearly V = f-l(f(x)) is a
local cc~plete intersection at x . As far as the classification of isolated
singularities is concerned, the problem is local. We may assome from now on
that M = ~n and N = ~k . The classification theory works particularly well
for the case k = 1 .
Two germs of smooth functions :f (~n,o) ~ (~,0) , :g (~n,o) ~ (6,0) are
equivalent, if they belong to the same orbit of the group of germs of holo-
morphic diffeomorphisms (~n,o) ~ (Gn,o). Two germs :f (~n,o) ~ (~,0) and
:g (~m,0) ~ (6,0) are stably equivalent, if they become equivalent after a
direct addition to both of nondegenerate quadratic forms (e.g., f(x) = x 3 is
stably equivalent to g(x,y) = x 3 + y2 but not to h(x,y) = x3). Two stably
equivalent germs on equidimensional spaces are equivalent. Clearly it suffices
to classify singularities up to stable equivalence. Before we do that, let us
first recall Milnor's results on the topology of hypersurface singularities.
Let :f u c~n+l~ be an analytic function on an open neighborhood U
of 0 in ~n+l . We denote
B z ¢ ~n+l ~}
£ = : LEzII <
z ~ ~n+l
S £ = ~B £ = : llzlI = £)
Then:
Theorem I.i For g > 0 small enough the mapping ~£: S - If = O
defined by ~g(z) = f(z)/If(z) I is a smooth fibration.
Theorem 1.2 For £ > 0 small enough and g >> ~ > 0 the mapping $~,~:
(Int B£) 0 f-l(~D~ ) ~ S 1 defined by ~E,~(z) = f(z)/If(z) I , where
~ = z ¢ 6: zI I = ~ , is a smooth fibration isomorphic to SE by an
isomorphism which preserves the argt~nents.
