Harmonic Maps: Proceedings of the N.S.F.-C.B.M.S. Regional Conference, Held at Tulane University, New Orleans December 15–19, 1980 PDF
Preview Harmonic Maps: Proceedings of the N.S.F.-C.B.M.S. Regional Conference, Held at Tulane University, New Orleans December 15–19, 1980
Lecture 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 This work is subject to copyright. All rights era ,devreser whether the whole or part of the lairetam is concerned, specifically those of translation, reprinting, re-use of illustrations, ,gnitsacdaorb reproduction yb photocopying machine or similar ,snaem dna storage in data .sknab Under § 54 of the namreG Copyright waL where copies era edam for other than private use, a eef is elbayap to tfahcsllesegsgnutrewreV" Wort", Munich. © yb galreV-regnirpS Berlin Heidelberg 2891 Printed in ynamreG Printing dna binding: Beltz Offsetdruck, .rtsgreB/hcabsmeH 012345-0413/6412 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.
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