HARMONIC MAPS
Photo by V. Scfobogna
JAMES EELLS
Opening a Workshop on Variational Analysis at the International Centre for Theoretical Physics (1986).
HARMONIC MAPS
Selected Papers of James Eells and Collaborators
Vfe V I
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V TABLE OF CONTENTS
Introduction [1] with J . H. Sampson, "Harmonic mappings of Riemannian manifolds", Amer. J. Math. 86 (1964) 109-160.
vii
1
[2] with J . H. Sampson, "Energie et deformations en geometric differentielle", Ann. Inst. Fourier 14 (1964) 61-69.
53
[3] with J . H. Sampson, "Variational theory in fibre bundles", Proc. U.S.-Japan Sem. Diff. Geom., Kyoto, 1965, pp. 22-33.
62
[4] with J . C . Wood, "Restrictions on harmonic maps of surfaces", Topology 15 (1976) 263-266.
75
[5] "The surfaces of Delaunay", Math. Intelligencer 9 (1987) 53-57.
79
[6] "Minimal graphs", Manuscripia Math. 28 (1979) 101-108.
84
[7] with L . Lemaire, "On the construction of harmonic and holomorphic maps between surfaces", Math. Ann. 252 (1980) 27-52.
92
[8] with L . Lemaire, "Deformations of metrics and associated harmonic maps", Geometry and Analysis, Patodi Memorial Volume, Tata Inst., 1981, pp. 33-45.
118
[9] with P. Baird, "A conservation law for harmonic maps", Springer Lecture Notes in Mathematics 894 (1981) pp. 1-25.
131
[10] with J . C . Wood, "Maps of minimum energy", J. London Math. Soc. (2) 23 (1981) 303-310.
156
[11] with J . C . Wood, "The existence and construction of certain harmonic maps", Symp. Math. Rome 26 (1982) 123-138.
164
[12] with J . C . Wood, "Harmonic maps from surfaces to complex projective spaces", Adv. tn Math. 49 (1983) 217-263.
180
[13] with L . Lemaire, "Examples of harmonic maps from disks to hemispheres", Math. Z. 185 (1984) 517-519.
227
[14] "Variational theory in fibre bundles: Examples", Proc. Diff. Geom. Math. Phys., M. Cahen et al. (eds.), Math. Phys. Studies 3, Reidel, 1983, pp. 149-158.
230
[15] with S. Salamon, "Constructions twistorielles des applications harmoniques", C. R. Acad. Set. Paris Ser. 1 Math. 296 (1983) 685-687. 240 [16] with J . C . Polking, "Removable singularities of harmonic maps", Indiana Univ. Math. J. 33 (1984) 859-871.
243
vi [17] "On equivariant harmonic maps", Proc. Shanghai -Hefei Symp. Geom. Dtff. Eq., 1981, Science Press, Beijing, 1984, pp. 55-73.
256
[18] "Regularity of certain harmonic maps", Global Riemanian Geometry, Proc. Durham, 1982, Eihs Horwood Series, 1984, pp. 137-147.
275
[19] "Gauss maps of surfaces", Perspeilives in Mathematics, Anniversary of Oberwolfach 1984, W. Jager, J . Moser and R. Remmert (eds.), Birkhauser, 1984, pp. 111-129.
286
[20] "Minimal branched immersions into three-manifolds", Proc. Univ. Maryland (1983-1984), Springer Lecture Notes in Mathematics 1167, 1985, pp. 81-94.
305
[21] with S. Salamon, "Twistorial construction of harmonic maps of surfaces into four-manifolds", Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 12 (1985) 589-640.
319
[22] "Certain variational principles in Riemannian geometry", Proc. V Int. Colloq. Diff. Geom. Santiago de Compostela, 1984, Pitman Research Notes No. 131, 1985, pp. 46-60.
371
[23] with K . - C . Chang, "Harmonic maps and minimal surface coboundaries", Lefschetz Centenary, Mexico City, 1984, Conlemp. Math. 58, III (1987) 11-18.
386
[24] with K . - C . Chang, "Unstable minimal surface coboundarios", ^Ida Math. Sinica 2 (1986) 233-247.
394
[25] with A. Ratto, "Harmonic maps between spheres and ellipsoids", Internal J. Math. 1 (1990) 1-27.
409
[26] with M. J , Ferreira, "On representing homo to py classes by harmonic maps", Bull. London Math. Soc. 23 (1991) 160-162.
436
Additional References (not in this volume) [27] with L . Lemaire, "A repoTt on harmonic maps", Bull. London Math. Soc. 10 (1978) 1-68. [28] with L. Lemaire, "Another report on harmonic maps", Bull. London Math. Soc. 20 (1988) 385-524.
vii
INTRODUCTION
The
Q u e s t i o n of E x i s t e n c e
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 :
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 \ 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 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].)
viii 2.
Even so, a delicate analysis of the heat equation
~T-
= T(4H) with
2
(based on Weitzenbock's method of expressing A ( | d ^ | ) ) established [ l , C h . 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 ( ) is defined for t
all t > 0 and subconverges to a harmonic map minimizing E and homotopic to <j>. Hartman showed that {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.
I n [4] and then [7] homotopy classes of maps between compact Riemann
surfaces were found containing no harmonic representative. T h e holomorphic/conformal structure on the domain was used essentially. For instance, 2
there is no harmonic map of degree ± 1 from a torus T no matter what Riemannian metrics are put on T
2
to the sphere 2
and S . 2
trast, [12] exhibits harmonic maps of all degrees of any T n
projective space CP
2
S,
B y way of coninto the complex
for n > 2.
We have no example of a homotopy class H of maps (j>: ( M , g) —* (N, k) between Riemannian manifolds with d i m M > 3, which does not contain a harmonic map. T h e 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 d i m M ^ 2. (That exception was expected on elementary grounds, for E is a conformal invariant of surface domains.) 4.
T h e case d i m M = 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 meth-
ods of attack. For instance, the Hopf fibration : S
3
2
—* S
is harmonic,
satisfying the system 2
A $ = \d4>\ 9. (Indeed [27, §8], its components are eigenfunctions of A . ) However, we do not know whether every map S
3
2
—* S
can be deformed to a harmonic map.
A n 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.
T h a t led to an ordinary differential
equation whose solution - with asymptotic limits - determines an equivariant harmonic map. n
maps S
n
—* S
In that way Smith proved the existence of harmonic
of all degrees, provided n < 7.
W . - Y . Ding [28, §§10.31 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
2
2
2
with p 4- q 4- 1 = n and a, b > 0. Assume b ja n
any map d>: Q (a,
n
b) —i Q (a,
2
: \x\ /a
2
+ \y\ /b 2
2
> (n - 3) /4(n
= 1} - 2). T h e n
b) can be deformed to a harmonic map.
Similarly, for any integers k, I there is a harmonic map (in fact, a har3
monic morphism) <j>: Q (a, 2
I
2
b) —* Q (a,
2
2
b) with Hopf invariant k.i iff b /a
2
jk . James
Eells
January
1991
=
1
H A R M O N I C MAPPINGS O F R I E M A N N I A N MANIFOLDS.* By J A M E S E E L L S , J E . and J . H . SAMPSON.
Introduction. With any smooth mapping of one Riemannian manifold into another it is possible to associate a variety of invariantly defined functionals. Each such functional of course determines a class of extremal mappings, in the sense of the calculus of variations, and those extremals, in the very special cases thus far considered, play an important r61e in a number of familiar differential-geometric theories. The present paper is devoted to a rather general study of a functional E of geometrical and physical interest, analogous to energy. Our central problem is that of deforming a given mapping into an extremal of E. Following an infinite-dimensional analogue of the Morse critical point theory, we construct gradient lines of E (in a suitable function space) ; and E is a decreasing function along those lines. With suitable metric and curvature assumptions on the target manifold (assumptions which cannot be entirely circumvented, in view of the examples of §§4E and 10D), we prove that the gradient lines do in fact lead to extremals (see Theorem 11A). If f:M—*M' is a smooth mapping of manifolds whose metrics are gijdx'dx! resp. g $'dy dy , then the energy E(f) is defined by the integral a
&
a
, 9f 9fi a
where the f are local coordinates of the point f(x), *1 being the volume element of M (assumed compact). Thus E(f) can be considered as a generalization of the classical integral of Dirichlet. The Euler-Lagrange equations for E are a system of non-linear partial differential equations of elliptic type:
A is the Laplace-Beltrami operator on M and the r y"* are the Christoffel symbols on M'. Although this system is suggestive of the simple equation Aw4-
