Photo by V. Scfobogna JAMES EELLS. Opening a Workshop on Variational Analysis at the International Centre for Theoretical Physics (1986).

Transcription

1 HARMONIC MAPS

2 Photo by V. Scfobogna JAMES EELLS Opening a Workshop on Variational Analysis at the International Centre for Theoretical Physics (1986).

3 HARMONIC MAPS Selected Papers of James Eells and Collaborators Vfe V I World Scientific Singapore New Jersey London Hong Kong

4 Published by World Scientific Publishing Co. Pie. Lid. P O Bo* 128, Fairer Road, Singapore 9128 USA office: Suite 18, 1060 Main Street, River Edge, NJ UK office: 73 Lynton Mead, Torteridge, London N20 SDH Hie Publisher thanks the following for permission to reproduce the papers on the pages staled in this volume. Academic Press (pp ) Academic des Sciences, Institut de France (pp ) American Mathematical Society (pp ) Birkhauser Verlag AG (pp ) Ellis Horwood Limited (pp ) Indiana University Mathematics Journal (pp ) Institut Fourier (pp 53-61) Istituto Nazionale di Alta Matematica Francesco Severi (pp ) The Johns Hopkins University Press (pp 1-52) Kiuwer Academic Publishers (pp ) The London Mathematical Society (pp , ) Longman Group (pp ) Professor Shingo Murakami (pp 62-73) Pergamon Press PLC (pp 75-78) Scuola Normale Superiore di Pisa (pp ) Springer-Verlag (pp , , , ) Tala Institute of Fundamental Research (pp ) HARMONIC MAPS Selected Papers of James Eells and Collaborators Copyright 1992 by World Scientific Publishing Co. Pte. Ltd. AH rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic ormechanical, including photocopying, recording orany information storage and retrieval system now known or to be invented, without written permission from the Publisher. ISBN Printed in Singapore by JBW Printers and Binders Pte. Ltd.

5 V TABLE OF CONTENTS Introduction vii [1] with J. H. Sampson, "Harmonic mappings of Riemannian manifolds", Amer. J. Math. 86 (1964) [2] with J. H. Sampson, "Energie et deformations en geometric differentielle", Ann. Inst. Fourier 14 (1964) [3] with J. H. Sampson, "Variational theory in fibre bundles", Proc. U.S.-Japan Sem. Diff. Geom., Kyoto, 1965, pp [4] with J. C. Wood, "Restrictions on harmonic maps of surfaces", Topology 15 (1976) [5] "The surfaces of Delaunay", Math. Intelligencer 9 (1987) [6] "Minimal graphs", Manuscripia Math. 28 (1979) [7] with L. Lemaire, "On the construction of harmonic and holomorphic maps between surfaces", Math. Ann. 252 (1980) [8] with L. Lemaire, "Deformations of metrics and associated harmonic maps", Geometry and Analysis, Patodi Memorial Volume, Tata Inst., 1981, pp [9] with P. Baird, "A conservation law for harmonic maps", Springer Lecture Notes in Mathematics 894 (1981) pp [10] with J. C. Wood, "Maps of minimum energy", J. London Math. Soc. (2) 23 (1981) [11] with J. C. Wood, "The existence and construction of certain harmonic maps", Symp. Math. Rome 26 (1982) [12] with J. C. Wood, "Harmonic maps from surfaces to complex projective spaces", Adv. tn Math. 49 (1983) [13] with L. Lemaire, "Examples of harmonic maps from disks to hemispheres", Math. Z. 185 (1984) [14] "Variational theory in fibre bundles: Examples", Proc. Diff. Geom. Math. Phys., M. Cahen et al. (eds.), Math. Phys. Studies 3, Reidel, 1983, pp [15] with S. Salamon, "Constructions twistorielles des applications harmoniques", C. R. Acad. Set. Paris Ser. 1 Math. 296 (1983) [16] with J. C. Polking, "Removable singularities of harmonic maps", Indiana Univ. Math. J. 33 (1984)

6 vi [17] "On equivariant harmonic maps", Proc. Shanghai -Hefei Symp. Geom. Dtff. Eq., 1981, Science Press, Beijing, 1984, pp [18] "Regularity of certain harmonic maps", Global Riemanian Geometry, Proc. Durham, 1982, Eihs Horwood Series, 1984, pp [19] "Gauss maps of surfaces", Perspeilives in Mathematics, Anniversary of Oberwolfach 1984, W. Jager, J. Moser and R. Remmert (eds.), Birkhauser, 1984, pp [20] "Minimal branched immersions into three-manifolds", Proc. Univ. Maryland ( ), Springer Lecture Notes in Mathematics 1167, 1985, pp [21] with S. Salamon, "Twistorial construction of harmonic maps of surfaces into four-manifolds", Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 12 (1985) [22] "Certain variational principles in Riemannian geometry", Proc. V Int. Colloq. Diff. Geom. Santiago de Compostela, 1984, Pitman Research Notes No. 131, 1985, pp [23] with K.-C. Chang, "Harmonic maps and minimal surface coboundaries", Lefschetz Centenary, Mexico City, 1984, Conlemp. Math. 58, III (1987) [24] with K.-C. Chang, "Unstable minimal surface coboundarios", ^Ida Math. Sinica 2 (1986) [25] with A. Ratto, "Harmonic maps between spheres and ellipsoids", Internal J. Math. 1 (1990) [26] with M. J, Ferreira, "On representing homo to py classes by harmonic maps", Bull. London Math. Soc. 23 (1991) Additional References (not in this volume) [27] with L. Lemaire, "A repott on harmonic maps", Bull. London Math. Soc. 10 (1978) [28] with L. Lemaire, "Another report on harmonic maps", Bull. London Math. Soc. 20 (1988)

7 vii INTRODUCTION The Question of Existence Harmonic maps pervade differential geometry and mathematical physics. They include geodesies, minimal surfaces, harmonic functions, abelian integrals, Riemannian nbrations with minimal fibres, holomorphic maps between Kahler manifolds, chiral models, strings. Their analytical framework is just as significant. Harmonic maps <f>: M * N are characterized as the smooth critical points of the energy functional (» = l/2 J \$pf, a variational problem of the simplest kind in geometry. Its Euler-Lagrange operator r is a semi-linear uniformly elliptic system in divergence form (whose principal part is diagonal and which is quadratic in the first derivatives). These geometric and analytic aspects are inseparable - an undiminisfiing source of fascination to me. Comprehensive accounts of the qualitative theory and applications of harmonic maps are given in [27] and [28]. My intention now is to indicate the wide variety of methods which have been brought to bear on the problem of existence and regularity - as well as some awful gaps which remain. 1. Here is a deceptively simple start. If <f>\ M * N is a map between Riemannian manifolds and N is isometrically embedded in a Euclidean space V: *\ n v then the Euler-Lagrangian r((p) is the projection of the tangent bundle of JV of the Laplacian A($), where $ denotes <f> viewed as a map M * V; and the normal component involves only the 1-jet of <j>. Thus are we encouraged to adapt linear methods applicable to A. There is a serious early warning [1, 4E]: the energy functional E usually does not achieve its minimum on homotopy classes of maps. (Characterization of that phenomenon is given in [28, 2].)

8 viii 2. Even so, a delicate analysis of the heat equation ~T- = T(4H) with <p t \ t=0 = 4> (based on Weitzenbock's method of expressing A( d^ 2 )) established [l,ch. II] that if M and N are compact and the sectional curvatures of N are nonpositive, then for any initial map <j>' Af -* N the solution (<f> t ) is defined for all t > 0 and subconverges to a harmonic map minimizing E and homotopic to <j>. Hartman showed that {<f>t) actually converges [27, 6]; and established uniqueness [27, 5]. Another illuminating proof of the existence of an E-miriimum homotopic to <j>, based on methods of partial regularity, has been given independently by Giaquinta-Giusti and Schoen-Uhlenbeck [28, 3]. Refinements by the latter provide substantially greater applicability. 3. In [4] and then [7] homotopy classes of maps between compact Riemann surfaces were found containing no harmonic representative. The holomorphic/conformal structure on the domain was used essentially. For instance, there is no harmonic map of degree ±1 from a torus T 2 to the sphere S 2, no matter what Riemannian metrics are put on T 2 and S 2. By way of contrast, [12] exhibits harmonic maps of all degrees of any T 2 into the complex projective space CP n for n > 2. We have no example of a homotopy class H of maps (j>: (M, g) * (N, k) between Riemannian manifolds with dimm > 3, which does not contain a harmonic map. The search for such a class should be reconsidered in terms of the following recent surprise [26]. For any Ti there is a metric g on M conformally equivalent to g and a harmonic map d>; (M, g) * (N, h) in H, provided dimm ^ 2. (That exception was expected on elementary grounds, for E is a conformal invariant of surface domains.) 4. The case dimm = 2 has several other special features which permit (a) synthetic constructions [7], [11], [12] of harmonic maps of M into manifolds with suitable symmetries; (b) twistor constructions [15], [20], [21], developing a technique in holomorphic geometry initiated by E. Calabi [28, 57]. Nonetheless, our knowledge of harmonic maps with surface domains remains embarrassingly inadequate. 5. Harmonic maps between Euclidean spheres have resisted general methods of attack. For instance, the Hopf fibration <f>: S 3 * S 2 is harmonic,

9 satisfying the system A$ = \d4>\ 2 9. (Indeed [27, 8], its components are eigenfunctions of A.) However, we do not know whether every map S 3 * S 2 can be deformed to a harmonic map. An important contribution was made in the thesis of R.T. Smith [27, 8] and [28, 10]. Working in an equivariant context, he constructed the harmonic join of suitable eigenmaps. That led to an ordinary differential equation whose solution - with asymptotic limits - determines an equivariant harmonic map. In that way Smith proved the existence of harmonic maps S n * S n of all degrees, provided n < 7. W.-Y. Ding [28, and 10.40] recast Smith's reduction as a variational principle for real functions on an interval - thereby permitting somewhat greater flexibility. For instance [25], let n > 3 and Q>, 6) = {far, y) R* +1 x R" +1 : \x\ 2 /a 2 + \y\ 2 /b 2 = 1} with p 4- q 4-1 = n and a, b > 0. Assume b 2 ja 2 > (n - 3) 2 /4(n - 2). Then any map d>: Q n (a, b) i Q n (a, b) can be deformed to a harmonic map. Similarly, for any integers k, I there is a harmonic map (in fact, a harmonic morphism) <j>: Q 3 (a, b) * Q 2 (a, b) with Hopf invariant k.i iff b 2 /a 2 = I 2 jk 2. James Eells January 1991