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dz 2 = _(¢*h)2,O
i(l¢x l2
l¢yl2 - 2i(¢x' ¢y»dz 2.
_~
/-~
Since the adjoint * is taken with respect to a Hermitian metri5,/we see that '1* S = 0 is
'" (e.J' e..,) is an •
J
phic normal
equivalent to (iJ/iJz)(
== 0, so that we obtain the welY-known
(9.9) PROPOSITION. (i) (¢*h)2,O
0 iff ¢ is conformal.
(ii) If ¢ is harmonic, then (¢*h)2,O is a holomorphic quadratic differential (i.e., a dif ferential adz 2 such that iJa/dz = 0).
(9.1 0) REMAR K. A posteriori it is not surprising that the stress-energy tensor leads to V f> 0 on a d od and hence
l
this result in dimension two. Indeed, minimal maps of a plane domain to Rn-and later of a surface to a manifold-were obtained as extremals of E for variations of ¢ and the metric (or
56
JAMES EELLS AND LUC LEMAIRE
conformal structure) of the domain. In [13 J this last variation is in fact obtained as are parametrization of the domain. But this is precisely what was done in §6 in all dimensions. (9.11) Proposition (9.9) is also the basis of the known results of nonexistence of harmonic representatives in homotopy classes of maps between surfaces [21, 19]. We pre sent two simple samples: Note that if M, g is a surface, the harmonicity of maps ¢: M, g -
N, h depends only
on the conformal class of g. This is an immediate consequence of the composition law (2.20) and the fact that a holomorphic map is harmonic.
(9.12) PROPOSITION. (a) Any harmonic map ¢from the sphere S2 to a manifold is
conformal.
(b) On the torus r2 = R 2 ! A, where A is a lattice, consider a global holomorphic
linear system of coordinates z = x -I.. iy. Then any harmonic map ¢>: r2 - N, h satisfies
for suitable a, b E R. By the above remark, the given metric h on r2 can always be replaced by a flat one, conformally eqUivalent to h. PROOF OF (a). Any holomorphic quadratic differential on S2 must be identically zero. Therefore (¢*h)2,O is null and ¢ is conformal. PROOF OF (b). In the Euclidean chart (x, y) on R2-the universal covering of r2_ the coefficient of (rp*h)2,O is a bounded holomorphic function, hence a constant. The following nonexistence theorem is a special case of the results of Eells-Wood [21]: (9.13) THEOREM. Any harmonic non ± holomorphic map ¢>: r2, g -
homotopic. In particular, there is no harmonic map of degree one from
r2
S2, h is null to S2.
RecalJ that for maps from an oriented surface to a sphere, the degree (8.11) parame trizes the homotopy classes. PROOF. Let ¢> be a harmonic non ± holomorphic map and consider again the Eucli dean chart (x, y) on r2. Then either ¢x or ¢>y has no zero on r2. For otherwise at a point where ¢>x = 0, we would have a ~ 0, b = 0 and where ¢y = 0, we would have a ;;;;. 0, b = 0, so that a = b = 0 and ¢ would be ± holomorphic. Assuming ¢>x 4= 0, we note that ¢x and J¢>x both defme nonzero sections of ¢> -1 rs2, so that this bundle is trivial. If cleW) E Z denotes the first Chern class of the line bundle W, we have
so that deg(¢» = O. If now ¢> is harmonic and of degree one, it must be holomorphic and hence a branched covering and a diffeomorphism (since deg(¢» = 1), which is impossible.
HARMONIC MAPS
57
d as a re dimensions. nCt; of )]. We pre
(9.14) PROPOSITION [19]. Let ¢: p2, g --'" S2, h be a harmonic map from the real projective plane to the sphere. Then ¢ is constant. In particular, the (single) nontrivial
homotopy class of maps from p2 to S2 contains no harmonic representative.
pends only on law (2.20)
Let S2, 11 *g --",1T p2, g be the universal Riemannian covering of p2 by S2. We define a map ¢ by the following composition
Another example is the
(
PROOF.
manifold is norphic h satisfies
, a flat one, entically zero.
Then ¢ is harmonic and hence conformal. Its degree is the image by ¢* of IH2(S2) in Z. But ¢* = 11* 0 ¢* is zero, for it is a homomorphism of Z to Z factoring through Z2' so that deg(¢) O. The map ¢ is conformal and of degree zero, and so must be constant, together with ¢. (9.15) In the remainder of this section we shall give a lower bound on the index of certain harmonic maps from a Riemann surface M, g to a Kahler manifold N, h. We shall consider only harmonic non ± holomorphic maps ¢: M, g --'" N, h, since ± holomorphic maps always have index zero (8.14). We first note for future reference that if z is a local complex coordinate on M, the equation 1{¢) o is equivalent to
Jg of T2_ :It.
a¢
(9.16)
vajaF az
Is-Wood [21]: S2, !J is null
S2.
o
or
where we go back to the notations of (9.3)-(9.5). Indeed, these are equivalent to va/axa¢/ax + va/aya¢/ay O. (9.17) Consider now a variation of a harmonic map ¢: M ---+ N by a complex para meter sEC. We have by (8.13) the equations
11) parame
aE
in the Eucli 'wise at a point ; a;;;;' 0, b = 0, sof¢-lTS2 ,
; line bundle W,
aE"
ax = 2 as
and
a 2E OSOS
so that it will suffice to estimate the second variation of E",
(9.18)
PROPOSITION,
02 E I/(¢)
osos
If ¢: M, g --'" N, h is harmonic, then
I s=o
¢_) z
where ( • > denotes the complex bilinear extension of h to TeN,
ence a branched
This formula was used by Suzuki [67,68J and Siu-Yau [64], and we shall follow the proof of [64]. Another form of (9.18) has been obtained by T. Ishihara [32].
58
JAMES EELLS Ai'1D LUC L':::MA!::>"E
PROOF.
We have lr
e (¢)
A
= <¢_, ¢->, z z
(9.19) Now we shall need the following two formulae:
wi
sil PROOF OF THE LEMMA. Using the properties of Kiihler manifolds, we shall consider
around a point Yo EN a system of holomorphic normal coordinates, so that at Yo' 0IJ. h _
=
a == 0_v hex,6_ and a'Yo 6 hex,6_ =
O. Using the formulae recalled in (9.1) for the
connectio~,6
fa
tu
and the curvature, we get
('1_¢_)"1 8
Z
m
= AS0_(¢2) + h'YP(o h _)¢::¢~ z ex {Jp S z
so that at the point Wo we have 51
= asa 0 -(¢'2) + h'P.E..(a h _)¢::¢~ s z as" ,6 p 8 2
(V '1_¢_)"1 s s z
Z
p j:
( t
which proves (i). Formula (ii) is established similarly. We now substitute (i) in (9.19) and obtain
Using the divergence theorem and (9.20) (ii), we note that
-J.,<'1 ¢-, v_~>-2i dzdz = fM(v. vsIL I/Ufdzdz .irA
8
S
Z
Z
~
S
Z
1..
HARMONIC MAPS
At s
=0
0, we have also
59
since ¢ is harmonic; thus we obtain Proposi
tion (9.18). (9.21) Note that
where HBRN denotes the holomorphic bisectional curvature of N Following [64J we shall now see that for special variations the formula reduces to a single term, so that it is possible to relate the second variation to geometry. The construction uses the following
lall consider .tYo,ap.h connection
(9.22) PROPOSITION. Let ¢: M, g - N, h be a harmonic map from a Riemann sur face to a Kahler manifold. Then the complex vector bundle ¢-l TIN - M admits a struc ture of holomorphic vector bundle such that morphic bundle TI*M ® ¢-l TIN PROOF.
a¢
is a holomorphic section of the holo
Recall that a complex vector bundle is holomorphic if its transition functions
for a suitable atlas are holomorphic. The operator a/af is well defined on the sections of such a bundle, because of that property. Now it follows from general results of Koszul-Malgrange [36] that for any complex vector bundle Von a surface, a complex connection
v on
V induces a holomorphic bundle
va/ az = a/az. Following J. C. Wood [71], we shall indicate a direct proof of this fact. It suffices to find in local charts of M some frame fields b (b I , ... , b n ) with the property that v%zb 0. Indeed, for any n-tuple of functions f i , we can set a(fibj)!az structure such that
j
= a/af and the transition functions for the trivialization given a az by these frame fields are hoiomorphic.
(ap/af)' bj , so that v /
To prove the existence of these b/s, let a = (aI' ... , an) be any local frame field around a point Xo EM, and write v . a A· a, where A is a matrix of complex func a10z tions. We shall look for the frame b ~ven in the form b = B . a, where B is a nonsingular matrix of complex functions in a possibly smaller neighborhood of xo' We require that = 0; i.e., that (aB/aZ) . a + B . A . a 0; it is known that such an equation can always be solved around a point (see e.g. [36, 50]). In local charts, we have
vo/ oz ¢. = 0
by (9.16), so that ¢z is holomorphic in the new
structure. Globally, this implies that a¢ is holomorphic. (9.23) For this structure, suppose that there exists a holomorphic section v of ¢-l TN _ M, and construct a variation ¢: C x M - N such that
,
as Is==0
== 0,
60
JAMES EELLS AND L UC LEMAIRE
v_v z
yields
= 0 and
= v_z ¢_s
O. For that variation Proposition (9.18)
( there e 1;:M 11 we obt
Thus we get the following result of Siu-Yau [64] :
(9.24) PROPOSITION. Suppose that the holomorphic bisectional curvature of the Kdhler manifold N is strictly positive and that ¢ is a harmonic non ± holomorphic map from the Riemann surface M to N. If there exists a nonidentically zero holomorphic variation v E C(¢ -I TN), then a 2 E"/asasls=o < 0; and the index of ¢ is positive.
P HBRN
¢: M
(: (!
«
Indeed, by hypothesis, ¢_ ::/:. 0 and ¢s ::/:. 0 on open dense sets, so that the proposition z
2
follows from the expression for a E"/asas and (9.21). Eells and Wood have observed that in certain cases the existence of such holomorphic variations can be guaranteed by the following
(~
compac phically
S. Mori (9.25) RIEMA..~N-RoCH THEOREM. Let W be a holomorphic vector bundle of com TJ plex rank n on the Riemann surface M of genus p. Denote by H*(M. W) the cohomology phic rna space of M with coefficients in the sheaf of sections of W, and by c 1 (W) [M] the first Chern map frc class of W evaluated on M Then (9.24) t dimcHO(M, W) - dimcHl(M, W) = c 1 (l\nW)[M] + n(l p). By a pn See e.g., [29, 31, 48].
We apply this to the case W = ¢ -1 T'N and obtain (see e.g. [31, p. 98])
cause of (9 strongly
so that
(9.26) PROPOSITION [23]. Let M be a Riemann surface ofgenus p and N a Kahler manifold of positive bisectional curvature. If ¢: M -l> N is harmonic and not ± hoiomorphic, then Index(¢) ~ (¢*cl(N)[M]
+ n(l
for JlI A The foll(
TH ¢:M-l> that K1V(
p»).
Indeed, by Proposition (9.24), any holomorphic vector field v along ¢ (Le., any ele ment of HO(M, ¢-I T'N)) provides a direction with negative Hessian. Therefore,
Siu
instance
TH Index(¢) ~ dim H°(lIJ,
-l>
N of degree deg(¢)
+ n(1
p»).
Let N = CP". Then for any harmonic nonholomorphic map ~
0, we have
Index(¢) ~ (deg(¢)(n
+ 1) + n(1
- p).
Then an) ApI
least two pact quol
icaOyeqt.;
r
I HARMONIC MAPS .osition (9.18)
're of the Jhic map from c variation
le proposition holomorphic tndle of com cohomology the first Chern
)
1- p).
nd N a Kahler t ± holomorphic,
61
(9.28) EXAMPLE. For any p;;;;' 1 and d with Idl ,,;; p 1 it was shown in [40] that there exists a Riemann surface M of genus p and a harmonic nonholomorphic map ¢: M ----+ S2 == CpI of degree d. We see that for d;;;;' p12, that map must have positive index. Notes and comments. (9.29) Using a complex version of Weitzenbock's formula (3.3), we obtain the following analogue of Corollary (3.4).
PROPOSITION [75,59]. Let M. N be complete Kahler manifolds with RicciM ;;;;, 0 and HBRN ,,;; 0 (the holomorphic bisectional curvature). Then for any holomorphic map ¢: M ---> N: (a) e(¢) is subharmonic; (b) if M is compact, then ¢ is totally geodesic; (c) if furthermore Ricci M '1- 0, then ¢ is constant. (9.30) Siu and Yau [64] use Proposition (9.24) to prove Frankel's conjecture that any compact Kahler manifold of strictly positive hoIom orphic bisectional curvature is biholomor phically equivalent to the complex projective space cp n (result which was also obtained by S. Mori in the framework of algebraic geometry [47]). That conjecture had previously been reduced to proving the existence of a holomor phic map ¢: S2 ~ N generating 1r2 (1V). Starting from the fact that an energy minimizing map from M = S2 or a union of spheres M generates 1r 2 (N), Siu-Yau use Proposition (9.24) to infer that this map is ± holomorphic on the various connected components of M. By a process of holomorphic deformation and by surgery, they conclude that M S2, be cause of the minimizing property of ¢. (9.31) Say that the curvature tensor R N of N, h is strongly seminegative (resp., strongly negative) at Yo EN if
for all A"', B"', C"', D'" E C with A"'B{J The following result is due to Siu [63]:
ex D{J '* 0 for at least one pair of indices (a, (3).
THEOREM. Let lVI, N be compact Kahler manifolds with R N strongly seminegative. If ¢ : M ---> N is a harmonic map and there is a point Xo E M at which rank d¢ ;;;;, 4 and such that RN (¢(x o)) is strongly negative, then ¢ is ± holomorphic.
[i.e., any ele
fore,
).
omorphic map
Siu also provided some important refinements and applications of that theorem. For instance THEOREM [63 J. Let M and N be as above with dimcM;;;;' 2 and RN strongly negative. Then any harmonic oriented homotopy equivalence is a biholomorphic diffeomorphism. ApPLICATION. If M and N are compact Kahler manifolds of complex dimension at least two having the same homotopy type, and if RN is strongly negative, or if N is a com pact quotient of a classical bounded symmetric domain, then M and N are ± biholomorph ically equivalent.
JAMES EELLS AND LUC LEMAIRE
(9.32) Generalizing results of [39, 64J we note the [22J. Let M be a compact Riemann surface and N a simply connected Kahler manifold with 1f 2 (N) generated by a holomorphic map cp l --» N. Then any map 1>: M --» N of minimum energy in its homotopy class is ± holomorphic. PROPOSITION
Some homotopy hypothesis is necessary, for there are K3 surfaces N which are sim ply connected Kiihler manifolds for which every holomorphic map 1> : M --» N is constant, and by a theorem of Sacks·Uh1enbeck [53 J maps. As an application, we cite the
1f 2 (N)
is harmonically generated by minimizing
COROLLARY [22]. If W:M --» T2 +n is a conformal immersion of a Riemann surface into a flat torus of dimension at least 3, then its Gauss map 1: M --» Qn == G°(2 + n, 2) has minimum energy iff M = S 2 or Wis a minimal immersion.
(9.33) In very broad terms, here is a statement of the main results of Eells-Wood [23]. Let M be a Riemann surface (open or closed) and 1> : M --» cp n a map into complex projec tive n-space. If H --» cp n denotes the Hopf bundle, we can defme a universal lift
en
y"v'
+ ly'
Using the Hermitian product ( , ), say that I/> is isotropic if (y I",cp, y "Pcp)
for all 1 ~
0: +~.
I
I
0
Say that a map into cp n is full if its image lies in no projective subspace.
CLASSIFICA TION THEOREM
t
[23]. Given an integer r (0 ~ r ~ n) and a full holomor· (fr- I 1\ gs- 1)1 : M --» cp n is a full isotropic harmonic
phic map f: M --» cp n , then I/> a map. Here f/i; is the kth associated curve of f [72], g f~_l and r + s == n. The corre spondence if, r) +--41/> is a bijection, where r is the y"-order of 1/>; i. e., the maximum dimen t sion of the subspaces spanned by y"i3cp, and where f is obtained from I/> by a procedure u somewhat similar to that giving I/> in terms off. If M is compact, then deg(l/» = degifr ) degifr _ I )· u
If M is the sphere S2, then any harmonic map I/> : S2 --» cp n is isotropic. If M is a torus T2, then any harmonic map of degree nonzero is isotropic. In particular, if n ;;;. 2, there are harmonic nonholomorphic maps I/> : M --» cp n of all degrees;;;' 0 if P = genus M is 0 or 1, and of degrees;;;' p + 1 in any case. An explicit con· struction can be given of a harmonic map I/> : T --» cp 2 of degree 1, in contrast to the case 1/>: T --» cp I = S2 treated irt-{~Finally, the estimate of the inctexln0i:1i) applies to these maps. REMARK. The classification of maps from S2 to Sn or RP n was obtained by Calabi [9] (see also [4 D. The present classification for cp n was initiated by work of Din-Zakrzew ski [15, 16] and Glaser-Stora [27J, and independently by D. Burns.
d n
h t1
e. it
T
I ly connected n any map
rhich are sim is constant,
Part II. Problems Relating to Harmonic Maps
y minimizing
Here is a list of unsolved problems (as of June 19, 1981; updated in January, 1983)
emann surface )(2 + n, 2)
drawn from the whole theory of harmonic maps. We have attempted a broad classification, but that was rather arbitrary; for instance, many of the problems in the first four sections have direct bearing on existence theory. Again, our background references are [18,41]; and
ells-Wood (23].
[97] for regularity theory.
)mplex projec-
I. Existence of harmonic maps.
I lift tP of ¢ to
(l.1) The basic existence problem is concerned with deforming a given map into a
I
split into its
harmonic map. It was proved in [20] that if M and N are compact Riemannian manifolds and
RiemN .;;;; 0, then for all !Po
E
a!p t
at
COO(M, 1'v) the heat equation = r(¢t),
has a solution
!Po'
(L2) Problem. Does the heat equation have a solution for all t;;;' 0, without curva ture restrictions on Jot? a fu:Z holomar The nonexistence examples [18, 11.7] show that we cannot always have a solution for Jic harmonic lctive subspace.
all t;;;' 0 which subconverges.
The carre On the other hand it might be possible to find examples where the response is affirma 1aximum dimen tive and yet the solution does not subconverge. (One might look at the case of warped prod procedure ) = degifr )
ucts [38], where existence for all t;;;' 0 has been established.)
Perhaps existence can be shown in the framework of stochastic Riemannian geometry utilizing the Gaussian measures on suitable Banach manifolds of maps due to P. Baxen
IC.
lie.
dale [79,80]. The study of the heat equation should be simplified in case M is a compact sym
-;. cp n of all
metric space of rank 1; for then the fundamental solution of its linear heat equation
,n explicit con
H: M x M x R (;;;' 0) -;. R (;;;' 0) has the form H(x. y. t)
ast to the case
the Riemannian distance function on M.
= k(dM(x. y), t), where d M is
(1.3) In case dim M == 1, the basic problem of the existence of a harmonic map in each homotopy class has an affirmative answer, and so does Problem (1.2). ,ned by Calabi
of Din-Zakrzew-
If dim M
2 and
11 2 (N)
0, each homotopy class contains a harmonic map [39,53], and
it seems reasonable to expect existence and subconvergence of the solution of the heat equation. 63
64
JAMES EELLS AND LUC LEMAIRE
In that order of ideas, it should be easy to extend the above existence result for dim M =: 2 and "2 (N) 0 to the case where N is complete and has suitable growth proper ties (as in [18, (5.2)]). ADDED IN PROOF. Indeed, the existence theorem remains true if we suppose that the complete manifold N is homogeneously regular and satisfies the following growth condition (analogous to [18, 5.2.b]): there exists a function f: R+ -+ R+ with lims_."J(s) 00 such that for some Yo N, the ball of center y and radius f(d(yo' y)) in N is contractible for each y. In special homotopy classes, this condition can be weakened or omitted. If dim M = 2 =: dim N, and we take M oriented and N the Euclidean 2-sphere S, then the degree parametrizes the homotopy classes of maps ¢: M S. For degree dq, ;;;" p genus M, the harmonic maps are holomorphic [21]; in particular, there is no harmonic map from any 2-torus T -+ S of degree 1. On the contrary, for Idl .;;;; p I and for metrles having sufficiently large groups of symmetries, there is a harmonic nonholomorphic map of degree d (which is not an absolute minimum of E in its homotopy class) [40 J • (1.4) Problem. Let M be a closed Riemann surface of genus p and CPo: M -+ S a map of degree Id I .;;;; p - 1. For an arbitrary metric h on S can a harmonic map cP: M S, h be
n a~
c(
OJ
oj
h( m
found which is homotopic to cpo?
One difficulty to be faced in that problem is that in some cases (e.g., for Idl = 1) there is no minimum of E in the given homotopy class [39]. For maps of a closed surface M into the real projective plane P, the homotopy classi fication (due to P. Olum) is known but much more subtle. We have certain existence and nonexistence results [I 9 J ; and have a problem analogous to (1,4). ADDED IN PROOF. See also: J. F. Adams, Maps from a surface to the projective plane. Bull. London Math. Soc. 14 (1982), 533-534. 0.5) In existence theory for dim M 2 and dim N;;;" 3, regularity is the same, but topological complexity increases. For instance [53], if 1f 2 (N) 0 then a basis of 1f 2 (N) modulo the action of 1f 1(N) can be harmonically represented. It is unlikely that all classes can be represented, in general; however, we pose the special (1.6) Problem. If N. h is a compact 3-manifold. can we represent harmonically each element of 1f2(N)? Perhaps the first case to consider is N = (Sl x S2) # (Sl x S2), with a
gu co ha: in hal an(
*"
suitable metric.
ha"
HOI
(1.7) Problem. If N, h is compact and simply connected, is every element of 1f 2 (N) harmonically representable?
We note that Futaki [91] has exhibited compact simply connected Kahler surfaces N (dimeN = 2) and classes in 1f2(N) which are not represented by E·rninimizing maps. (1.8) The theorem of Ruh·Vilms (see Part I (2.25)) produces harmonic maps :
m=
pIe:
M -+ Qn -2 = d,:.2 of surfaces into the complex quadric, from solutions to the following
0.9) Problem. Find isometric immersions M -+ Tn with constant mean CUrJiature. (Here Tn is a flat n·torus.) For instance, see the helicoidal surfaces in T3 given in [86]. (1.10) Suppose now that dim M;;;" 3. The question of regularity appears to be much more complicated-in such a manner that we can expect a much wider variety of nonexis tence examples for harmonic maps. It is therefore a paradox for the moment that we have
cont esser limit Bern
r
I 65
HARMONIC MAPS
lit for wth proper ose that the h. condition (s) 00 such ctible for
l. here S, then 'rp ;;. p
=
[monic map , metrics ,hic map of
r--..s a map 'vi --.. S, h be
Idl
= 1)
topy classi· stence and Ijective plane, l
no nonexistence examples for dim M;;' 3 and aM = ¢; and no especially promising methods at hand. However, for m ;;. 3: (a) J. C. Wood [115] has shown that if ¢: D m --.. N, h is harmonic and ¢l ilDm is constant, then ¢ is a constant map. (The case m = 2 was proved earlier by Lemaire [39], using different methods.) Here D m denotes the flat m-dimensional disk. (b) There exists a weakly harmonic Li-map D m --.. Sm which is not continuous [98]. (c) There is no suspension of the identity map Sm -1 --.. Sm -1 as a harmonic map of Sm to a long ellipsoid [111,112]. We might suspect that nonexistence could occur because of local excess of curvature of N, without any topological restriction required. We are led therefore to the (1.11) Problem [41]. Let M, N be compact manifolds of dimension;;' 3, and H a homotopy class of maps M --.. N, in which all maps have maximal rank;;' 3. Do there exist metrics on M and N with respect to which H contains no harmonic representative? Again, the idea might be to put an excess of positive curvature around a point of N. Special situations-with possible topological obstructions- should be more accessible: (1.12) Problem. For m ;;. 3 is there a harmonic map of degree 1 from T m to Sm? We have no idea whether nonexistence of such a map when m 2 should serve as a guide. For instance, we have seen in Part I (9.14) that every harmonic map ¢: p2 --.. S 2 is constant, where p2 denotes the real projective plane. By way of constrast, R. Wood [18, §8] has noted that an isoparametric example of E. Cartan produces a harmonic map p4 --.. S4 in the nontrivial homotopy class of such maps; and as we have seen in (5.5), a nonconstant harmonic map pm -+ Sm has positive index (for m ;;. 3). Thus matters are not so simple, and we pose the (1.13) Problem. Is the generator of
same, but
[pm, Sm] = Z
of "2(N) l t all classes nically each < S2), with a
= Z2
for m odd, for m even
harmonically representable for m ;;. 3? (1.14) Problem [111, Il2]. Which classes of [S3, S2] have harmonic representatives? That question was posed by Smith [111, Il2], who constructed harmonic maps whose Hopf invariant is k 2 , for any given integer k. (1.15) Problem. Is every harmonic map cpm -+ Cpn constant for m > n?
».
r surfaces N
maps. : maps ~ following : curvature. 1in [86]. 5 to be much of nonexis that we have
Certainly every such ± holomorphic map is constant (see Part I (8.21 The case m = n + 1 is especially interesting, for we know the homotopy classification, and it is sim ple: [Cpn ... 1 , Cpn] = 0 for n odd,
Z2
for n even.
(1.16) Smith [112] has shown that for m .;;;; 7 every homotopy class of maps Sm -+ Sm contains a harmonic representative. We have no idea whether that dimension restriction is essential. However, in the delightful company of Giusti and Miranda it was found that the limitation of Smith's method was just that which appears as limitation in the solution of the Bernstein problem.
66
JAMES EELLS AND LUC LEMAIRE
flJthough several interesting harmonic maps Sm ~ Sm are known for m
> 7 (see
[18,
is
§8]), no general existence theorem is known in these dimensions. (1.17) Problem. Discuss the existence of harmonic maps Sm
~ sm for m > 7. (1.18) Existence theory is especially important and interesting for domain manifolds
ane to
M with boundary. See [18, § 12; 99 and 115] for an account of the main results obtained to date. In that case, one can consider homotopy classes relative to Dirichlet, Neumann or mixed boundary problems. Apart from the results mentioned above, it is not clear what kind of conditions on, say, Dirichlet data would imply existence or nonexistence of a solu tion.
C1 me diff
2. Regularity problems.
(2.1) The fundamental regularity theorem for harmonic maps is that a continuous weakly harmonic map of class L~ is smooth [18,97]. As already noted in (1.3), if dim M == 2 certain existence results can be obtained from a theorem of Morrey to the effect that an L -map from M ~ N, h is smooth, provided it minimizes E over all sufficiently
i
can
an
small disks.
(2.2) Problem. Let M be a closed Riemann surface and if>: M ~ N, h an Li-harmonic map. Is if> continuous? A partial result has been obtained by Gruter [95], who has shown that an Li-harmonic map if> which is weakly conformal (in the sense that (¢*h)2,0 == 0) is continuous. Grilter's proof utilizes a property of approximate differentiability of q-maps from surfaces (due to H. Federer). Problem (2.2) is a special case of the following conjecture of S. Hildebrandt [97, §2], dim which we formulate as (2.3) Problem. Let M be a surface and rp: M, g ~ N, h an q-extremal of a func tional
F(if» == f'l1f(x, rf>(x), \lqi(x))vg
is
where f has quadratic growth in its third variable. If F conformally invariant with respect to M, is rp continuous? (2.4) For any manifolds M and N, Schoen-Yau (107) have indicated how an ac tion of an Li-map on classes of loops defines a conjugacy class of homomorphisms 'lT (M) ~ 'IT 1 (N), which coincides with the usual class if the map is continuous. Together l with Morrey's regUlarity theorem, this leads to an alternative proof of existence of harmonic maps, provided dim M = 2 and 'lT2(N)
appt
= O.
If dim M;;;' 3 let a be a given conjugacy class of homomorphisms
'IT 1 (M) ~ 'IT 1 (N);
Geo
let F denote the set of Lf-maps inducing a. It follows from [107) that E takes an absolute minimum at some map 0 E F. That led Schoen to pose the
Sch(
strai (2.5) Problem [105]. Is that absolute minimum 0 continuous? If not, is if>o contin rp:k uous off a subset of codimension 2? tinuc ADDED IN PROOF. This problem has been solved by Schoen-Uhlenbeck, see (2.10) below. (2.6) Problem. Let
~m
1>: it!, g
C
h(!.>
~ N, h be a continuous Li-map between two manifolds
of the same dimension, and S¢ its stress-energy tensor. Suppose (1) that the Jacobian of if>
The redu
T
! 67
HARMONIC MAPS
, 7 (see [18,
n> 7. manifolds
is defined and positive almost el'erywhere; (2) div So == 0 in a distributional sense. Is ¢ harmonic? It is Li-harmonic? As Sealey has emphasized [59], there is special interest in the case dim M = 2 dim N and (Scl»2,O == _(¢*h)2,o holomorphic, for a positive answer would give a positive solution
ts obtained
to Shibata's problem [109; see §5 below]. The answer is yes if ¢ E C2 [70]; and if ¢ is a
eumann or
C1-diffeomorphism between compact surfaces [59]. We also call attention to the announce
~ar
ment [108].
what
ADDED IN PROOF.
: of a solu-
Problem (2.6) has been solved affirmatively by Sealey for dim M =
= 2, see (5.2) below. (2.7) Instead of regularity for the second order system defining harmonic maps, we
dim N Iltinuous
can pose the following, which may well be easier:
(2.8) Problem. Let M and N be Kahler (or even Hermitian) manifolds and ¢: M
), if tc the effect
~
N
= O. Is ¢ continuous (and therefore holomorphic)? (2.9) In most discussions of regularity problems, the space L i(M, N) is defined (by
an Li-map such that E"(¢;)
'ficiently
means of an embedding of N in Rq) as
.i-harmonic {E Li(M, Rq): (M) CNa.e.}.
Li-harmonic s. Gruter's
Problem. Is the space C""(M, N) dense in Li(M, N)? Equivalently, is CaCM, N)
n
lces (due to
Li(M, N) dense in q(,tf, N)?
A modification of an example of [98] suggests that the answer could be negative for ldt [97, §2], dim M ~ 3, but says nothing about dim M = 2. ADDED IN PROOF. Problem (2.9) was solved by Schoen-Uh1enbeck (Boundary regu of a func larity and miscellaneous results on harmonic maps). Using the above-mentioned example, they show that C""'(M, N) is not always dense in Li(M. N) when dim M = 3. On the other hand, they prove that it is dense if dim M = 2. ADDED IN PROOF. with respect (2.10) Important regularity results have been recently obtained: M. Giaquinta and E. Giusti, On the regularity of the minima of variational integrals, how an ac Acta Math. 148 (1982),31-46. phisms The singular set of the minima of certain quadratic functionals, Analysis (to . Together appear). ; of harmonic ~ 1f 1 (N);
es an absolute
R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom. 17 (1982), 307-335. (2.11) Giaquinta-Giusti consider the case of maps from a manifold to Rn, whereas Schoen-Uhlenbeck study maps between two manifolds, realizing the range as a set of con
straints in a Euclidean space. This put aside, the main result in common is that an L; -map , is ¢o contin ¢: M ~ N which minimizes E locally is HOlder-continuous on M\SIj> = {x EM: ¢ is con see (2.10)
tinuous in a neighborhood of x}, where the singular set S
~
two manifolds facobian of ¢
The proofs are sinllilar: both appeal to Morrey's Dirichlet growth lemma, to scaling and to a re'duction theorem of Federer.
68
JAMES EELLS AND Lue LEMAIRE
(2.12) Schoen-Uhlenbeck refine the method in the following way: assume that N has
the property that for some integer k ;;;: 3 there is no nonconstant radial harmonic map
RiVQ;
-+
N
1-/
5j -
1
which minimize E on compact sets in Ri (3 0:;;; j 0:;;; k). If ¢: .M -+ N is an Li-map mini mizing E focaf(v, then the Hausdorff dimension of 59 is 0:;;; m S9 is a discrete set. If m
(2.13)
EXAMPLE.
< k + 1,
k
1. If m
k
+ 1, then
then ¢ is smooth.
Schoen-Uhlenbeck note that if the universal cover of N supports a
strictly convex function, then every harmonic map Si- 1 -+ N is constant for j;;;: 3 (Part I, Proposition (3.13)), so that the assumption on N in (2.12) is fulfilled. Consequently, any
Li-map ¢:
,~f --+ N which minimizes E locally is smooth.
This provides an entirely new proof-and substantial generalization-of the existence reo suIt of Eells-Sampson [20], by the method outlined in Problem (2.4). Note also that a com· plete simply connected manifold without focal point supports a strictly convex function (P. Eberlein, When is a geodesic flow of Anosov type? II, J. Differential Geom. 8 (1973), 565-577; also Y. L Xin, Nonexistence and existence for harmonic maps in Riemannian
manifolds).
(2.14)
EXAMPLE.
Let N. h be a closed surface with nonpositive Euler characteristic.
By Part I, (9.12), any harmonic map from S2 to N is constant, so that we find as above that
any homotopy class of maps from a 3-dimensional manifold to N contains a minimizing harmonic map.
a f
(2.15) These results apply also to manifolds M with boundary. Indeed, R. Schoen and K. Uhlenbeck (Boundary regularity and miscellaneous results on harmonic maps) have
Li(kf, N) Ii c2+~(aAI. N) is an E'minimum, then S9 is a compact subset of ivf\aM An explanation of the fact that S9 Ii aii-I is empty is a result of Wood [115,
Si
established that if ¢
see (I. 10 (a)) above] . Boundary regularity has also been obtained by J. Jost and M. Meier. (2.16) The methods of the papers above do not involve a priori estimates. Much pro gress in that direction has been made by M. Giaquinta and S. Hildebrandt, A -priori estimates
IT
for harmonic mappings, J. Reine Angew. Math. 336 (1982), 124-164. a
3. Holomorphic and confonnal maps.
(3.1) We have seen (part I (8.15)) that every holomorphic map ¢;: M
---+
N between
a1
Kahler manifolds is harmonic with respect to any admissible Kahler metrics on lvl and N. Of course, it is especially interesting to find conditions to insure the converse. The first of these occurred in [21], and was the source of nonexistence examples. Other occurrences form the
m
bases of striking applications by Siu [63] and Siu-Yau [64], in the presence of curvature reo
co
strictions on N; see Part I, (9.30) and (9.31). The following conjecture is due to S.·T. Yau [117].
[2
T
J 69
HARMONIC MAPS
that N has :. map
Ulp mini
+ 1,
then
(3.2) Problem. Let M, g and lV, h be compact Kahler manifolds of strictly negative
sectional curvature and of complex dimension;;;' 2. If ¢: M --+ N is a harmonic homotopy
equivalence, then ¢ is ± biholomorphic.
Under the further condition that N, h has strongly negative curvature and with no cur vature restriction on M, g, this is Siu's result [63]. Also in a related direction, Yau [117] has found the following result: Let dimcM 2 = dimcN and suppose that N is holomorph ically covered by a disk in C 2 . If ¢: M --+ N is an oriented homotopy equivalence, then ¢ can be deformed to a biholomorphic equivalence. On the other hand, there are compact Kahler surfaces N, h with strongly negative curvature which are not covered by a disk in C 2 •
. supports a ;:. 3 (part I, ,ndy, any
(3.3) Problem. Let ¢o'
--+
N be homotopic holomorphic maps between com
pact Kahler manifolds. Under what conditions on M, N will
There is a related question of T. Sunada [104J: In the above situation, let N be hy
) that a com
perbolic and ¢o surjective; then is
unction (P.
bounded domain. (Brody [82] has characterized compact hyperbolic manifolds as those
:1973), lannian
compact complex manifolds N for which every holomorphic map C --+ N is constant.)
aracteristic.
map of Morse index O. Is ¢ ± holomorphic?
P. F. Leung has posed the following
(3.4) Problem. Let M be a compact Kahler manifold and ¢: M as above that
limizing
--+
Cpn a harmonic
In the special case where M is a Riemann surface of genus p, then the response is affirmative if deg ¢;;;' n(p - l)/(n
+ 1)
[23]. In the general case, we expect some sort of
further topological restrictions on ¢ to be necessary . .. Schoen
laps) have a compact
(3.5) Problem. If ¢: M --+ N is a harmonic map between Kahler mamfolds, then its stress-energy tensor S1> satisfies 17'*S2,O
Wood [115,
+ 17"*SI,1
0,
(see Part I (9.7));lor dimcM;;;' 2 do these equations imply sigmficant global restrictions on ¢, as they do in case dimcM = 1 (Part I, (9.11)-(9.14))? . Much pro
ori estimates
(3.6) The following was suggested by J. C. Wood, and has served as motivation for much in [23]:
(3.7) Problem. If M is a closed Riemann surface of genus M = p and ¢: M --+ Cpn a harmonic map of degree ¢ ;;;. p, is ¢ weakly conformal? As we have already noted, the answer is yes if n = 1, with any metric on Cpl. It is
rv between
also yes if p = 1 [116]. On the other hand, for 0 ~ deg(
\1 and N. Of
nonconformal maps ¢: M --+ Cpn.
first of these [lCe1
form the
curvature re-
(3.8) Problem. Let M be a Riemann surface and N a simply connected Riemannian manifold. If ¢: M --+ N is a map of minimum energy in its homotopy class, is ¢ weak{Y conformal? If this is not troe, what examples can be given? An affirmative answer is known if N = S2 [39 J , and for a class of Kahler manifolds [22]. In these cases, ¢ is ± holomorphic. As we saw in (9.12), the answer is yes when M = S2.
70
JAMES EELLS AND LUC LEMARIE
We can ask the same question more generally, requiring rp to be a harmonic map of index O. 4. Construction/classification of hannonic maps. (4.1) An eigenmap rp: M -
S" of a compact manifold is one whose R"+ l-compo
nents are eigenfunctions of the Laplacian of M, all with the same eigenvalue X. Then (letting <1>: M R"+ 1 denote the composition of ¢ with the inclusion map S" c... R"+ 1)
- ~
+ X =
0
and
e¢>
= X/2.
In particular, ¢ is harmonic. Cheng [85] has noted in the case of S2 that the presence of such eigenmaps can restrict the possible metrics on M In many nongeneric cases the eigenspaces of
~
are of dimension
~
2. and provide ex
amples of harmonic maps [5]; however, we cannot prescribe the dimensions of the ranges. We limit the dimension in the following
(4.2) Problem. Characterize those compact M, g for which there is an eigenmap ¢: M -
S" with m ~ n.
(4.3) In particular, let <1>: R m
-
R" be a map whose components are harmonic homo
geneous polynomials of the same degree k, such that the restriction <1>1
sm-l
=,/,: Sm 'I'
1
_S"-I.
Then ¢ is a harmonic map of constant energy density. A few such maps are known, through the work of Hopf, E. Cartan, Lam, Adem; see [18, §8], The general problem was posed by
R. T. Smith [111]: (4.4) Problem. Classify the harmonic k-homogeneous polynomial maps <1>: Rm _ R" carrying sm-l - S"-I, Is there a general formula for the degree of ¢ =
-
lyl2, 2f(x, y»)
S". Smith also posed (4.6) Problem. Classify the orthogonal multiplications f: RP x RP - R". ClassIfy the (left and right) irreducible orthogonal multiplications f: RP x Rq - R". ADDED IN PROOF. For p q = 2 and 3, the orthogonal multiplications have been completely described by M. Parker (Orthogonal multiplications in small dimensions, Bull. London Math. Soc.). For p = q = 2 the possible dimensions of the image are 2 and 4, and q = 3 they are 4, 7, 8 and 9. for p (4.7) Problem. For which such multiplications f: RP x RP - R n is the induced map ¢: S2P-l _Sn a harmonic morphism? produces a harmonic quadratic polynomial map ¢: S2P-l -
Multiplication in the real division algebras provides examples for p = n
(4.8) Problem. Classify the harmonic morphisms ¢: Sm -
N, h.
1, 2, 4, 8.
I
I
71
HARMONIC MAPS
That should be within reach, at least for those ¢ which are also submersions (every
; map of
where).
I-compo Then (letting 1)
'* °
(4.9) The harmonic maps 1/>: T ----- Cpn of degree have been classified ([23] ; see Part I, (9.33» via their directrices. In particular, they exist for all degrees, provided n ~ 2. Let us take a harmonic map ¢: T ----- Cp2 of degree k bundle over Cp2:
Wk
---+)
pull back the Hopf circle
S5
1
'resence of
> 0, and
JSI
T-O:;,..·__) Cp2
l provide ex the ranges.
°
The total space Wk I/> IS5 is a compact oriented 3·manifold with lIiWk) =: for i,* 1. In fact, Wk is a nilmanifold, expressible in the form Wk =: r k \N, where N is the Heisenberg group
lfmonic homo
lOwn, through
and
rk
is the subgroup
was posed by
:fJ: Rm -R" 'm-I'
when
R" with onic map
Thus Wk has the structure of a reductive homogeneous space, carrying a unique N-invariant connection for which parallel transport is given by multiplication in N. Also, dx 1\ dy 1\ dz is the unique translation invariant measure on Wk of volume 1. With respect to that measure we have the decomposition
L2e Wk ; C)
posed R". Classify
~ions, Bull.
? and 4, and
+ t) = e 2 'ffip t[ex,
ep
{G E CO(C, C): GU + 1)
Then we have the linear isomorphism
he induced 1, 2, 4, 8.
Hp,
y, z). Let Cp := CO(Wk ; C) n Hp' Define the space of continuous theta functions on C as
where [E Hp if [(x, y, z ; have been
EB pEZ
= Gcn Mp:
and
ep ----- Cp
G(~
+ i) :::
defined by
Mp(G)(x, y, z):= e21riPZ-1rpy2G(x For references we recommend [77, 78].
+ iy).
e-'ffip(H-I
)Gcn}.
72
JAMES EELLS AND Lt:C LEMAIRE
(4.10) Problem. Find maps ljJ: Wk ---+ 55 C C 3 which cover harmonic maps ¢: T ---+ Cp2 of degree k, and which are expressed in terms of theta functions.
(4.11) Consider the smooth fib rations of Euclidean spheres ¢ ~hi 57 ---+ 54 with fibre 53 and structural group 50(4). Their Euler numbers W(~h,j)[54J h +j 1; and their Pontrjagin numbers PI (~h,j)[54J = ± 2(2h - I), with h(h - 1) 0 mod 56; see [88]. (4.12) Problem. If 9 is a harmonic morphism, is ¢ the Hopf fibration (11 = 0 or I)?
If not, then perhaps its class [¢] E 117(5 4 ) has no harmonic representative.
We have seen (Part I (5.17)) that any nonconstant harmonic map ¢: 57 ---+ 54 has
Morse index greater than its rank. (4.13) Problem. There is an example (based on a construction of Serre) of a smooth locally trivial fibration
which cannot be made into a Riemannian fibration with totally geodesic fibres. In general, can we endow such an M' with a Riemannian metric so that ¢ is a Riemannian fibration with minimal fibres? If so, then index(¢) ~ 4 [65]. General conditions have been given by RummIer [106] and Sullivan [113]. The exis tence of such a metric seems unlikely. 5. Properties of harmonic maps.
(5.1) Substantial applications underline the importance of knowing that certain har monic maps are diffeomorphisms, or have positive Jacobians. We begin with a question, a positive answer to which would provide the foundation for TeichmUller theory in the L context. (5.2) Problem. Let M, N be closed oriented surfaces of the same genus P. endowed with metrics g, h. Let H be a component of C(M, N) containing a diffeomorphism. Does H contain a harmonic diffeomorphism? The answer is yes if p = 0 or if Riem h ~ 0 [18, § 11.14]. For p > 0 we know that H contains a harmonic map 9. If ¢ is locally bijective then its Jacobian l¢ 0 on M, from which we can conclude that 9 is a diffeomorphism. (See (5.6) below.) Thus Problem (5.2) is solved if we can show that some harmonic map 9 E H is locally bijective. An important attack on that problem has been made by Shibata [109], with clarifica tions and modifications by Sealey [59]. However, we believe that the problem remains open. It has been reduced to the regularity question discussed in (2.6). ADDED IN PROOF. Problem (5.2) has now been solved affirmatively -by Sealey (The regularity of quasiconformal 5-harmonic mappings) who completed Shibata's program, using a theorem of Seretov [108] to solve Problem (2.6) in dimension two. See also H. Sealey, Harmonic diffeomorphisms of surfaces, Harmonic Maps Tulane (1980), Lecture Notes in Math., vol. 949, Springer-Verlag, Berlin and New York, 1982, pp. 140-145; and On the existence of harmonic diffeomorphisms of surfaces (preprint).
i·
'*
73
HARMONIC MAPS
-by 1. lost and R. Schoen (On the existence of harmonic diffeomorphisms between
ps
surfaces, Invent. Math. 66 (1982), 353-359), using a previous result of Jost (Univalency of -;. S4 with
harmonic maps between surfaces, 1. Reine Angew. Math. 324 (1981), 141-153).
1; and
see [88].
= 0 or I)?
(5.3) The following conjecture has been around for several years. Sampson has con sidered it, with the hope of finding a proof of Mostow's rigidity theorem via harmonic maps; it has been formally posed by Lawson-Yau.
(5.4) Problem. Let M, g and N, h be compact manifolds with strictly negative curva ture, and ¢: M ....... N a harmonic homotopy equivalence. Is ¢ a diffeomorphism?
f a smooth
In general, fibration
The answer is yes if dim M 2 dim N, and in the case of flat manifolds. However, as we have noted in [89], Calabi [84] has given examples of metrics g on the torus T m (m ~ 3) such that a harmonic map of Tm, g to a flat torus T m cannot be a diffeomorphism. (5.5) Problem. Are those maps homeomorphisms? Note that with respect to the local problem, H. Lewy [101] has shown that any harmonic homeomorphism between open sets of R2 is a diffeomorphism. By way of contrast, J. C. Wood [70] has observed that the map ¢: R3 ....... R3 given by (x, y, z) ....... (x 3
-
3xz 2
+ yz,
y - 3xz, z)
is a harmonic polynomial homeomorphism, with lacobian determinant
3x 2 •
m
. The exis
(5.6) It is well known that if U and V are domains in C and ¢: U""'" V is a holo morphic homeomorphism, then its Jacobian determinant =1= 0; in particular, ¢ -1: V ....... U is hoiomorphic. (5.7) Problem. If ¢: m ....... Cm is a complex polynomial map with Jacobian deter minant =1= 0, is ¢ biholomorphic?
c
ertain har luestion, a n the
Li
, endowed ism. Does
Bieberbach has given examples when m = 2 to show that such an assertion is not true for hoi om orphic maps, in general. Apparently it is known that in (5.7) we have dim(C m ¢(C m » .;:;; m - 2. See [102, 110]. The next problem is due to Lawson: (5.8) Problem. Let ¢: (Dm+ 1, Sm) ....... (Dm+ 1, Sm) be a harmonic map such that
¢Ism is a homeomorphism. If m .;:;; 5, is ¢ a diffeomorphism?
know that on M, from :oblem (5.2) nth c1arifica remains
completed dimension 1aps Tulane York, 1982, es (preprint).
The basic question of rank of harmonic maps has been studied by Sampson [54] : (5.9) Problem. Let ¢: M""'" N be a harmonic map and U an open subset of M such 0 or that rank ¢I u .;:;; k. Then does ¢ have rank ¢ .;:;; k on all ,M? The answer is yes if k k = 1 [54]; and of course, if M and N are both real analytic. This question might be examined in the framework of differential forms: If w is a
harmonic I-form with vector bundle values and wx: TxM""'" Vx has rank';:;; k on U, does w have rank';:;; k on M? Looking more closely at the singular set of ¢ (Le., the set of points of M at which rank ¢ is not maximal), we formulate the (5.10) Problem. Let ¢, l/I be harmonic (resp., holomorphic) maps between Riemann ian (resp., Kdhler) manifolds. If they have the same singular structure-in a sense to be made precise-do they essentially coincide?
74
JAMES EELLS AND LUC LEMAIRE
(5.11) J. C. Wood described completely the possible singularities of harmonic maps between surfaces [70j. For a map I/> with nonvanishing Jacobian of a closed Riemann sur face M of genus p to a flat torus, he showed that I/> has at most 2p 2 branch points,
(2p - 2)(6p - 6) general folds,
6p - 6 meeting points of general folds.
(5.12) Problem. Are these bounds attained? Are there similar bounds for maps onto a surface of higher genus, with a metric of constant negative curvature?
de: thE (p(
Such a restriction on the metric of the range is necessary, for without it a construction of R. T. Smith could produce harmonic maps with arbitrarily many folds.
(5.13) J. C. Wood has shown that for a nonconstant harmonic map 1/>: M, g -+ N, h
tria
between compact real analytic surfaces,
f
q,(JIf)
KNvh ;;. 21TX(I/>(M»,
where KN denotes the Gauss curvature of N [70j. He asks:
(5.14) Problem. Is there an analogous inequality for higher dimensional range? (5.1 5) Problem. What sort of Runge approximation theorem can be established for harmonic maps? For instance, let K be a compact subset of a complete manifold M, and 1/>0 a harmonic map of a neighborhood of K into N. Wben can we approximate 1/>0 uniformly on K by a harmonic map 1/>: M -+ N? We should expect topological restrictions on M - K (e.g., that M - K should have no compact components). See [81 J for the linear case. (5.16) Problem. Let K be a compact subset of M and 1/>: M - K -+ N a harmonic map. Under what conditions can we assert that I/> has an extension to a harmonic map 1/>: M -+ N? For instance, if D is a 2-disk and 1/>: D {O} -+ N a harmonic map of finite energy, then I/> extends to a harmonic map 1/>: D -+ N [53j. On the other hand, the analogous as sertion for m-disks (m ;;. 3) is false without further growth restrictions at O. See [18; 10.15, 12.10J. In general, we should expect restrictions on the capacity or Hausdorff measure of K. (5.17) Problem. Calculate the index of harmonic maps 1/>: Sm -+ Sn. If m;;' 3, we have seen in Part I (5.17) that index(l/» ;;. max . rank ¢; + 1. See also
[59J for special maps. (5.18) Problem. Let M be a closed Riemann surface of genus p and ¢;: M -+ CpI a harmonic map. What is the value of index(¢;)? If ¢; is holomorphic, and in particular if degree(¢;);;' p. then its index is zero. For 0< degree I/> < p - 1, on the other hand, nonholomorphic examples exist [40J, and by [23] . their index is ;;. (degree( 0) + 1 - p) for degree I/> ;;. p /2. 6. Spaces of maps.
into clos
han sal ( Rier:
ham exiSI
Ph. J tence. ;;. 3, gies.
also. , necte is of· Lie gl
(6.1) Let M, N be compact. The space H(M, N) of harmonic maps is locally com pact and locally finite dimensional.
carrie! gree p
75
HARMONIC MAPS
,nic maps :'11ann sur
(6.2) Problem. Is HOII, N) an absolute neighborhood retract? If RiemN ,.;;;; 0 and C is a component of C(1);!, N). then H ::::: C() H(fvi, IV) is a compact deformation retract of C. Therefore the answer is yes in that case. Also, for any point a EM the evaluation map eVa: H -+ N is an immersion onto a totally geodesic submanifold of N (possibly with Lipschitz boundary); see [66,57]. Furthermore,
r maps onto
1TlH) =
°
1T 1 (H)
centralizer of ¢* 1T 1 (M)
for i =F I, in 1T 1 (N) [96].
construction
,g-""N,h
range? Iblished for
Po
a harmonic
on K by a
K (e.g., that
(6.3) Problem. If M, g and N. h are compact and real ana(ytic, is the space H(M, N)
triangulable?
The answer is yes if Riemh ,.;;;; 0, as in (6.2). Also, the space of holomorphic maps be
tween compact complex manifolds is an analytic space [87], and hence triangulable.
(6.4) Problem. Under what conditions can we deform the components of C(M, N) into H(M, N)? (6.5) It is well known that every compact Riemannian manifold N, h has a nontrivial closed geodesic.
(6.6) Problem. Characterize those manifolds N, h for which there exists a nontrivial harmonic map ¢: S2 -+ IV. As we saw in (9.12), such a map is a minimal branched immersion. When the univer sal cover Fl of N is noncontractible, such a map exists [53]. On the other hand, if RiemJli ,.;;;; 0, then every harmonic map S2 -+ N is constant [20]. Another question analogous to one for closed geodesics:
harmonic )nic map
'1
(6.7) Problem. For a given metric h on S3, are there at least 4 geometrically distinct
harmonic maps? Are there 4 such maps which are embeddings? ADDED IN PROOF. For the existence of one embedding, see: F. R. Smith, On the
finite energy,
existence of embedded minimal 2-spheres in the 3-sphere, endowed with an arbitrary metric,
anabgous as
Ph. D. Thesis, Melbourne.
See [18; 10.15,
(6.8) Problem. Find topological restrictions on a manifold N. h to insure the exis tence of infinitely many harmonic maps S2 -+ N with distinct images? (6.9) As we have seen in (5.11) of Part I, if M is a product of spheres of dimensions ~ 3, then there are maps ¢: M -+ M homotopic to the identity with arbitrarily small ener
ff measure of K. . 1. See also
gies. N. Koiso posed the following
(6.10) Problem [105]. Characten'ze those compact mamfolds M with that property; also, characten'ze their homotopy types. ; zero. For
.(jj, and by
ADDED IN PROOF. Min-Oo has shown that this property is satisfied by all simply con nected compact Lie groups, a key ingredient in the proof being that the cut-locus of a point is of codirnension at least three (Maps of minimum energy from compact simply connected
Lie groups). locally com-
(6.11) Results of [21] show that a closed orientable surface M of odd genus p ;;;;;. 3 carries two metrics go and gl relative to which there is a harmonic map M, go ~ S2 of de gree p, but no such map M, gl -> S2. Similarly:
76
JAMES EELLS AND LUC LEivlAIRE
(6.l2) Problem. Give an example of a harmonic map rp: il{ g --- N, h between com pact manifolds such that the metric h can be deformed into a metric hI such that rp is not homotopic to a harmonic map M, g --- N, hI' (6.13) Problem. Is there a generic set of mettlcs II on 11i for each of which there exist infinite{v many geometrical{v distinct harmonic maps 52 --- N, h?
fok!
is ar
In the case of closed geodesics, see [76].
(6.l4) Calabi [9,83] has completely classified the harmonic maps 52 --- Rpn, real
projective n-space with its standard metric h of constant curvature. (6.15) Problem. Is that classification reflected when the metric h is perturbed? 7. Noncompact domains. (7.1) Of course, general existence questions arise when the domain M is noncompact a harmonic map being characterized as an extremal of £ for all compactly supported varia tions. See [56,100].
(7.2) Problem. Classify the harmonic maps ¢: R m --- Rn which have maximal rank almost everywhere. If m = 1, those maps are affine. If m == 2 nand ¢ is injective, then again ¢ is af fine. If m = 2 and n = 3, the only known (nonplanar) injections are the catenoid and the helicoid. There are many immersions. Calabi has asked (7.3) Problem. Does there exist a nonplanar harmonically embedded (or properly immersed) surface in R3 with no tangent plane passing through the origin? (7.4) Problem. Is there a harmonic map from R2 to the hyperbolic plane H2 of rank 2 almost everywhere?
dim tion
ham diffc beh,a (pan
shou
Certainly such a map ¢ must have £(¢) == 00. Furthermore, ¢ cannot have bounded dilatation. (7.5) Problem. R. Osserman [103] has asked for a classification of the injective
If k Llonj
{z E C: PI < /z/ < P2} with PI > 0 and E == harmonic maps : A --- E, where A {z E C: 0 < Iz/ < I}. Nitsche has exhibited an interesting example. (7.6) Problem. Let M, g be a complete noncompact manifold with dim M;;;' 3 and Riemg ;;;. O. Under what conditions on N, h can we conclude that a harmonic map
mani
¢: M, g --- N, h with maximal rank;;;' 3 must have £(
consi
The case dim M == 2 or max rank(¢) == 2 should be excluded, since any harmonic map
l/;: S2 --- N induces by restriction a harmonic map ¢: R2 == S2\{00} --- N of finite energy.
f!l=
for h
and ~
borde
8. Variations on a theme. The following are suggestions for possible extensions of the theory of harmonic maps. Their interest will depend on successful applications and geometric interpretation. (8.1) The Teichmuller map between closed Riemann surfaces appears as a harmonic map with respect to a "Riemannian metric" on the range which is degenerate at certain points (corresponding to zeros of a holomorphic quadratic differential). See [18, §ll].
That fined
77
HARMONIC MAPS
ween com !t if! is not h there
(8.2) Problem [105]. Develop an existence theory for harmonic maps between mani
folds with "Riemannian metrics" admitting certain (controllable) degeneracies. (8.3) Given a positive real valued function f on M, an f-harmonic map [18, § 10; 20] is an extremal of
Rpn. real An example of an [-harmonic map is given by the Kelvin transform [24]. If m
rbed?
dim M,* 2, they are harmonic maps for the metric g
= f 2 /(m-2)g;
For m
2 the situa
tion is different, but the known existence theorems for harmonic maps extend easily to f harmonic maps. oncompact Hted varia·
(8.4) Problem. Can a class of maps be specified as solutions to a first order system of differential equations, behaving with respect to [-harrnonicity the way ± holomorphic maps behave with respect to harmonicity? Does this lead to nonexistence results? (8.5) We have seen that a map ¢ is harmonic iff its differential satisfies b.d¢
ain ¢ is af· Jid and the
properly
=- 0
(part I (2.15». (8.6) Problem. Study the solutions of the equations I::>.d¢ = "d¢ for" E R, from the viewpoint of geometrical interpretations. It is important to keep in mind that the operator I::>. itself depends on ¢, so that we should not think of this problem as looking for a spectral decomposition of a fixed operator. (8.7) A polyharrnonic map of order k is an extremal of
: bounded
njecr've dB
If k > m12, then Fk satisfies Condition (C) of Palais-Smale; therefore, there is a polyhar monic map of order k in every homotopy class. (8.8) Problem. Study the existence of polyharmonic maps in the critical dimension
m = 2k. More precisely, what are the existence and nonexistence results analogous to those
'lie map
for harmonic maps in dimension 2? (8.9) The Plateau problem requires an extremal coboundary for a given closed sub manifold, without specifying its topological type. That suggests the following:
;6] .
Problem [90]. Given a compact orientable Riemannian manifold N and an integer m, consider pairs (¢, M), where M is a compact oriented Riemannian m-manifold with volume I
larmonic map
finite energy.
and ¢: M - N a smooth map. Say that two such pairs (¢o. Mo) and (¢)' M) bordant if there is
:ensions of
nons and
(1) a compact oriented Riemannian (m + i)-manifold W whose oriented boundary aW = M) - Mo has Riemannian structure that of Mo' M) ; (2) a smooth map ¢: W - N such that
.w;;:;:: 3 and
¢I A1k
a harmonic
= ¢k
for k
= 0,
are co
1.
1t certain
That cobordism is an equivalence relation on those pairs; and the energy functional is de
18, § 11].
fined on pairs.
78
JAMES EELLS AND LUC LEMAIRE
(8.l0) Problem. Under what conditions is there an extremal pair in a given cobordism class? (8.11) Dropping all orientability assumptions in (8.9), we can formulate an analogous problem in homology with Z2 -coefficients:
A class J.1. E Hm(N; Z2) is realized by a smooth map rf;: M -+ N of a compact m manifold M if the induced homomorphism 9*: Hmf.M; Z2) -+ Hm(N, Z2) carries the funda mental class of M onto J.1.. Thorn [114, Chapter III, 3] has shown that every class J.1. is so realized.
(8.12) Problem. When is a class J.1. E Hm(N; Z2) realized by a harmonic map 9: M, g -+ N, h of some M, g? (8.13) Problem. Develop a decent theory of harmonic maps between Riemannian piecewise linear mamfolds. Such spaces have canonically defined Lipschitz structures, and piecewise linear maps between them are Lipschitz. Stochastic Riemannian geometry might provide an interesting approach; see Part I (2.34).
fel
CO 1;:
21 tur
Ap SOl
din: Hol
tial! vo1. sc.'~
Nuc
419
r
ven cobordism ;;':1
analogous
npact m nes the funda ;lass 11 is so Bibliography for Part I ~onic
map
:iemannian linear maps :m interesting
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80
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Mec: and
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Inst.
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81
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57. - - , Compact group actions and the topology of manifolds with non-positive curvature, Topology 18 (1979),361-380; 21 (1982),483. 58. G. Stolzenberg, Volumes, limits and extensions of analytic varieties, Lecture Notes in Math., vol. 19, Springer· Verlag, Berlin and New York, 1966. 59. H. C. Sealey, Some properties of harmonic mappings, Thesis, Warwick Univ., 1980. 60. - - , Some conditions insuring the vanishing of harmonic differential forms (to appear). 61. - - , Harmonic maps of small energy, Bull. London Math. Soc. 13 (1981), 405-408. 62. 1. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105. 63. Y.-T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kahler manifolds, Ann. of Math. (2) 112 (1980), 73-111. 64. Y.-T. Siu and S.-T. Yau, Compact Kahler manifolds of positive bisectional curva ture, Invent. Math. 59 (1980), 189-204. 65. R. T. Smith, The second variation formula for harmonic mappings, Proc. Amer. Math. Soc. 47 (1975), 229-236. 66. T. Sunada, Rigidity of certain harmonic mappings, Invent. Math. 51 (1979), 297 307. 67. O. Suzuki, Theorems on holomorphic bisectional curvature and pseudoconvexity on Kahler manifolds, Anal. Funct. Kozubnik 1979, Lecture Notes in Math., vol. 798, Springer Verlag, Berlin and New York, 1979, pp. 412-428. 68. - - , Pseudoconvexity and holomorphic bisectional curvature on Kahler mani folds. 69. R. O. Wells, Jr., Differential analysis on complex manifolds, Prentice-Hall, Engle wood Cliffs, N. J., 1973; Graduate Texts in Math. 65, Springer-Verlag, Berlin and New York, 1980. 70. J. C. Wood, Harmonic mappings between surfaces, Thesis, Warwick Univ., 1974. 71. - - , Holomorphicity of certain harmonic maps from a surface to complex pro jective n-space, J. London Math. Soc. (2) 20 (1979), 137-142. 72. H. H. Wu, The equidistribution theory of holomorphic curves, Ann. of Math. Studies, no. 64, Princeton Univ. Press, Princeton, N. J., 1970. 73. Y. L Xin, Some results on stable harmonic maps, Duke Math. J. 47 (1980), 609 613. 74. K. Yano, On harmonic and Killing vector fields, Ann. of Math. (2) 55 (1952), 38-45. 75. S.·T. Yau, A general Schwarz lemma for Kahler manifolds, Amer. J. Math. 100 (I 978), 197-203.
Soc Mat fw.(
an.. 182
appt New 235 ~opi
(Pap
Press Math
d'un
Pate'( Sem.
in-positive £~ture
Notes
: Univ., 1980. ~l forms (to (1981), ~)
88 (1968),
igidity of anal curva
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Hall, Engle d New York, niv., 1974. omplex pro f Math. (980),609 i (1952),
4ath.
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•c. Beijing harmonic : of surfaces compact )p.23-26. mappings,
1
~,
Ann. Sci.
stem for )Vic1~nce,
r
leurs appli
~es
and the :. of Math. ormal map
.;.5 (1963), )rem, Proc. p.91-95. k, 1972. ABCDEFGHIJ -AMS-89876543
