数理解析研究所講究録
第 1605 巻 2008 年 151-169

151

Some Recent Developments in the study of minimal
2-spheres in spheres
J. Bolton, L. Fern\’andez and J.C. Wood*

Abstract
We discuss recent progress in the study of the space of harmonic maps from the 2sphere to the unit n-sphere in Euclidean $(n+1)$ -space. We consider the structure ofthis space as an algebraic variety, the existence of non-manifold points in this space,
and the relation between this question and the integrability of Jacobi fields along
harmonic maps. One of the main tools used is that of the twistor lift of a harmonic
map, which replaces a harmonic map by a holomorphic horizontal map into a K\"ahler

manifold.

Key words: Minimal surface; harmonic map; moduli space; infinitesimal deformation.
Subject class:1$53C42,53C43$ .

Introduction

A smooth map : $Marrow W$ between Riemannian manifolds $AI$ and $W$ is harmonic if it isof a smooth map : $Marrow W$
an extremal of the energy functional. Here, the energy
between compact Riemannian manifolds is given by$\phi$$\mathcal{E}(\phi)$$\mathcal{E}(\phi)=\frac{1}{2}/M|d\phi|^{2}\omega$where is the volume form on $M$ andeach point by$\omega$$|d\phi|$$\phi$,(1)is the Hilbert-Schmidt nom of$d\phi$given at$d\phi_{x}(e_{i})\rangle$$|d \phi_{x}|^{2}=\sum_{i}\langle d\phi_{x}(e_{i}),$of the tangent spaceof $M$ at . Equivalently, the map
for any orthonormal basis
is harmonic if it satisfies the Euler-Lagrange equations for the energy functional. Theseis a vector field along the map called
equations may be expressed as $\tau(\phi)=0$ , where$\tau(\phi)=$. Here denotes the connection
trace
the tension field, which is defined byon the bundle $T^{*}M\otimes\phi^{-1}TW$ induced from the Levi-Civita connections on $M$ and $W$ .
$\{e_{i}\}$$T_{x}\Lambda:[$$x$$\phi$$\tau(\phi)$$\nabla d\phi$$\nabla$*Gulbenkian Visiting Professor, University of Lisbon, during the preparation of this paper.

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For more details and an extensive survey of harmonic maps, with many references, see thearticles [16, 18].
, and hence
From now on, we assume that $M$ is 2-dimensional. In this case,$M$, and, if is conformal,
harmonicity of , depends only on the conformal structure of
the energy is equal to the area of the image of . lf the domain surface $M$ is the unit
, then an argument due to Hopf [32] involving holomorphic differentials
insphereis weakly conformal, and hence a map
shows that a non-constant harmonic map frombranched [30] immersion.
aminimalis harmonic if and only if it isfrom
has a long history
into the unit sphere
The case of harmonic maps fromwhich contains many beautiful and interesting results (see, for example, [10, 13, 14, 2]).
Although this is a special case of the more general case of harmonic maps of a Riemann, for reasons to do with the general theory of singularities of harmonic maps
surface into
[42, 43], it is arguably the most important case. It also has a wealth of interesting features.
forhas area
For instance [13], the area of the image of a harmonic 2-sphere insome integer . Further, if the map is full, that is to say its image is not contained in a
, then $m=2n$ for some integer , and $d\geq n(n+1)/2$ ,proper vector subspace ofIn 1975, Lawson [35] posed the problem of studying the structure of the space. In the present
with induced area
into
Harm $d(S^{2}, S^{2n})$ of harmonic maps ofarticle, we shall give a brief survey of some recent results we have obtained in this area; it
may be regarded as a sequel to [6], which appeared in the report of the first MathematicalSociety of Japan International Research Institute held at Tohoku University in 1993.
It was conjectured in [6] that $Harm_{d}(S^{2}, S^{2n})$ is a complex algebraic variety of dimension$2d+n^{2}$ , and this was proved by Fern\’andez in 2006. We give a brief account of the methodof proof in Section 7.
At the 1993 MSJ conference, Leon Simon asked about the singular points of the algebraic variety $Harm_{d}(S^{2}, S^{2n})$ . It is not hard to show that a non-full harmonic 2-spherewhich is the limit of a l-parameter faimly of full ones is singular, but the question ofwhether any full harmonic maps are singular points remains. In , it is shown that the‘inis a manifold for $d\leq 5$ ,of $fuU$ harmonic 2-spheres of area
space
iswhile recent work of Bolton and Fern\’aiidez, see Section 4, shows that
$d<6$
8,
this
is
,
see
Sectionalso a manifold. As the case $d=6$ is somewhat different fromperhaps rather a surprising result.
One way of understanding the space of harmonic maps is to look at their infinitesimal
deformations, or Jacobi fields; in particular, if they are all integrable, the space of harmonic
maps is a manifold with the Jacobi fields giving the tangent spaces. For $m=4$ , this has
been recently addressed by Lemaire and Wood [38], and a brief account of this work is
given in Section 8. The paper ends with applications of this to calculating the nullity ofthe energy, and a comparision with the nullity of the area functional.
$\mathcal{E}(\phi)$$i$$\phi$$\varphi$$\phi$$S^{2}$$\mathbb{R}^{3}$$\phi$$\phi$$S^{2}$$S^{2}$$\mathbb{R}^{m+1}$$S^{m}$$S^{2}$$S^{m}$$4\pi d$$S^{m}$$d$$\mathbb{R}^{m+1}$$n$$S^{2}$$4\pi d$$S^{2n}$$[9|$$4\pi d$$Harm_{d}^{fu1}$ $(S^{2}, S^{4})$$S^{4}$$Harm_{6}^{fu11}(S^{2}, S^{4})$Remark 1 Similar questions may be asked about the space of harmonic 2-spheres in complex space forms. This has been studied in [15, $36|$ for the case of harmonic 2-spheres in. In this case, the components of this space consist of the holomorphic and antiholoand energy $4\pi|d|$ , together with harmonic maps of degree andmorphic maps of degree$\mathbb{C}P^{2}$$\pm d$$d$153energy $4\pi E$ , where $E=3|d|+4+2r$ for some non-negative integer . It is shown in [36]that these components are smooth manifolds, of dimension $6|d|+4$ in the holomorphic andantiholomorphic cases and $2E+8$ in the other cases; in [37] it is shown that the tangent$r$bundle is given precisely by the Jacobi fields.

As in the talk on which it is based, the aim of this article is to give an overview and aflavour of the topic. The interested reader should refer to the papers cited in the text for
further details.

2

Early results

It is clear ffom the characterization of harmonic 2-spheres inas minimal branchedilmnersions that all great 2-spheres are harmonic. Rather more interestingly, we recall
that for each positive integer , the spaceof full harmonic 2-spheres inof area
is non-empty for each $d\geq n(n+1)/2$ .
In fact, some interesting special cases were studied in 1933 by Boruvka [10], who foundfull harmonic 2-spheres of constant curvature $K= \frac{2}{n(n+1)}$ in. The particular case$S^{m}$$d$$S^{2n}$$Harm_{d}^{fu11}(S^{2}, S^{2n})$$4\pi d$$S^{2n}$of $n=2$ gives the Veronese surface in$S^{4}$, given by$\phi(x, y, z)=(xy,$ $xz,$ $yz,$ $\frac{1}{2}(x^{2}-y^{2}),$ $\frac{x^{2}+y^{2}-2z^{2}}{2\sqrt{3}})$ ,$x^{2}+y^{2}+z^{2}=3$.

These Boruvka spheres all have the smallest possible area among full harmonic 2-spheres, namely $4\pi n(n+1)/2$ . However, in 1975 Barbosa [2] gave examples of full harmonicin
2-spheres inof area
for each $d\geq n(n+1)/2$ . Barbosa also showed that if $d=$$n(n+1)/2$ , then Harm $dfu11(S^{2}, S^{2n})=O(2n+1;\mathbb{C})$ .
The space $Harm_{d}(S^{2}, S^{2})$ consists of those maps $homS^{2}$ to itself which are holomorphic$(d\geq 0)$ or antiholomorphic $(d\leq 0)$ of degree, while there are no full harmonic 2-spheresin . Thus the first case where there are full harmonic maps of interest is Harm $d(S^{2}, S^{4})$ ,which may be studied using the the twistor fibration described in the next section.
$S^{2n}$$S^{2n}$$4\pi d$$d$$S^{3}$

3

The twistor fibration

We first recall the definition of the twistor fibration :. Regarding
as a left
quaternionic vector space, this is obtained by composing the Hopf map :given by$\rho([z_{1}, z_{2}, z_{3}, z_{4}])=[z_{1}+z_{2}j, z_{3}+z_{4}j]$ ,with the canonical identification ofandgiven by stereographic
projection of
from $(0,0,0,0, -1)$ onto the equatoria14-plane inwhich is includedin
by $[q]\mapsto[q, 1]$ . We recall [7, 11] that is a Riemannian submersion whenisgiven the lfubini-Study metric of constant holomorphic sectional curvature 1.
$\pi$$\mathbb{C}P^{3}arrow S^{4}$$\mathbb{H}^{2}$$\mathbb{C}P^{3}arrow \mathbb{H}P^{1}$$\rho$$\mathbb{H}P^{1}$$S^{4}\subset \mathbb{H}\oplus \mathbb{R}=\mathbb{R}^{5}$$S^{4}$$\mathbb{H}$$\mathbb{R}^{5}$$\mathbb{H}P^{1}$$\pi$$\mathbb{C}P^{3}$

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is said to be horizontal if its image is everywhere orthogonal to the
A map intoand full if its image is not contained in a totally geodesic
fibres of. It is easy tois holomorphic and horizontal thensee that if :harmonic,
isbut thecrucial result, as formulated by Bryant [11], is that:$\mathbb{C}P^{3}$$\mathbb{C}P^{2}$$\pi_{7}$$S^{2}arrow \mathbb{C}P^{3}$$\psi$$\pi\circ\psi$

Theorem 1 Every full harmonic map$\phi$:

is given by$S^{2}arrow S^{4}$(2)$\phi=\pm(\pi 0\psi)$. Every nonfor some uniquely-determined full honzontal holomorphic map :is the projection of a uniquefull (and hence totally geodesic) harmonic map :$\psi$$\phi$

honzontal totally geodesic$\mathbb{C}P^{1}$in$\mathbb{C}P^{3}$$S^{2}arrow \mathbb{C}P^{3}$$S^{2}arrow S^{4}$.

We $caU$ the sign in (2) the spin of . In some sense, this result reduces the study
to that of the space HHol
of Harmof horizontal holomorphic maps:
. As we shall see below, the latter space is much easier to work with, as it iscontained in the projectivization of a finite-dimensional vector space.
With the above as motivation, we now give an elementary description of the elements. Regardingof HHolasin the usual way, a map:
isholomorphic if and only if it may be written as$\phi$$(S^{2}, S^{4})$$(S^{2}, \mathbb{C}P^{3})$$\psi$$S^{2}arrow \mathbb{C}P^{3}$$(S^{2}, \mathbb{C}P^{3})$$S^{2}$$\mathbb{C}\cup\{\infty\}$$\psi$$S^{2}arrow \mathbb{C}P^{3}$(3)$\psi(z)=[f_{1}(z), f_{2}(z), f_{3}(z), f_{4}(z)]$where $f_{1}(z),$are polynomials which we may assume have no common zeros. The
degree of is then the maximum of the degrees of the polynomials $f_{1}(z),$,.
In this way, we identify the space Holof holomorphic 2-spheres of degreeinwith the projectivization of a dense open subset $V$ of the vector space,
whereis the vector space of complex polynoniials in with degree less than or equalto . It is easy to see [11] that a map of the form (3) is horizontal if and only if$\ldots,$$d$$f_{4}(z)$$\psi$$\ldots$$d(S^{2}, \mathbb{C}P^{3})$$f_{4}(z)$$d$$\mathbb{C}P^{3}$$(\mathbb{C}[z]_{d})^{4}$$\mathbb{C}[z]_{d}$

$z$

$d$

$f_{1}f_{2}’-f_{1}’f_{2}+f_{3}f_{4}’-f_{3}’f_{4}=0$,

in which case the corresponding harmonic map $\phi=\pi 0\psi$ has area

4

The structure of(4)$4\pi d$.

$HHo1_{d}(S^{2}, \mathbb{C}P^{3})$It follows from the previous section that, and hence $Harm_{d}(S^{2}, S^{4})$ , may be
given the stmcture of a complex algebraic variety in the projectivization of the vector space. By counting the number of constraints imposed by the horizontality condition(4), one might expect that the dimension of this algebraic variety should be$HHo1_{d}(S^{2}, \mathbb{C}P^{3})$$(\mathbb{C}[z]_{d})^{4}$$4(d+1)-(2d-1)-1=2d+4$ .
This was confirmed independently by Verdier and Loo [39, 45, 46, 47], who both madea detailed study of this variety, and, in particular, proved the following.

155
Theorem 2 (Verdier 1985, Loo 1989) For any positive integer $Harm_{d}(S^{2}, S^{4})$ is a connected algebraic variety of dimension $2d+4$ . When $d=1,2$ , it is irreducible; when $d\geq 3$ ,it has three irreducible components, namely the subset of non-full maps and the closures ofthe subsets of full maps of positive and negative spin.
$d,$

Of course, it is clear from the description above in terms of polynomials thatis empty for $d=1,2$ .
It is natural to ask if $Harm_{d}(S^{2}, S^{4})$ has any singular points. Non-full harmonic 2spheres inwhich are the limits of a l-parameter family of full ones are singular (seeSection 8); on the other hand, it is shown in [9] thathas no singularpoints for $d\leq 5$ and hence is a manifold. This uses the twistor correspondence describedin Section 3 to identify $Harm^{fu11}(S^{2}, S^{4})$ as a double cover of; in [5], itis shown that the compact-open topology on Har
coincides with that comingfrom the complex algebraic variety stmcture on. In fact, Lemaire andthe third author [38, \S 2] have shown that the correspondence is real analytic.
We now outline a proof of the fact thathas no singular points forsince the techniques will be useful later on. We letbe the dense open subset
of consisting of quadruplets of linearly independent polynommials. The condition (4) forhorizontality motivates our definition of$Harm_{d}^{fu11}(S^{2}, S^{4})$$S^{4}$$Harm_{d}^{fu11}(S^{2}, S^{4})$$HHo1^{fu11}(S^{2}, \mathbb{C}P^{3})$$m^{}$$(S^{2}, S^{4})$$HHo1^{fu11}(S^{2}, \mathbb{C}P^{3})$$Harm_{d}^{ful1}(S^{2}, S^{4})$$d\leq 5_{7}$$V_{0}$$V$$Q:V_{0}arrow \mathbb{C}[z]_{2d-2}$as$Q(f_{1}, \ldots, f_{4})=f_{1}f_{2}’-f_{1}’f_{2}+f_{3}f_{4}’-f_{3}’f_{4}$.

(5)We hope to show that the zero polynomial in
is a regular value of , so that
$Q^{-1}(0)$ is a manifold. Since
may be identified with the projectivization of$Q^{-1}(0)$ , it then follows that, and hence its double cover,is a manifold, in fact, by [38] a real-analytic submanifold of a suitable space of smooth
mappings fromto .
However, the dimensions of the spaces involved are quite high! For instance, if $d=5$
then the domain has dimension 24 and the codomain 9, so verifying that $dQ$ has maximal$\mathbb{C}[z]_{2d-2}$$Q$$HHo1_{d}^{fu11}(S^{2}, \mathbb{C}P^{3})$$HHo1_{d}^{fu11}(S^{2}, \mathbb{C}P^{3})$$S^{2}$$Harm_{d}^{fu11}(S^{2}, S^{4})$$S^{4}$

rank at all points of $Q^{-1}(0)$ is quite daunting.
We now describe how we may simplify the problem by using two natural group actions
on . Firstly, the standard action of the complexified symplectic group Sp 2,on
induces a natural action on
via $Af(z)=A(f(z))$ , and $Q$ is constant on the orbits of thisaction. Secondly, for each positive integer , a M\"obius transformation $\mu=(\alpha z+\beta)/(\gamma z+\delta)$
induces a diffeomorphism :given by$V_{0}$$($$\mathbb{C})$$\mathbb{C}^{4}$$V_{0}$$k$$\tilde{\mu}$$\mathbb{C}[z|_{k}arrow \mathbb{C}[z]_{k}$$(\tilde{\mu}f)(z)=(\gamma z+\delta)^{k}(f(\mu(z))$.

This, in turn, induces a diffeomorphism, also denoted , fromthat if $f=(f_{1}, f_{2}, f_{3}, f_{4})\in V_{0}$ , then$\tilde{\mu}$$Q(\tilde{\mu}f)=(\alpha\delta-\beta\gamma)\tilde{\mu}(Qf)$,$V_{0}$

to$V_{0}$. It is easily checked

156
so that the rank of $dQ$ at is equal to the rank of $dQ$ at Ajtf.
This reduces the problem to showing that the rank of $dQ$ is maximal at certain special
elements of . For instance, for $d=4$ it is shown in [8] that if $f\in V_{0}$ satisfies (4) thensuch that
there exists a M\"obius transformation and an element of Sp 2,$f$$V_{0}$

$A$

$\mu$

$($

$A(\tilde{\mu}f)(z)=(1,2z^{4}, -4z, z^{3})$$\mathbb{C})$.

Hence it is enough to show that $dQ$ has maximal rank at$(f_{1}(z), f_{2}(z), f_{3}(z), f_{4}(z))=(1,2z^{4}, -4z, z^{3})$,and this is easy to see.

For $d=5$ it tums out to be sufficient to consider the case$(f_{1}(z), f_{2}(z), f_{3}(z), f_{4}(z))=(a0+a_{1}z, b_{4}z^{4}+b_{5}z^{5}, c_{1}z+c_{2}z^{2}, d_{3}z^{3}+d_{4}z^{4})$,where horizontality reduces to the system of equations:$2a0b_{4}+c_{1}d_{3}=0$,$5a_{0}b_{5}+3a_{1}b_{4}+3c_{1}d_{4}+c_{2}d_{3}=0$ ,$2a_{1}b_{5}+c_{2}d_{4}=0$.

This was done by Bolton and Woodward [8], who thus showed that Harmis

$5fu1l(S^{2}, S^{4})$a manifold.
The third author of this article pointed out that the case $d=6$ may be worth invesare the limits of sequences oftigating because some harmonic 2-spheres of degree 6 in.
points of Harm
regular,notand hence areharmonic 2-spheres which are full in
Taking up the challenge, and using similar methods (and, initially, Mathematica) the firstis a manifold. In line with thetwo authors of this article have proved that Harm
$d=5$
is the following.
simplifying
result, the crucial
method used for $d=4$ and$S^{4}$$6(S^{2}, S^{6})$$S^{6}$$6\iota_{u11}(S^{2}, S^{4})$Proposition 1 Let $f\in V_{0}$ Then there exists a Mobius transformation$A$of Sp 2, such that
$($

$\mu$

and an element$\mathbb{C})$$A(\tilde{\mu}f)(z)=(a_{0}+a_{1}z+a_{2}z^{2}, b_{4}z^{4}+b_{5}z^{5}+b_{6}z^{6}, c_{1}z+c_{2}z^{2}+c_{3}z^{3}, d_{3}z^{3}+d_{4}z^{4}+d_{5}z^{5})$

,

$or$

$A(\tilde{\mu}f)(z)=(a_{0}+a_{1}z+a_{2}z^{2}, b_{4}z^{4}+b_{5}z^{5}+b_{6}z^{6}, c_{1}z+c_{2}z^{2}+c_{4}z^{4}+c_{5}z^{5}, d_{3}z^{3})$,with, in both cases, $a_{0}b_{6}\neq 0$ , and, in the second case, $d_{3}\neq 0$ , and where both right handsides satisfy (4).

157

5

Full harmonic maps fromspheres.

to $evenrightarrow dimensional$$S^{2}$As mentioned earlier, if a harmonic maps fromto a sphere is full, then the codomain
sphere is even-dimensional [13]. The study of harmonic maps $homS^{2}$ tofor general
has many common features with the case $n=2$ . The twistor fibration explained above
is a particular case of the general construction that appeared in [13, 2]. Recall that thetwistor space of the $2n$ -sphere, denoted, is defined as the subvariety of Gr(theGrassmanian of n-dimensional subspaces in) consisting of totally isotropic subspaces
respect to the complex-bilinear extension of the usual dot product. In other words,$S^{2}$$S^{2n}$$n$$(n, \mathbb{C}^{2n+1})$$\mathcal{Z}_{n}$$\mathbb{C}^{2n+1}$$lt\dot{q}th$$\mathcal{Z}_{n}=\{P\in Gr(n, \mathbb{C}^{2n+1}) :(u, v)=0\forall u, v\in P\}$,

where $( u, v)=\sum_{i=1}^{2n+1}u_{i}v_{i}$ for $u=(u_{1}, \cdots, u_{2n+1})$ and $v=(v_{1}, \cdots, v_{2n+1})$ in.
There is a projection
defined as follows: given, andan orthonormal basis of $P,$ $\pi(P)$ is defined as the (unique) real vector such that the basisofgiven byis orthonormal and positively oriented.
As in the $n=2$ case, we have the following [2, 13, 27]:$\mathbb{C}^{2n+1}$$\pi:\mathcal{Z}_{n}arrow S^{2n}$$\mathbb{C}^{2n+1}$$\bullet$$\{\pi(P), E_{1}, \ldots, E_{n}, \overline{E}_{1}, \ldots, \overline{E}_{n}\}$Given a harmonic and full map $\phi:S^{2}arrow S^{2n}$ there exists a unique holomorphic andhorizontal map :(the twistor lifl of ) such thatis either or.
$\psi$$\bullet$$\bullet$$\{E_{1}, \ldots , E_{n}\}$$P\in \mathcal{Z}_{n}$$S^{2}arrow \mathcal{Z}_{n}$Conversely, if :is harmonic and full.
$\psi$

The area of
the image of$\phi(S^{2})$$S^{2}arrow \mathcal{Z}_{n}$$\pi\circ\psi$$\phi$

is holomorphic, horizontal and full, thenis equal to$4\pi d$$1\in \mathbb{Z}\simeq H_{2}(S^{2}, \mathbb{Z})$, whereis the algebraic degree ofunder the map$d$$-\phi$$\phi$$\pi\circ\psi$$\psi$: $S^{2}arrow S^{2n}$(or equivalently,.

$\psi_{*}:H_{2}(S^{2}, \mathbb{Z})arrow \mathbb{Z}\simeq H_{2}(Z_{n}, \mathbb{Z}))$

An iimnediate consequence of this is that(i.e. the set of harmonic,
full maps from) can be identified with two copies ofto, whereHHodenotes the variety of holomorphic, horizontal, full maps of degree from.
to
Therefore, from now on, we will concentrate in the study of. For theparticular case of $n=2$ , recall that, and that the horizontality condition,
is justwritten in homogeneous coordinates in, is given by equation (4).
For general , it is certainly not the case thatis isomorphic to a complex projective
space. However, the variety
is birationally equivalent to(note that thedimension ofis $n(n+1)/2)$ . The idea would then be: Fix a birational map $bhom$. Then, for eachto, define the map:. This should give some variety of maps frominto
satisfying some
sort of horizontality’ condition. Then, instead of studying, study theset of such maps.
Of course this is $aU$ wishful thinking: the idea of the previous paragraph, althoughplausible, is full of obstacles. Several things can go wrong:$Harm_{d}^{fu11}(S^{2}, S^{2n})$$S^{2}$$S^{2n}$$HHo1_{d}^{ful1}(S^{2}, \mathcal{Z}_{n})$$1_{d}^{ful1}(S^{2}, \mathcal{Z}_{n})$$S^{2}$$d$$\mathcal{Z}_{n}$$HHo1_{d}^{ful1}(S^{2}, \mathcal{Z}_{n})$$\mathbb{C}P^{3}$$\mathcal{Z}_{2}$$\mathbb{C}P^{3}$$n$$\mathcal{Z}_{n}$$\mathbb{C}P^{n(n+1)/2}$$\mathcal{Z}_{n}$$\mathcal{Z}_{n}$$\mathbb{C}P^{n(n+1)/2}$$\mathbb{C}P^{n(n+1)/2}$$\mathcal{Z}_{n}$$\psi\in HHo1_{d}^{fu11}(S^{2}, \mathcal{Z}_{n})$$S^{2}$$b^{-1}\circ\psi$$\mathbb{C}P^{n(n+1)/2}$$\psi\in HHo1_{d}^{iul1}(S^{2}, \mathcal{Z}_{n})$$S^{2}arrow$158
1. Since a birational map is only defined outside of a codimension 2 subvariety, the map
will not be defined at all if the image of lies entirely in the subvariety whereis not defined.
$b^{-1}o\psi$$\psi$$b^{-1}$will translate into some condition for maps into
2. The horizontality condition in. But this condition may be much harder to work with than the original.
$\mathcal{Z}_{n}$$\mathbb{C}P^{n(n+1)/2}$

is defined, we also have to take into account that we want the degree
3. Even if
is not the same asof maps to be preserved. In other words, if the degree of.
variety
study
the
to
able
the degree of we will not be
Fortunately all the possible things that can go wrong either go right or not terribly wrong.
But, before giving the answer to these questions, we need to give an explicit description of.
andsome birational maps between
the canonical Hermitian product) $\beta=$
torespect(with
Given an orthonormal basis:by, define a birational mapof$b^{-1}\circ\psi$$b^{-1}\circ\psi$$HHo1_{d}^{ful1}(S^{2}, \mathcal{Z}_{n})$$\psi$$\mathbb{C}P^{n(n+1)/2}$$\mathcal{Z}_{n}$$\mathbb{C}^{2n+1}$$\{E_{0}, E_{1}, \ldots, E_{n}, \overline{E}_{1}, \ldots , \overline{E}_{n}\}$$[s: \alpha_{1}:.

.

.

:\alpha_{n}:\tau_{12}:.

.

$\mathbb{C}P^{n(n+1)/2}arrow \mathcal{Z}_{n}$:\tau_{n-1,n}]arrow^{b\rho}1^{n-p1anegenerbythevectors}\frac{\alpha_{\ell}}{s}E_{0}+E_{\ell}+\sum_{k=1}^{n}(-\frac{\alpha_{\ell}a_{k}ated}{2s^{2}}+\frac{\tau_{\ell k}}{2s})\overline{E}_{k},1\leq\ell\leq n\}$.

:the idea would be to define the map
and study its properties. The questions about what can go wrong are solved as

Then, given$\mathbb{C}P^{n(n+1)/2}$.

$b_{\beta}$$\tilde{\psi}_{\beta}=b_{\beta}^{-1}\circ\psi$$\psi\in HHo1_{d}^{iul1}(S^{2}, \mathcal{Z}_{n})$$S^{2}arrow$follows:
$1’$

where
is not defined. A complete
cannot lie in the subvariety ofisis that the map
proof
the
ingredient
of
key
[24].
The
in
proof of this appears
full.

. The image of$\mathcal{Z}_{n}$$\psi$$b_{\beta}^{-1}$$\psi$$2’$. The fact that the map
differential system:
Writing a map from$\psi$

is horizontal translates into the following relatively niceas $[s : \alpha_{1} : . . . : \alpha_{n} : \tau_{12} :. . . : \tau_{n-1,n}]$tois horizontal translates into the map(in homogeneous coordinates), the fact that
: $S^{2}arrow \mathbb{C}P^{n(n+1)/2}$ satisfying the differential system given by$S^{2}$$\mathbb{C}P^{n(n+1)/2}$$\psi$$b_{\beta}^{-1}\circ\psi$$\alpha_{i}’\alpha_{j}-\alpha_{i}\alpha_{j}’=s\tau_{ij}’-s’\tau_{ij}$,$1\leq i,$ $j\leq n$,(6)where, as usual, the dashes denote differentiation with respect to a conformal param. Note that this reduces to equation (4) when $n=2$ .
eter onThis differential system was actually found by Bryant in [12], although in a different
form. It also appears in $[31|$ in the form presented here.
$S^{2}$$3’$

is not equal to the degree of .
. There are examples for which the degree ofsame,
we are trying to study the set
since
is
the
degreemaps
theAlthough for most, it seems that the original idea willof all holomorphic and horizontal maps into
as follows.
beovercome
not work. However, this problem can$b_{\beta}^{-1}\circ\psi$$\mathcal{Z}_{n}$$\psi$

159
Define the varieties$PD_{d}^{fu11}(S^{2}, \mathbb{C}P^{n(n+1)/2})$$\{maps[s:\alpha_{1}:$=$holomorphic of degree$d$satisfying

.

.

.

\alpha_{n}:\tau_{12}:

.

.

.

:\tau_{n-1,n}]:S^{2}arrow \mathbb{C}P^{n(n+1)/2}$$\alpha_{i}’\alpha_{j}-\alpha_{i}\alpha_{j}’=s\tau_{ij}’-s’\tau_{ij}$,and$( \frac{\alpha_{i}}{s})’$independent

Notice that, since these are maps fromtoof degree , each homogeneouscomponent of one such map can be regarded as a polynomial of degree in one complex
variable . We define the follorving open subset of PDdfull:$\mathbb{C}P^{n(n+1)/2}$$S^{2}$$d$$d$$(S^{2}, \mathbb{C}P^{n(n+1)/2})$$z$$PD_{d_{7}0}^{fu11}(S^{2}, \mathbb{C}P^{n(n+1)/2})$$\{[s :$=$

\alpha_{1}:\ldots]\in PD_{d}^{tul1}(S^{2}, \mathbb{C}P^{n(n+1)/2})$$s= \prod_{\ell=1}^{d}(z-s_{\ell}),$$s_{\ell}$distinct, andThe following proposition gives the key for the subsequent study of
proof is long and technical; details can be found in [26].
Proposition 2 For all
set
with

$\psi_{0}\in HHo1_{d}^{full}(S^{2}, \mathcal{Z}_{n})$$\mathcal{U}_{\beta}\subset HHo1_{d}^{full}(S^{2}, Z_{n})$$\psi_{0}\in \mathcal{U}_{\beta}$$\psi\in \mathcal{U}_{\beta}\subset HHo1_{d}^{fvll}(S_{1}^{2}\mathcal{Z}_{n})$with$\alpha_{1}(s_{\ell})\neq 0_{i}\forall\ell\}$$HHo1_{d}^{fu11}(S^{2}, \mathcal{Z}_{n})$. there exists a birational map

such that the map$arrow$$b_{\beta}$

.

. The

and an open$\tilde{\psi}_{\beta}=b_{\beta}^{-1}\circ\psi\in PD_{d,0}^{full}(S^{2}, \mathbb{C}P^{n(n+1)/2})$

is an algebraic isomorphism.

In other words, although PD
is really not isomorphic to HHoldfull,
we can completely cover the latter variety with patches algebraically isomorphic to theformer.
$d_{1}r_{u11}0(S^{2}, \mathbb{C}P^{n(n+1)/2})$

6$(S^{2}, \mathcal{Z}_{n})$An algebraic construction of harmonic maps fromto$S^{2}$$S^{2n}$In view of Proposition 2, in order to study local characteristics ofwe
can study PD $df,0(S^{2}, \mathbb{C}P^{n(n+1)/2})$ instead. To this end, we have to analyse the system
where is a polynomial with distinct roots$1\leq m\leq d$ , andare polynomials of degree less than or equal to , withfor all $m$ .
The obvious idea would be to write the polynontials in the usual basis, substitute into
system (6) and obtain algebraic equations on the coefficients. The equations obtained,
however, turn out to be too entangled and they are too hard to analyse.
Instead, one can write the polynomials as follows: since has distinct roots$1\leq$$m\leq d$ , the polynonials
form a basis for the space of polynomials of$HHo1_{d}^{fu11}(S^{2}, \mathcal{Z}_{n})$$\alpha_{i}’\alpha_{j}-\alpha_{i}\alpha_{j}’=s\tau_{ij}’-s’\tau_{ij}$$\alpha_{i},$

$s$

$d$

$s_{m},$$d$$\tau_{jk}$$\alpha_{1}(s_{m})\neq 0$$s$$\{s, \frac{s}{z-\epsilon_{1}}, \ldots, \frac{S}{z-s_{m}}\}$$s_{m},$

160
degree . Thus we can write$d$$s= \prod_{m=1}^{d}(z-s_{m})$,$\alpha_{i}=a_{i0}s+\sum_{m=1}^{d}a_{im}\frac{S}{z-s_{m}}$$\tau_{ij}=t_{ij0^{S}}+\sum_{m=1}^{d}t_{ijm}\frac{s}{z-s_{m}}$Using this representation, the system (6) turns into the following algebraic equations (see[26] for details):(7)$a_{im} \sum_{k\neq m}\frac{a_{jk}}{(s_{m}-s_{k})^{2}}-a_{jm}\sum_{k\neq m}\frac{a_{ik}}{(s_{m}-s_{k})^{2}}=0,1\leq m\leq d$

and
.

(8)

$\tau_{ij}=t_{ij0^{S}}+s/\frac{\alpha_{i}’\alpha_{j}-\alpha_{i}\alpha_{j}’}{s^{2}}dz$(Equation (7) guarantees that the integrand in equation (8) has no residues andpolynommial of degree at most )It is useful to write (7) in the following matrix form:$\tau_{ij}$

is a

$d.$

$..\cdot.\cdot$$\frac\frac{(s1s_{d})^{2}\overline{1}1}{(s_{2}-,..s_{d})^{2}}\lambda_{d}$$( \frac{\lambda_{1}1}{(s_{2}-,.s_{1})^{2}}\frac{1}{(s_{d}-s1)^{2}}$$(a_{12}a_{11}a_{1d}a_{22}a_{21}a_{2d}$$\frac{1}{(s_{\overline{\lambda}_{2}}1^{s)^{2}}2}\frac{1}{(sd-s2)^{2}}$$..\cdot.$$a_{n2}a_{nd}a_{n1}$(9)$=0$ ,.
where$\lambda_{m}=-\frac{1}{a_{1m}}\sum_{k\neq m}\frac{a_{1k}}{(s_{m}-s_{k})^{2}}$.

This approach immediately gives an interesting result: it provides an algebraic ‘recipe’(and hence any harmonic map
to construct any linearly full harmonic map fromtoto a sphere): First findfromso that the nullity of the matrix
and$S^{2}$$S^{2}$$s_{m}$$S^{2n}$$\alpha_{1m}$(10)$\frac\frac{1}{(s1-s_{d})^{2},(s_{2}-s_{d})^{2}1}\lambda_{d}$$( \frac{\lambda_{1}1}{(s_{2}-s_{1})^{2}}\frac{1}{(s_{d}-s_{1})^{2}}$$\frac{1}{(s1-,.s2)^{2}\lambda_{2}}\frac{1}{(s_{d}-s2)^{2}}$$.\cdot\cdot.\cdot$$2\leq i\leq n,$ $1\leq m\leq d$ , so
is at least . Then findwherethat equation (9) is satisfied, and so that the second matrix in that equation has maximal
rank (this $wiU$ guarantee that the map is full). Then use equation (8) to find theandfollow the procedure above backwards to obtain a harmonic map. Of course, there will besome free choices on the way, such as the choice of a basis as explained in Section 5.
$($$\lambda_{m}=-\frac{1}{a1m}\sum_{k\neq m(s_{m}-s_{k})^{2}}\ovalbox{\tt\small REJECT}^{a})$$n$$a_{im},$$\tau_{ij}$$\beta$161

7$Harm_{d}(S^{2}, S^{2n}(1))$

has dimension $2d+n^{2}$ .

In this section we sketch the proof of the following:

Conjecture (Bolton-Woodward, 1993, First MSJ htemational Research Institute, Tohoku
University [6] : $Hann_{d}(S^{2}, S^{2n}(1))$ is an algebmic variety of dimension $2d+n^{2}$ .
$)$The algebraic construction of the previous section allows for a very detailed analysis of
the variety PD. In particular, it is possible to give a constructive proof
that there is a $2d+n^{2}$ -dimensional variety inside PD, which shows thatthe dimension of PD, and hence of, is at least $2d+n^{2}$ .
The main steps are as follows:$d_{2}0fu11(S^{2}, \mathbb{C}P^{n(n+1)/2})$$d_{I}f0(S^{2}, \mathbb{C}P^{n(n+1)/2})$$dr,0(S^{2}, \mathbb{C}P^{n(n+1)/2})$$Harm_{d}^{ful1}(S^{2}\}S^{2n})$

1. Show that the variety of thosesuch that the matrix (10) has
nullity has dimension at least $2d-n(n+1)/2[26|$ .
$(s_{1}, \ldots, s_{d}, \lambda_{1}, \ldots, \lambda_{d})$$n$2. Assuming that the nullity of the matrix (10) is , it is not hard to see that thedimension of the set of solutions$1\leq i\leq n,$ $0\leq m\leq d$ , of equation (9),is$n^{2}+n$ .
$n$$a_{im},$3. Finally, theare completely determined by (8), but each has one degree of heedom(the), giving $n(n-1)/2$ dimensions more.
$\tau_{ij}$$t_{ij0}$

4. Add up: $2d-n(n+1)/2+n^{2}+n+n(n-1)/2=2d+n^{2}$ , as desired. Hence$\dim(Harm_{d}^{ful1}(S^{2}, S^{2n}))\geq 2d+n^{2}$.

To finish the proof of the conjecture stated at the beginning of the section, it only
remains to show that Harm $dfu1|(S^{2}, S^{2n})$ has dimension at most $2d+n^{2}$ . This is actuallyeasier, and it was essentially known to Bolton and Woodward. It was also proved by Kotani(see the last corollary in [34]). Another proof of this fact, using different techniques, appearsin [25] for the particular case $n=3$ .
The proof we sketch here is very similar to that in [34], but we use the algebraic construction explained above. The key point is to note that there are well-defined projections$p_{n}:PD_{d,0}^{fu11}(S^{2}, \mathbb{C}P^{n(n+1)/2})arrow PD_{d,0}^{f}(S^{2}, \mathbb{C}P^{(n-1)n/2})$

defined by deleting the$\alpha_{n+1}$

and the$p_{n}([s:\alpha_{1} :.

.

$=[s:\alpha_{1} :.

.

$\tau_{i,n+1}$::\alpha_{n}:\tau_{12}:.

.

.

,

.

.

:\tau_{n-2,n-1}:\tau_{1_{1}n}\cdots:\tau_{n-1,n}])$:\alpha_{n-1}:\tau_{12} :.

.

.

.

:\tau_{n-2,n-1}])$This map has the following properties:$\bullet$Its image has codimension at least 1. This is expected but not quite trivial. See [26]for the detailed proof.

162$\bullet$

The fibre over a generic point has dimension at most $2n$ . This is not hard to see if wesuch that thelook at equation (9). The points in the fibre are essentially those$(a_{n1}, \ldots, a_{nd})$ is in the kernel of the matrix (9). Since this matrix has nullity
vector, we have degrees of freedom. In addition, we have 1 degree of freedom from theand $n-1$ degrees of freedom from the choice of $t_{in0},1\leq i\leq n-1$ .
choice of
Therefore the fibre has dimension $n+1+(n-1)=2n$ .
$\alpha_{n}$

$n$

$n$

$a_{n0}$.
Note that
Then proceed by induction on the dimension of PD, which corresponds to the case $n=1$ , is the set of holomorphic mapsPDof degree : this has dimension $2d+1$ , so of course the conjecture is truetofromin this case.
If the conjecture is true at level $n-1$ , then$d_{)}0fu11(S^{2}, \mathbb{C}P^{n(n+1)/2})$$df,0(S^{2}, \mathbb{C}P^{1})$$S^{2}$$\mathbb{C}P^{1}$$d$$\dim(PDd_{1}r_{u11}0(S^{2}, \mathbb{C}P^{n(n+1)/2}))$) $+\dim$ (Fibre of$\leq$

dim(Image of$\leq$$(\dim(PDd,0f(S^{2}, \mathbb{C}P^{(n-1)n/2}))-1)+2n$$\leq$$p_{n}$$p_{n}$)$2d+(n-1)^{2}-1+2n$$=2d+n^{2}$ ,as desired. Therefore we have proved the following.
Theorem 3 The (pure) dimension of$Harm_{d}^{ful}’(S^{2}, S^{2n})$is $2d+n^{2}$ .

Maybe the curious thing about these proofs is that numbers match very well, but it is
not clear why things work (or at least the second author does not completely understand
why they work). There are also many relationships between the last part of this section
and integrability of Jacobi fields, as $weU$ as the extm eigenfunctions (see below) whichcorrespond, in our setting, to maps for which the matrix (9) has nullity greater than .
$n$

The role of Jacobi fields

8

A Jacobi field is an infinitesimal deformation of a harmonic map. We can make this more
precise in two ways.
The first way is by means of the the second variation as follows. Let $\phi:Marrow W$ be a
be vector fields along ,harmonic map between compact Riemannian manifolds. Let$w\in\Gamma(\phi^{-1}TW)$ . Choose a smooth two-parameter variationof withi.e,$v,$ $w$$\phi$$\{\phi_{t.s}\}$$v,$$\frac{\partial\phi_{l_{\backslash }s}}{\partial t}(t,s)=(0,0)=v$

and$\frac{\partial\phi_{t,s}}{\partial s}(t.s)=(0,0)=w$

then the second vareation or Hessian of the energy at$\phi$;

is defined by$H_{\phi}(v, w)= \frac{\partial^{2}}{\partial t\partial s}E(\phi_{t_{2}s})(t)s)=(0,0)$$\phi$

163
It is given by the second variation formula (see, for example, [17]):$H_{\phi}(v, w)=/M\langle J_{\phi}(v),$ $w\rangle\omega_{g}$where$J_{\phi}$: $\Gamma(\phi^{-1}Tf4^{\gamma})arrow\Gamma(\phi^{-1}TW)$ is the self-adjoint linear operator defined by$J_{\phi}(v)=\Delta^{\phi}v$–Thr $R^{W}(d\phi, v)d\phi$ .

Heredenotes the Laplacian on $\phi^{-1}TW$ andthe curvature operator of $W$ (conventions as in [17] . The operator
is called the Jacobi opemtor; a vector field $v\in\Gamma(\phi^{-1}TW)$
is called a Jacobi field if it satisfies the Jacobi equation $J_{\phi}(v)=0$ . By standard eUiptic
theory, the setof Jacobi fields along a given harmonic map is a finite-dimensional
vector subspace of.
A second way to understand the Jacobi operator is as (ninus) the linearization of thetension field as follows (see [37]).
$\Delta^{\phi}$$R^{W}$$)$$J_{\phi}$$kerJ_{\phi}$$\Gamma(\phi^{-1}TW)$

Proposition 3 Let: $Marrow W$ be harmonic and let $v\in\Gamma(\phi^{-1}TW)$ . Letsmooth (one-pammeter) variation ofwhich is tangent to , i. e., withThen$\phi$$\phi$$v$,$J_{\phi}(v)=- \frac{\partial}{\partial t}\tau(\phi_{t})_{t=0}$

$i.e$

Thus

$v$

is a Jacobi field along

be a

.

(11)

., the components of each side with respect to a local frame on $\phi^{-1}TW$ satisfy$-(\partial/\partial t)\tau(\phi_{t})^{\alpha}|_{t=0}(\alpha=1, \ldots, m)$$\{\phi_{t}\}$$\partial\phi_{t}/\partial t|_{t=0}=v$$J_{\phi}(v)^{\alpha}=$.

$\phi$

if and only if$\tau(\phi)=0$

and$\frac{\partial}{\partial t}\tau(\phi_{f})_{t=0}=0$

.

(12)

Note that equation (11) and condition (12) are independent of the local frame chosen.
We shall call a smooth variation $\{h\}$ harmonic to first order if it satisfies condition (12).
Thus a smooth vantationof a harmonic map is harmonic to first order if and onlyisittangent
toaJacobi field along .
if
In particular, if
is a smooth variation of with eachharmonic, its variation
vector field
is a Jacobi field. We now ask whether every Jacobi field arises
this way; to discuss this, we make the following definition.
$\{\phi_{t}\}$$\phi$$\phi$$\{\phi_{t}\}$$\phi$$\phi_{t}$$v=\partial\phi_{t}/\partial t|_{t=0}$

Definition 1 A Jacobi field along a harmonic map : $Marrow W$ is called integmble if
it is tangent to a smooth variationof through harmonic maps, i.e., there exists a
one-parameter familyof harmonic maps with
and
.
$v$

$\phi$$\{\phi_{t}\}$$\phi$$\{\phi_{t}\}$$\phi_{0}=\phi$$\partial\phi_{t}/\partial t|_{t=0}=v$

Proposition 4 [1] Let: $Marrow W$ be a harmonic map between compact real-analytic
Riemannian manifolds. Then all Jacobi fields along
are integrable if and only if thespace of harmonic maps
to is a smooth manifold whose tangent space at
is exactly the spaceof Jacobi fields along it.
$\phi$$\phi$$(C^{2,\alpha_{-}})close$$kerJ_{\phi}$$\phi$$\phi$

164
The converse is false: there are examples where the space of harmonic maps is a smoothmanifold, but the space of Jacobi fields contains non-integrable ones which are not in atangent space, see [37] and below.
Now, to analyse Jacobi fields along harmonic maps from, one idea is to use
toreplacethem with infinitesimal deformations of the twistor
the twistor construction to
lift. This works well in the case $m=4$ , as we now describe. Given a holomorphic map
along an infinitesimal $h_{07}\tau zontal$ holomorphic:, we call a vector field
deformation (IHHD) if it is holomorphic, i.e, tangent to a curve of holomorphic maps, andpreserves horizontality ’to first order’. Representing by a quadruplet of polynomials as
in Section 3, $f=(f_{1}, f_{2}, f_{3}, f_{4})$ :so that is represented by a holomorphic, the latter condition reads
map$S^{2}$$\psi$$S^{2}arrow \mathbb{C}P^{3}$$u$$S^{m}$$\psi$$\psi$$\mathbb{C}-\mathbb{C}^{4}\backslash \{0\}$$u$$U:\mathbb{C}arrow \mathbb{C}^{4}$$dQ_{f}(U)=0$ .

(13)Given an infinitesimal horizontal holomorphic deformation of , it is easy to see fromthe composition law for harmonic maps [16, \S 4] that $v=d\pi\circ u$ is a Jacobi field along$\phi=\pi\circ\psi$ .
The inverse construction is harder because of the presence of branch points,
however, we can show [38]:$u$$\psi$

be a full harmonic map with twistor lifl :Proposition 5 Let :$v=d\pi\circu$Then setting
defines one-to-one correspondence between IHHDs
Jacobi fields along .
$\phi$$S^{2}-arrow S^{4}$$\psi$

$a$

$v$

.
and

$S^{2}arrow \mathbb{C}P^{3}$

$u$

of

$\psi$$\phi$

This reduces the problem of finding Jacobi fields along to solving equation (13). In
particular, we see that, if has maximal rank at $F$ , then, not only is the space of harmonic
maps a smooth manifold at $\phi=\pi 0\psi$ , but also the Jacobi fields along are all integrable
and form the tangent space to that manifold at . If does not have maximal rank, thenthere will be non-integrable Jacobi fields along .
As we saw earlier, if $d\leq 6$ then $Q$ is always submersive, so that all Jacobi fields areintegmble and form the tangent space to the smooth manifold.
It is not known whether Proposition 5 generalizes to higher dimensions; the argument
establishing extension over branch points is special to four dimensions. Note that for any$m\geq 4,$ $d\geq 3$ , the space of all (i.e. full and non-full) harmonic maps fromtois
not a manifold –indeed, harmonic maps can collapse to a non-full harmonic map, see thework of N. Ejiri and M. Kotani [20, 21, 22, 34]. For $d\geq 3$ , some non-full maps
are the limits of a famuily of full harmonic maps into , we shall call such maps collapse
points: see [38] for an analysis of those, especially for $d<6$ . When $d\geq 6$ , a non-full map
might also occur as the limit of $fuU$ maps into higher-dimensional spheres; see [34] for someresults on collapsing in higher dimensions.
Let be a non-full (non-constant) harmonic map $homS^{2}$ towith $m=3$ or 4; we
examine the Jacobi fields along . Note first that has image in a totally geodesic.
From this, it is easy to see that the space of non-full maps is a smooth manifold. Now any
Jacobi field along such a map decomposes into components tangential and normal to the. The tangential component is a conformal vector field, so we concentrate on theimage$\phi$$Q$$\phi$$\phi$$Q$$\phi$$Harm_{d}^{ful1}(S^{2}, S^{4})$$S^{2}$$S^{m}$$S^{2}arrow S^{4}$$S^{4}$$S^{m}$$\phi$$\phi$$S^{2}$$\phi$$S^{2}$

165
normal component. This may be tangent to the space of non-full maps; if it is not, thenit is called extm. Take a parallel basis for the normal bundle of the image. Then the
Jacobi equation assumes a simple form: a vector field along is Jacobi if and only if its$S^{2}$$\phi$$n-2$ components$v_{i}$

satisfy the genemlized eigenvalue (Schrodinger) equation:$\Delta v_{i}=|d\phi|^{2}v_{i}$.

The coordinate functions of considered as a map into
span a 3-dimensional space oftrivial solutions to this equation; any other solution is called an extm eigenfunction (of ).
It is asy to see that a Jacobi field is extra if and only if at least one of its components is
an extra eigenfunction.
Now, it can be shown that a non-full harmonic map fromtohas an extra Jacobi
field if and only if it is a collapse point. But then one of the components of is an
extra eigenfunction; this gives an extra Jacobi field of considered as a map intowhich
cannot be integrable, since all harmonic maps into
are non-full. So we see that the
space of harnonic maps fromtois a smooth manifold, however those harmonic maps
which are collapse points when considered as maps into
have non-integmble$\mathbb{R}^{3}$$\phi$$\phi$$e$$S^{2}$$S^{4}$$v$$v$$S^{3}$$\phi$$S^{3}$$S^{2}$$S^{3}$$S^{2}arrow S^{3}$$Ja\omega bi$$S^{4}$

fields.

Thus, integmbility of all
fields implies that the space of harmonic maps is asmooth manifold, but not conversely.
$Ja\omega bi$

9Area and nullity

The nullity of (the energy) of a harmonic map is the real dimension of the space of Jacobi
fields along it. Since the Jacobi fields are the solutions to equation (13), we obtain
Theorem 4 Let :be a harmonic map of twistor degree . Then the nullity ofis greater than or equal to $4d+8$ with equality if and only if is a regular point of $Q$ .
$\phi$$S^{2}arrow S^{4}$$d$$\phi$$\phi$

Recalling the results of Bolton-Woodward and Bolton-Fern\’andez cited in Section 3,
we deduce:
Corollary 1 The nullity of a full harmonic map$4d+8$ .

$\phi$: $S^{2}arrow S^{4}$ of degree$\leq 6$is exactly

We can consider instead the second variation of the area. This depends only on normal
vector fields. Results of N. Ejiri and M. Micallef [23] imply that, for any non-constantharmonic map from the 2-sphere, the map $v\mapsto the$ normal $\omega mponent$ of is a surjective
linear map from the space of Jacobi fields for the energy to the space of Jacobi fields forthe area, with kemel the tangential conformal fields.
S. Montiel and F. Urbano [40, Corollary 7] show that the nullity of the (second variationof the) area of a (full or non-full) minimal innnersion ofinof twistor degree isexactly $4d+2$ . Since the tangential conformal fields form a space of (real) dimension 6,$v$$S^{2}$$S^{4}$$d$166
the nunity of the energy is precisely $4d+8$ ; we deduce that any immersive harmonic map
is a regular, and so a smooth, point of $Harm_{d}(S^{2}, S^{4})$ .

References[1] D. Adams and L. Simon. Rates of asymptotic convergence near isolated singularities ofgeometric extrema. Indiana J. of Math., 37 (1988), 225-254.
[2] J.L. Barbosa. On minimal immersions of(1975), 75-106.

$S^{2}$

into$S^{2m}$. Trans. Amer. Math. Soc,, 210[3] J. Bolton, G.R. Jensen, M. Rigoli and L.M. Woodward. On conformal minimal immerinto. Math. Ann., 279 (1988), 599-620.
sions of$S^{2}$$\mathbb{C}P^{n}$[4] J. Bolton, W.M. Oxbury, L. Vrancken and L.M. Woodward. Minimal immersions of. In: Spntnger Lecture Notes in Mathematics 1481 (1991), 18-27.
into$\mathbb{R}P^{2}$$\mathbb{C}P^{n}$[5] J. Bolton and L.M. Woodward. Moduli spaces of harmonic 2-spheres. In: Geometry
and Topology of Submanifolds IV, World Scientific, (1992), 143-151.
[6] J. Bolton and L.M. Woodward. The space of harmonic maps of. Geometry
intoand global analysis (Sendai, 1993), 165-173, Tohoku Univ., Sendai, 1993.
$S^{2}$$S^{n}$[7] J. Bolton and L.M. Woodward. Higher singularities and the twistor fibration. Geom. Dedicata, 80 (2000), 231-246.

$\pi:\mathbb{C}P^{3}arrow$$S^{4}$[8] J. Bolton and L.M. Woodward. Linearly full harmonic 2-spheres in
Intemat. J. Math., 12 (2001), 535-554.

$S^{4}$of area $20\pi$ .

[9] J. Bolton and L.M. Woodward. The space of harmonic two-spheres in the unit foursphere. Tohoku Math. J. (2), 58 (2006), 231-236.
[10] 0. Boruvka. Sur les surfaces repr\’esent\’ees par les fonctions sph\’eriques de premi\‘ere
esp\‘ece. J. Math. Pures
12 (1933), 455-473.
$\mathcal{A}ppl.,$[11] R.L. Bryant. Conformal and minimal immersions of compact surfaces into the 4sphere. J. Diff Geom., 17 (1982), 455-473.

[12] R. L. Bryant. Lie groups and twistor spaces. Duke Math. J., 52 (1985), 223-261.
[13] E. Calabi. Minimal immersions of surfaces into Euclidean spheres. J. Diff. Geom., 1(1967), 111-125.

[14] S.S. Chem. On the minimal immersions of the two-sphere in a space of constant, Princeton University Press, (1970), 27-40.
curvature. In: Problems in$\mathcal{A}nalysis$

167[15] T. A. Crawford. The space of harmonic maps from the 2-sphere to the complex projective plane. Canad. Math. Bull., 40 (1997), 285-295.

[16] J. Eells and L. Lemaire. A report on harmonic maps. Bull. London Math. Soc., 10(1978), 1-68.
[17] J. Eells and L. Lemaire. Selected topics in harmonic maps. C.B.M.S. Regional Conf.
Series 50, Amer, Math. Soc., 1983.

[18] J. Eells and L. Lemaire. Another report on harmonic maps. Bull. Lond. Math. Soc.,
20 (1988), 385-524.
[19] D. Eisenbud and J. Harris. Divisors on general curves and cuspidal rational curves.
Invent. Math., 74 (1983), 371-434.

[20] N. Ejiri. Minimal deformation of a nonfull minimal surface in $S^{4}(1)$ . Compositio Math.,
90 (1994), no. 2, 183-209.
[21] N. Ejiri. The boundary of the space of full harmonic maps ofeigenfunctions. Japan. J. Math. (N.S.), 24 (1998), 83-121.

$S^{2}$

into $S^{2m}(1)$ and extra[22] N. Ejiri and M. Kotani. Minimal surfaces in $S^{2m}(1)$ with extra eigenfunctions. Quart.
J. Math. Oxford Ser., (2) 43 (1992), 421-440.
[23] N. Ejiri and M. Micallef. Comparison between second variation of area and second
variation of energy of a minimal surface. Preprint, arXiv:0708.2188 (2007).
[24] L. Fern\’andez. On the moduli space of superninimal surfaces in spheres. Int. J. Math.
Math. Sci., 44 (2003), 2803-2827.

[25] L. Fern\’aiidez. The dimension of the space of harmonic 2-spheres in the 6-sphere. Bull.
Lond. Math. Soc., 38 (2006), 163-176,[26] L. Fern\’andez. The dimension and structure of the space of harmonic 2-spheres in the$2n$ -sphere. Preprin t.
[27] M. Furuta, M.A. Guest, M. Kotani and Y. Ohnita. On the fundamental group of thespace of harmonic 2-spheres in the n-sphere. Alath. Z., 215 (1994), 503-518.

[28] P. Griffiths and J. Harris. Principles of Algebmic $G\omega metry$ , Wiley (1978).
[29] M.A. Guest and Y. Ohmita. Group actions and deformations for harmonic maps. J.
Math. Soc. Japan, 45 (1993), 671-704.

[30] R.D. Gulliver, R. Osserman and H.L. Royden. A theory of branched immersions of. J. Math., 95 (1973), 750-812.
surfaces.
$\mathcal{A}mer$168[31] J. I. Hano. Conformal immersions of compact Riemann surfaces into the $2n$ -sphere. Nagoya Math. J. 141 (1996), 79-105.
(n$\geq 2)$[32 H. Hopf. \"Uber Fl\"achen mmit einer Relation zwischen den Hauptkr\"ummungen. Math.
Nachr., 4 (1950-51), 232-249.
$|$[33] M. Kotani. Connectedness of the space of minimal 2-spheres in $S^{2m}(1)$ . Proc.
Math. Soc., 120 (1994), 803-810.
[34] M. Kotani. Harmonic 2-spheres with
Math. Soc., 125 (1997), 2083-2092.

$r$$\mathcal{A}mer$.

pairs of extra eigenfunctions. Proc. Amer.

[35] H.B. Lawson. Minimal varieties. Proc. Symp. Pure Math., 27 (1975), 143-175.
[36] L. Lemaire and J. C. Wood. On the space of harmonic 2-spheres in
Math., 7 (1996), 211-225.

$\mathbb{C}P^{2}$[37] L. Lemaire and J. C. Wood, Jacobi fields along harmonic 2-spheres ingrable. J. London Math. Soc.(2), 66 (2002), 468-486.
[38] L. Lemaire and J. C. Wood, Jacobi fields along harmonic 2-spheres in
not all integrable. Preprint, ArXiv:0709.1417.
[39] B. Loo. The space of harmonic maps of(1989), 81-102.

$S^{2}$

into$S^{4}$. Intemat.

$\mathbb{C}P^{2}$$S^{3}$.

$J$

are inte-

and$S^{4}$are. Trans. Amer. Math. Soc., 313[40] S. Montiel and F. Urbano. Second variation of superminimal surfaces into self-dual. Math. Soc., 349 (1997), 2253-2269.
Einstein four-manifolds. Trans.
$\mathcal{A}mer$[41] M. Mukai. On the connectedness of the space of harmonic 2-spheres in quatemionicprojective spaces. Tokyo J. Math., 17 (1994), 241-252,[42] L. Simon. Singularities of Geometric Variational Problems. In: Nonlinear partial differential equations in differential geometry (Park City, $UT$, 1992), 185-223, IAS/Park
City Math. Ser., 2, Amer. Math. Soc., Providence, RI, 1996.
[43] L. Simon. Theorems on regularity and singularity of energy minimizing maps. Lecturesin Mathematics, ETH Z\"urich, Birkh\"auser (1996).
Comprehensive Introduction to Differential Geometry, Vol. 4. Publish[44] M. Spivak.
or Perish, (1975).
$\mathcal{A}$. In: Geometry today (Rome,dans[45] J.-L. Verdier. Applications harmoniques de
Birkh\"auser
Boston, Boston, MA, 1985.
1984), 267-282, Progr. Math., 60,$S^{2}$[46] J.L. Verdier. Applications harmoniques de60 (1985), 267-282.

$S^{2}$

dans$S^{4}$$S^{4}$

I. In: Birkhauser Math. Series

169[47] J.L. Verdier. Applications harmoniques de
II. In: Harmonic Mappings,
dansTwistors and -models, World Scientific (1988), 124-147.
$S^{2}$$S^{4}$$\sigma$

J. Bolton, Dept of Mathematical Sciences, University of Durham, Durham $DHl3LE_{7}UK$ .
E-mail: john.bolton@durham.ac.uk

L. Fem\’andez, Dept. of Mathematics and Computer Science, Bronx Community College(CUNY), New York, $NY$ 10453, USA.
E-mail: luis. femandezOl@bcc. cuny. $edu$
J. C. Wood, School of Mathematics, University of Leeds, Leeds $LS29JT,$ $UK$.
. c.wood@leeds.ac.uk$j$E-mail: