Josai Mathematical Monographs vol 3 (2001), pp. 93-103

ON ANAND CONJECTURE

YuXin Dong

Institute of Mathematics, Fudan University, Shanghai 200433 , P.R. China

1. Introduction

Unitons are harmonic maps φ : S^{2} → U(N), which are important object in mathematics and physics. The construction of unitons was investigated from different perspectives by Uhlenbeck, Valli, Segal, Wood, Guest-Ohnita, Gu-Hu, Anand, and Burstall-Guest. For recent progress in this area see [12], [2], [11], [5] and their references.

In [1,2], Anand showed that unitons are equivalent to holomorphic ‘uniton bundles’, with energy corresponding to the bundles’ second Chern class. Using monad representation, he obtained a simple formula for the unitons and proved that 2untions have normalized energy at least four. This bound is sharp. In a personal communication with the author, Anand proposed the following conjecture:

Conjecture (Finite Gap): m-unitons have energy at least m^{2}.

The purpose of this note is to give a positive answer to this conjecture. It is known that harmonic maps of simply connected Riemann surfaces into U(N) can be lifted into holomorphic maps into the based loop group ΩU(N), which are called extended harmonic maps. Unitons have algebraic extended harmonic maps Φ : S^{2} → Ω_{alg}U(N) ( see [20] for details). Our idea is to deform a given extended harmonic map into a simpler one, which allows us to confirm the above conjecture of Anand.

2. Harmonic maps and extended harmonic maps

Let φ : R^{2} →U(N) be a smooth map. Define A = \frac{1}{2}φ^{−1}dφ, and decompose it as follows:

A = A_{z}dz + A\overline{z}d\overline{z} with (A_{z})^{∗} = −A\overline{z},

where z = x + iy ∈ R^{2}. Then A satisfies the Maurer-Cartan equation:

(1)  \overline{∂}A_{z} − ∂A\overline{z} + 2[A\overline{z}, A_{z}] = 0.

1991 Mathematics Subject Classification. 58E20,58D27.

Supported by NSFC, SFZP and MCSEC

In the case of maps into a matrix group, with the standard (left-invariant) metric, the energy takes the form

E(φ) = \frac{1}{2} ∫_{R^{2}}(|φ^{−1} \frac{∂φ}{∂x}|^{2} + |φ^{−1} \frac{∂φ}{∂y}|^{2})dx ∧ dy

= −4i ∫ trA_{z}A\overline{z}dzd\overline{z}. (2)

It is known that φ is harmonic if and only if A is co-closed; i.e.,

(3)  d^{∗}A = \frac{∂}{∂\overline{z}} A_{z} + \frac{∂}{∂z} A\overline{z} = 0.

Untions are harmonic maps from S^{2} to U(N). From [19], we know that harmonic maps from R^{2} → U(N) extend to S^{2} iff they have finite energy, and that such maps are always smooth. In the following, we will use this fact and work in terms of coordinates x and y on R^{2}.

The Lax pair of the harmonic maps is ([20]):

(4)  \{ \begin{matrix}\overline{∂}Φ & = (1 − λ)ΦA\overline{z}, \\ ∂Φ & = (1 − λ^{−1})ΦA_{z}, \end{matrix}

whose integrability conditions are (1) and (3). A nondegenerated N × N matrix solution Φ(λ) := Φ_{λ} to (4) is called extended solution or extended harmonic map. It can be normalized so that

(5)  Φ_{1} = I,  Φ_{λ}Φ_{σ(λ)}^{∗} = I,

where σ(λ) = (\overline{λ})^{−1}. Moreover, we have Φ_{−1} = Qφ for some constant Q ∈ U(N).

Let ΩU(N) = {γ : S^{1} → U(N)|γ is smooth, γ(1) = I} be the group of smooth based loops in U(N). By (5), we can regard an extended harmonic map as a smooth map into ΩU(N). Denote by Ω_{alg}U(N) the subgroup of ΩU(N) consisting of algebraic loops. One of the main results of Uhlenbeck [20] concerning harmonic maps φ : S^{2} → U(N) is :

Theorem 2.1[20]. Let φ : S^{2} → U(N) be harmonic. Then there exists an algebraic extended harmonic map Φ : M → Ω_{alg}U_{n} of φ.

It is easy to see that the algebraic extended harmonic map Φ can be expressed as the following type after multiplication on the left by a suitable constant loop γ ∈ ΩU(N):

(6)  Φ = ∑_{α=0}^{m} λ^{α}T_{α}.

According to [20], we call the harmonic map φ : S^{2} → U(N) and the extended harmonic map Φ : S^{2} → ΩU(N) a uniton and an extended uniton respectively. We say that φ or Φ has finite uniton number m. The minimal such number m is called the minimal uniton number of φ or Φ. It was shown in [20] that an extended uniton Φ can be normalized further so that

(7)  V_{0}(Φ) := Span{Im_{z∈S^{2}}T_{0}(z)} = C^{N},

and if an extended uniton Φ satisfies the condition (7 ) then m is just the minimal uniton number. An extended uniton Φ with the property (7) will be called the standard extended uniton of φ.

lemma 2.2. Let Φ_{λ} = ∑_{α=0}^{m} λ^{α}T_{α} : S^{2} → ΩU(N) be an extended uniton. Then ImT_{0} : S^{2} → G_{k}(C^{N}) is a holomorphic map , where rank(ImT_{0}) = k . Furthermore, Φ_{λ} is standard if and only if ImT_{0} is a linearly full holomorphic curve in G_{k}(C^{N}). Similarly, ImT_{m} is an anti-holomorphic map.

Proof. We have two unitary connections on the trivial bundle \underline{C}^{N} : d_{A} = d + A and the trivial connection d. They determine their corresponding Koszul-Malgrange holomorphic structures on \underline{C}^{N} by [15]. From (4), we have

\overline{∂}T_{0} = T_{0}A\overline{z}

which means that T_{0} is a global holomorphic section of Hom[(\underline{C}^{N}, d_{A}) , (\underline{C}^{N}, d)]. Thus \underline{Im}T_{0} is a holomorphic subbundle of (\underline{C}^{N}, d), i.e., ImT_{0} is a holomorphic map to G_{k}(C^{N}). Φ_{λ} is standard if and only if it satisfies (7), which means that ImT_{0} is a linearly full holomorphic map. By (4), we have also

∂T_{m} = T_{m}A_{z}

We can prove similarly that ImT_{m} is an anti-holomorphic map. □

We introduce various fundamental loop groups as follows:

ΛGL(N, C) ={γ : S^{1} → GL(N, C)|γ is smooth},
Λ^{+}GL(N, C) ={γ ∈ ΛGL(N, C)|γ extends continuously to
a holomorphic map D^{+} → GL(N, C)}

where D^{+} = {λ | |λ| < 1}. We recall the Iwasawa decomposition:

Theorem 2.3[18]. Each γ in ΛGL(N, C) can be uniquely decomposed as

γ = γ_{u}γ_{+},

where γ_{u} ∈ ΩU(N) and γ_{+} ∈ Λ^{+}GL(N, C).

By the theorem, we have an identification

(8)  ΩU(N) ≅ ΛGL(N, C)/Λ^{+}GL(N, C)

and thus, a natural action of the complex group ΛGL(N, C) on the complex manifold ΩU(N) ([13]): if γ ∈ ΛGL(N, C) and δ ∈ ΩU(N) , then

(9)  γ · δ = (γδ)_{u}.

Let us choose a maximal torus T of U(N) and a fundamental Weyl chamber W_{0} in the Lie algebra t of T. Denote by I′ the intersection of the integer lattice I = (2π)^{−1} exp^{−1}(e) ∩ t with W_{0} . Then we know that I′ (∋ ξ) parametrizes all the conjugacy classes of homomorphisms γ_{ξ} : S^{1} → U(N). By [18], we have the following Bruhat decomposition :

(10)  Ω_{a lg}U(N) = ⊔_{ξ∈I′}Λ_{a lg}^{+}GL(N, C) · γ_{ξ},

where Λ_{a lg}^{+}GL(N, C) = Λ_{a lg}GL(N, C) ∩ Λ^{+}GL(N, C).

There is a beautiful Morse theoretic explanation of these decompositions. Let H : ΩU(N) → R be the usual energy functional on paths, H(γ) = ∫_{S^{1}} |γ′|^{2}. This is a Morse-Bott function. The critical manifolds are the conjugacy classes of homomorphisms. We write

(11)  Ω_{ξ} = {gγ_{ξ}g^{−1}| g ∈ U(N)}

for the conjugacy class of the homomorphism γ_{ξ}.Denote by U_{ξ} the unstable manifold of Ω_{ξ} with respect to the flow of −gradH (i.e., U_{ξ} is the stable manifold of Ω_{ξ} with respect to the flow of gradH). Then we have ([17])

(12)  U_{ξ} = Λ_{a lg}^{+}GL(N, C) · γ_{ξ}.

In [17], the energy functional of loops H is viewed as a Hamiltonian function associated to a symplectic action of the group S^{1}on ΩU(N), which is given by

(13)  μ · γ(λ) = γ(μλ)γ(u)^{−1}, μ ∈ S^{1}, γ ∈ ΩU(N).

It was observed by Terng (see sect.7 of [20]) that this action preserves the extended harmonic equation (4). We have the following action of C^{∗} on Ω_{alg}U(N):

(14)  μ · γ(λ) = γ(μλ)Λ^{+}GL(N, C), μ ∈ C^{∗}, γ ∈ Ω_{alg}U(N),

which is the complexification of the S^{1}−action. It turns out that the flow line of gradH starting at a point γ ∈ Ω_{alg}U(N) is given by the action of subsemigroup (0, 1] of C^{∗}.

For an extended untion Φ : S^{2} → Ω_{alg}U(N), we cannot conclude that Φ(S^{2}) ⊆ U_{ξ} for some ξ ∈ I′ in general. However, we have

proposition 2.4[5]. Let Φ : S^{2} → Ω _{alg} U(N) be an extended uniton. Then there exists some ξ ∈ I′, and some discrete subset D of S^{2} , such that Φ(S^{2} − D) ⊆ U_{ξ}.

3. Deformations of S^{1}− invariant unitons

In the classification theory of unitons in [5], a fundamental role is played by the following special extended unitons:

Definition 3.1. An S^{1}−invarinat extended uniton is an extended uniton Φ such that ImΦ ⊆ Ω_{ξ}, for some ξ ∈ I′; or equivalently Φ is invariant under the S^{1}−action (13). In this case, Φ_{−1} is also called S^{1}−invariant uniton.

By [20], we know that Φ is an S^{1}−invarinat extended uniton into ΩU(N) if and only if it can be expressed as

(15)  Φ = ∑_{α=0}^{k} Π_{α}λ^{α}

where f_{Φ} := (Π_{0}, Π_{1}, ..., Π_{k}) is a super-horizontal holomorphic map into a complex flag manifold F = F (n_{0}, n_{1}, ..., n_{k}) with n_{α} = rankΠ_{α} (α = 0, 1, ..., k) and N = n_{0} +n_{1} +⋯+n_{k}. We say that f_{Φ} is standard if the extended untion Φ is standard.

In the following, we will investigate the deformation of the S^{1}−invariant harmonic map. We know that F (n_{0}, ..., n_{k}) can be identified with the complex homogeneous space U(N)/U(n_{0}) × ... × U(n_{k}). A point B of F may be viewed as a (k + 1)− tuple of orthogonal subspaces B_{0} ⊕ ... ⊕ B_{k} where dim(B_{i}) = n_{i}, or , equivalently, as a flag

0 ⊂ V_{0} = B_{0} ⊂ V_{1} = B_{0} + B_{1} ⊂ ... ⊂ V_{k} = B_{0} + B_{1} + ... + B_{k} = C^{N}.

Let {V_{0}, ..., V_{k}} denote the flag corresponding (B_{0}, ..., B_{k}). An element of F (n_{0}, ⋯, n_{k}) is written as B = (B_{0}, B_{1}, ..., B_{k}) or \{V_{0}, V_{1}, ..., V_{k}\}. Set J = \{0, 2, ..., 2[\frac{k}{2} ] \} and J^{c} = {0, 1, ..., k} \ J. Define the projection

(16)  π : F (n_{0}, n_{1}, ..., n_{k}) ∋ (B_{0}, B_{1}, ..., B_{k}) → ⨁_{j∈J^{c}} B_{j} ∈ Gr_{t}(C^{N})

where t = ∑_{j∈J^{c}} n_{j}. It is known that π is the so called canonical twistor fibration(see [6] or [9]).

Define the following tautological complex vector bundles for B ∈ F (n_{0}, n_{1}, . .., n_{k}) : (V_{0})_{B} = B_{0}, (V_{1})_{B} = B_{0} ⊕ B_{1}, ..., (V_{k})_{B} = ⨁_{i=0}^{k} B_{i}. A map f : S^{2} → F (n_{0}, n_{1}, ..., n_{k}) is called super-horizontal holomorphic map if

(17)  \{ \begin{matrix}∂′′C^{∞}(f^{−1}V_{i}) & ⊂ C^{∞}(f^{−1}V_{i}), \\ ∂′C^{∞}(f^{−1}V_{i}) & ⊂ C^{∞}(f^{−1}V_{i+1}), i = 0, 1, ..., k, \end{matrix}

where we set V_{k+1} = C^{N}.

Hence a uniton φ : S^{2} → Gr_{t}(C^{N}) is S^{1}-invariant if and only if there exists a super-horizontal holomorphic map f_{Φ} : S^{2} → F such that φ = π ∘ f_{Φ}. The group GL(N, C) acts on F (n_{0}, n_{1}, ..., n_{k}) in the natural way : for A ∈ GL(N, C) and V_{0} ⊂ V_{1} ⊂ ⋯ ⊂ V_{k}, we have AV_{0} ⊂ AV_{1} ⊂ ⋯ ⊂ AV_{k}. It is easy to see that this action preserves the super- horizontality and holomorphicity conditions. Thus it induces an action on the S^{1}−invariant untions.

Let G = U(N) and g = u(N) denote the unitary group and its Lie algebra. Then we can regard F = F (n_{0}, n_{1}, ..., n_{k}) as an “Ad-orbit” of U(N) as follows: If we let (B̂_{0}, B̂_{1}, ..., B̂_{k}) be a fixed element of F (n_{0}, n_{1}, ..., n_{k}) and set ξ = \sqrt{−1} ∑_{i=0}^{k} π_{W_{i}} where W_{i} = ⨁_{α=0}^{i} B̂_{α} for i = 0, 1, ...k, then we have F (n_{0}, n_{1}, ..., n_{k}) ≅ Ad(G)ξ. Here π_{V} denotes the Hermitian projection in C^{N} onto a complex subspace V.

Fix an element L ∈ CP^{N−1} and put Q = \sqrt{−1}π_{L} ∈ u(N). For a X = \sqrt{−1} ∑_{i=0}^{k} π_{V_{i}} ∈ Ad(G)ξ, where (V_{0}, V_{0}^{⊥} ∩ V_{1}, ..., V_{k−1} ∩ V_{k}^{⊥}) ∈ F (n_{0}, n_{1}, ..., n_{k}), we define the height function h^{Q} : Ad(G)ξ → R by

(18)  h^{Q}(X) = (X, Q).

Here (,) is an Ad(G)−invariant inner product on g. Then it is known that h^{Q} is a Morse-Bott function. Let gradh^{Q} be the gradient of h^{Q} with respect ot the Kähler metric. The following fact is due to Frankel [16] : the flow of −gradh^{Q} is given by the natural action of \{exp \sqrt{−1}tQ\}. A point (V_{0}, V_{0}^{⊥} ∩V_{1}, ...., V_{k−1}^{⊥} ∩V_{k}) is a critical point if and only if [\sqrt{−1} ∑_{i=0}^{k} π_{V_{i}}, \sqrt{−1}π_{L}] = 0 , i.e., the line L is contained in one of V_{0}, V_{0}^{⊥} ∩ V_{1}, ..., V_{k−1}^{⊥} ∩ V_{k}.

lemma 3.1. There are k + 1 connected no-degenerate critical manifolds of h^{Q} :

F_{0} = {{V_{0}, V_{1}, ..., V_{k}} ∈ F (n_{0}, n_{1}, ..., n_{k})|L ⊂ V_{0}} ≅ F (n_{0} − 1, n_{1}, ..., n_{k}),
F_{1} = {{V_{0}, V_{1}, ..., V_{k}} ∈ F (n_{0}, n_{1}, ..., n_{k})|L ⊂ V_{0}^{⊥} ∩ V_{1}} ≅ F (n_{0}, n_{1} − 1, ..., n_{k}),

......

F_{k} = {{V_{0}, V_{1}, ..., V_{k}} ∈ F (n_{0}, n_{1}, ..., n_{k})|L ⊂ V_{k−1}^{⊥} ∩ V_{k}} ≅ F (n_{0}, n_{1}, ..., n_{k} − 1).

We set G_{Q} = {A ∈ GL(N, C)|A(L) = L}. It is known that the stable manifold for a non-degenerate critical manifold N is given by S^{Q}(N) = G_{Q}X for X ∈ N. In this case, we have

lemma 3.2. The corresponding stable manifolds are

S^{Q}(F_{0}) = F_{0},
S^{Q}(F_{1}) = {{V_{0}, V_{1}, ..., V_{k}} ∈ F (n_{0}, n_{1}, ..., n_{k})|V_{0} ∩ L = {0}, L ⊂ V_{1}},
......
S^{Q}(F_{k}) = {{V_{0}, V_{1}, ..., V_{k}} ∈ F (n_{0}, n_{1}, ..., n_{k})|V_{k−1} ∩ L = {0}, L ⊂ V_{k}}.

Let φ : S^{2} → Gr_{t}(C^{N}) be a S^{1}−invariant harmonic map, and f_{Φ} : S^{2} → F be a super-horizontal holomorphic map corresponding to φ. If f_{Φ}(S^{2}) ⊂ S^{Q}(F_{k}), then \{(exp \sqrt{−1}tQ) ∘ f_{Φ}\} provides a continuous deformation to a super-horizontal holomorphic map into F_{k}(≅ F (n_{0}, n_{1}, ..., n_{k} − 1)). We shall show that there exists some L ∈ CP^{N−1} such that f_{Φ}(S^{2}) ⊂ S^{Q}(F_{k}) if n_{k} ≥ 2.

Set Y = {L ∈ CP^{N−1}|f_{Φ}(z) ∉ S^{Q}(F_{k}) for some z ∈ S^{2}}. Then we have Y = {L ∈ CP^{N−1}|L ⊂ V_{k−1}(z) for some z ∈ S^{2}}. It suffices to show that Y cannot be equal to CP^{N−1}.

Fix z ∈ S^{2}, we set Y_{z} = {L ∈ CP^{N−1}|L ⊂ V_{k−1}(z)}. It is easy to see that

(19)  dim_{C} Y_{z} ≤ ∑_{i=0}^{k−1} n_{i} − 1.

Since dim_{C} S^{2} = 1, we have

(20)  dim_{C} Y ≤ ∑_{i=0}^{k−1} n_{i}.

It follows that, if n_{k} ≥ 2, the space Y cannot be equal to CP^{N−1}. It suffices to choose L ∈ CP^{n−1}\Y . Hence , by induction, we obtain a map whose image lies in F (n_{0}, n_{1}, ..., n_{k−1}, 1) . By repeating the argument with f ^{∗} = {V_{k}^{⊥}, ..., V_{0}^{⊥}, C^{n}} ∈ F (1, n_{k−1}, ..., n_{0}), we can similarly deform f _{Φ} into {{V_{0}, V_{1}, ..., V_{k}} ∈ F (n_{0}, n_{1}, ..., n_{k−1}, 1)| C^{n_{0}−1} ⊆ V_{0}} ≅ F (1, n_{1}, ..., n_{k−1}, 1). Thus we obtain the following proposition.

proposition 3.3. If n ≥ 3, then any super-horizontal holomorphic map f_{Φ} into F (n_{0}, n_{1}, ..., n_{k}) can be deformed continuously through super-horizontal holomorphic maps to a super-horizontal holomorphic map fΦ̃ into F (1, n_{1}, ..., n_{k−1}, 1). Furthermore, if f_{Φ} is standard , then fΦ̃ is also standard in its corresponding space.

Proof. We only need to prove the second part of the proposition. Assume that f_{Φ} is standard, i.e., the corresponding extended uniton Φ = ∑_{α=0}^{k} Π_{α}λ^{α} satisfies the following condition

V_{0}(Φ) = Span{Im_{z∈S^{2}}Π_{0}(z)} = C^{N}

Let Φ^{t} = ∑_{α=0}^{k} Π_{α}^{t}λ^{α} be the continuous deformation obtained in the above. It is clear that dim V_{0}(Φ^{t}) = const. So, V_{0}(Φ^{t}) = C^{N}. In particular, we have

V_{0}(Φ^{∞}) = C^{N}.

The above deformation shows that f_{Φ^{∞}}(S^{2}) ⊂ {{V_{0}, V_{1}, ..., V_{k}} ∈ F (n_{0}, n_{1}, ..., n_{k})|

C^{n_{0}−1} ⊆ V_{0}, C^{n_{k}−1} ⊆ V_{k−1}^{⊥}∩V_{k}} ≅^{i} F (1, n_{1}, ..., n_{k−1}, 1), where i denotes the natural

isomorphism. Let fΦ̃ = i ∘ f_{Φ^{∞}} : S^{2} → F (1, n_{1}, ..., n_{k−1}, 1). We see that

V_{0}(Φ̃) = CÑ

where Ñ = N − (n_{0} + n_{k} − 2). Hence Φ̃ is a standard extended U(Ñ)−uniton, i.e., fΦ̃ is standard. □

We know from [8] that any harmonic map S^{2} → CP^{N−1} corresponds to a superhorizontal holomorphic map into F (n_{0}, 1, n_{2}) with n_{0} + 1 + n_{2} = N. Hence we obtain:

Corollary 3.4[13]. If N ≥ 3, then any harmonic map φ : S^{2} → CP^{N−1} can be deformed continuously through harmonic maps to a harmonic map ψ : S^{2} → CP^{2}.

4. Anand’s conjecture

We now give an application of the above deformations for unitons.

Theorem 4.1. Let φ : S^{2} → U(N) be a harmonic map with minimal uniton number m. Then

(21)  E(φ) ≥ 4πm^{2}.

Furthermore, the equlity holds if and only if φ can be deformed continuously to a harmonic map ψ which is one of the following two cases :

(i) ψ : S^{2} → CP^{1}(⊂ U(2) ⊂ U(N)) is a totally unramified holomorphic map ; (ii) ψ : S^{2} → Gr_{2}(C^{3}) is a harmonic map with minimal uniton number m = 2

and ψ^{⊥} : S^{2} → CP^{2}(⊂ U(3) ⊂ U(N)) is a totally unramified, not±holomorphic , full harmonic map.

Proof. Let Φ = ∑_{α=0}^{m} T_{α}λ^{α} be the standard extended harmonic map of φ. By Proposition 2.3 we can express Φ on M − D as follows:

(22)  Φ = Aγ_{ξ}B

where A, B : M − D → Λ_{alg}^{+} GL_{n}(C) and γ_{ξ} is the geodesic given by

γ_{ξ}(λ) = diag(λ^{m}, ..., λ^{m}, ...., λ, ..., λ, 1, ..., 1).

It is easy to see that

(23)  T_{0} = A(0)γ_{ξ}(0, ..., 1, ..., 1)B(0).

on M − D. Now we deform the extended harmonic map Φ by the gradient flow of the Morse-Bott function H : ΩG → R. From the description of this flow in Sec.2, we have Φ̂ = lim_{t→+∞} Φ^{t}, where Φ^{t} is the extended harmonic map e^{−t} · Φ. The points in D are the possible bubble points in the deformation. If bubbling occurs, then E(φ) > E(Φ̂_{−1}). So, in general , we have E(φ) ≥ E(Φ̂_{−1}). Since ImΨ ⊆ Ω_{ξ},the extended uniton Ψ is an S^{1}−invariant uniton. By [20] Φ̂ has the following expression:

(24)  Φ̂ = ∑_{α=0}^{m} Π_{α}λ^{α}

where (Π_{0}, Π_{1}, ..., Π_{m}) is a super-horizontal holomorphci map into F (n_{0}, n_{1}, ..., n_{m}) with n_{α} = rank(Π_{α}) (α = 0, 1, ..., m). From the definition of the action e^{−t} ∈ C^{∗} on extended uniton and (22), we have

Φ̂ = A(0) · γ_{ξ}
= A(0)γ_{ξ}C_{0}.

We see that π_{0} = A(0)diag(0, ..., 0, 1, ...1)C_{0}. It follows that

Imπ_{0} = ImA(0)diag(0, ..., 0, 1, ..., 1)
= ImT_{0}.

Thus Φ̂ is a standard extended uniton with minimal uniton number m.

By Proposition 3.3, Φ̂ can be deformed continuously to an S^{1}−invariant extended harmonic map Ψ which corresponding a standard super-horizontal holomorphic map into F (1, n_{1}, ..., n_{m−1}, 1). Hence E(Φ̂_{−1}) = E(Ψ_{−1}) and the S^{1}− invariant extended harmonic map Ψ is given by

(25)  Ψ = ∑_{α=0}^{m}(π_{α} − π_{α−1})λ^{α}

with rank(π_{0}) = 1. From Lemma 2.2, we see that ψ_{0} := Im(π_{0}) : S^{2} → CP^{n} is a full holomorphic map , where n = 1 + ∑_{i=1}^{m−1} n_{i} . The holomorphic map ψ_{0} generates the following harmonic sequence (see [4], [22]):

(26)  ψ_{0} →^{∂_{0}} ψ_{1} →^{∂_{1}} ⋯ →^{∂_{N−2}} ψ_{n},

with ⨁_{j=0}^{n} \underline{ψ}_{j} = \underline{C}^{n+1} , where ∂ denotes the fundamental collineation defined by Chern and Wolfson(see [7]). It is easy to see from the section10 of [20] that

(27)  \underline{π}_{0} = \underline{ψ}_{0}, \underline{π}_{1} = ⨁_{j=0}^{i_{1}} \underline{ψ}_{j}, ⋯, \underline{π}_{m−1} = ⨁_{j=0}^{i_{m−1}} ψ_{j}, \underline{π}_{m} = \underline{C}^{n+1},

with 0 = i_{0} < i_{1} < ... < i_{m−1} < i_{m} = n. It follows from Uhlenbeck’s factorization and Valli’s formula ([21]) that

E(Ψ_{−1}) = 4π ∑_{α=0}^{m} deg π_{α}

= 4π ∑_{α=0}^{m−1} deg(⨁_{j=0}^{i_{α}} \underline{ψ}_{j}). (28)

Since \underline{π}_{α} corresponds to the i_{α}−th osculating curve of the holomorphic curve ψ_{0}, we have by [4] that

deg(⨁_{j=0}^{i_{α}} \underline{ψ}_{j}) = (i_{α} + 1)(n − i_{α}) + \frac{n − i_{α}}{n + 1} ∑_{k=0}^{i_{α}−1}(k + 1)r(∂_{k})

+ \frac{i_{α} + 1}{n + 1} ∑_{k=i_{α}}^{n−1}(n − k)r(∂_{k})

≥ (i_{α} + 1)(n − i_{α}), (29)

where r(∂_{k}) denotes the ramification index of ∂_{k}. It follows from (28) and (29) that E(Ψ_{−1}) ≥ 4πmn. The minimal uniton number m ≤ n by [20], so we obtain E(φ) ≥ 4πm^{2}.

Set ψ = Ψ_{−1}. If the equality holds, then we see from (28) and (29) that

r(∂_{k}) = 0

for k = 0, 1, ..., n − 1 and

(30)  (i_{α} + 1)(n − i_{α}) = n

for α = 0, 1, ..., m − 1. It follows from (30) that either m = 1 and n = 1, i.e., ψ belongs to the case (i) or m = 2 , n = 2 and

ψ = π_{0} − π_{1} + π_{2}
= π_{0} + π_{2} − π_{1},

i.e., ψ belongs to the case (ii) . Actually, we see via the Cartan embedding that ψ = ψ_{0} in the case (i) ; and ψ = ψ_{0} ⊕ ψ_{2} , ψ^{⊥} = ψ_{1} in the case (ii). Conversely, if ψ is one of the cases (i) and (ii), it is easy to verify that the equality in (21) holds. □

Remark 4.1. The above proof shows a little more, namely that if φ : S^{2} → U(N) has a standard extended harmonic map Φ = ∑_{α=0}^{m} T_{α}λ^{α} with max_{z∈S^{2}} rank(T_{0}) = 1, then E(φ) ≥ 4πm(N −1). We can normalize the energy to be Ẽ(φ) = \frac{1}{4π} E(φ). So, the Anand conjecture states that m− unitons have energy at least m^{2}.

Acknowledgments: The author wishes to thank Professor C.K. Anand for drawing his attention to the question considered here. He thanks also Professors C.H. Gu, H.S. Hu for stimulating discussions and constant encouragement.

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