aa r X i v : . [ m a t h . DG ] D ec DEFORMATION OF BRODY CURVES AND MEAN DIMENSION
MASAKI TSUKAMOTO ∗ Abstract.
The main purpose of this paper is to show that ideas of deformation theorycan be applied to “infinite dimensional geometry”. We develop the deformation theoryof Brody curves. Brody curve is a kind of holomorphic map from the complex plane tothe projective space. Since the complex plane is not compact, the parameter space ofthe deformation can be infinite dimensional. As an application we prove a lower boundon the mean dimension of the space of Brody curves. Introduction
Main results.
Let z = x + y √− C .For a holomorphic curve f = [ f : f : · · · : f N ] : C → C P N with holomorphic functions f , f : · · · , f N , we define the pointwise norm | df | ≥ | df | = 14 π ∆ log (cid:0) | f | + | f | + · · · + | f N | (cid:1) (∆ := ∂ ∂x + ∂ ∂y ) . We call f a Brody curve if it satisfies | df | ≤ M ( C P N ) be the spaceof Brody curves in C P N with the compact-open topology. Then M ( C P N ) becomes aninfinite dimensional compact space and it admits a natural C -action:(2) ( f ( z ) , a ) f ( z + a ) for a Brody curve f ( z ) and a ∈ C . This paper studies the “mean dimension” dim( M ( C P N ) : C ). Mean dimension isa notion defined by Gromov [5] (see also Lindenstrauss-Weiss [8] and Lindenstrauss [7]).Mean dimension is a “dimension of an infinite dimensional space”. Intuitively (the precisedefinition will be given in Section 2),“ dim( M ( C P N ) : C ) = dim M ( C P N ) / vol( C )” . When we study the space of holomorphic maps from a compact Riemann surface, its(virtual) dimension can be derived from the deformation theory (and the index theorem).
Date : October 25, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Brody curve, deformation theory, mean dimension, the Nevanlinna theory. ∗ Supported by Grant-in-Aid for JSPS Fellows (19 · The main purpose of this paper is to develop a new deformation theory which can beapplied to the computation of dim( M ( C P N ) : C ).For a Brody curve f we define the Shimizu-Ahlfors characteristic function T ( r, f ) by T ( r, f ) := Z r dtt Z | z |
0. Let e ( C P N ) ell be the supremum of e ( f ) over elliptic Brody curves f in C P N .Obviously 0 < e ( C P N ) ell ≤ e ( C P N ). Using the argument in Tsukamoto [10, Section 4],we can prove that e ( C P N ) ell and e ( C P N ) asymptotically become equal to 1:(3) lim N →∞ e ( C P N ) ell = lim N →∞ e ( C P N ) = 1 . Our main result on the mean dimension is the following inequality:
Theorem 1.1. e ( C P N ) ell ( N + 1) ≤ dim( M ( C P N ) : C ) ≤ e ( C P N ) N. This theorem has the following two consequences:
Theorem 1.2. e ( C P ) ell ≤ dim( M ( C P ) : C ) ≤ e ( C P ) . Theorem 1.3. ≤ lim inf N →∞ dim( M ( C P N ) : C )) /N ≤ lim sup N →∞ dim( M ( C P N ) : C ) /N ≤ . Theorem 1.2 is the special case of Theorem 1.1. Theorem 1.3 comes from (3). The pointof Theorem 1.3 is that the estimate is explicit. (The mean dimension dim( M ( C P N ) : C )is a very transcendental object.)Theorem 1.2 leads us to the following conjecture (actually a second main purpose ofthis paper is to propose this conjecture to the mathematical community): Conjecture 1.4. e ( C P ) ell = e ( C P ) . EFORMATION OF BRODY CURVES 3
If this is true, then we get the following (index-theorem-like) result:(4) dim( M ( C P ) : C ) = 4 e ( C P ) . I think this formula is (if it is true) astonishing because the definitions of the left-hand-sideand right-hand-side of (4) are very different. (Mean dimension is a topological quantity ofthe space, and mean energy is defined by using the energy distribution of Brody curves.)Note that Conjecture 1.4 itself is a purely function-theoretic problem. It does not containa notion in the mean dimension theory.The upper bound, dim( M ( C P N ) : C ) ≤ e ( C P N ) N , in Theorem 1.1 is already provedin Tsukamoto [12, Theorem 1.4 and 1.5] by using the Nevanlinna theory . The task ofthis paper is to prove the lower bound: dim( M ( C P N ) : C ) ≥ e ( C P N ) ell ( N + 1). In orderto prove this, we will develop a deformation theory of Brody curves. This deformationtheory is a step toward the “infinite dimensional geometry”: The parameter space of thedeformation can be infinite dimensional. (But this is very natural because the space ofBrody curves is an infinite dimensional space.)A technical new feature of our deformation theory is the following: Usually we constructdeformation theory within the framework of “ L -theory” (or sometimes L p -theory for p < ∞ ). But (I think that) L -theory is not suitable for our purpose and it is better toconstruct the theory in the settings of “ L ∞ -theory”. (The fact that L ∞ is suitable forthe mean dimension theory is also suggested by [11]. In [11] it is shown that the meandimension of the unit ball in ℓ p (Γ) is zero, where 1 ≤ p < ∞ and Γ is a finitely generatedinfinite amenable group.) But the analysis in the L ∞ -settings is more complicated thanthat of L , and it is the main technical task of the paper.1.2. Remark on Conjecture 1.4.
An elliptic function f constructed below might bea good candidate for the function which attains the supremum of e ( f ). Actually thefollowing f is an extremal function of the Bloch-constant-type problem solved in Bonk-Eremenko [1]. Put e := 1 / √ , e := e π √− / / √ , e := e π √− / / √ , e := ∞ . These four points become the vertices of a regular tetrahedron inscribed in the Riemannsphere S = C P . Let ω be a positive real number (which will be fixed later) and set ω := ω exp( π √− / ⊂ C be the regular triangle whose vertices are 0, ω , ω and,˜∆ ⊂ C P the spherical regular triangle whose vertices are e , e , e . From the Riemannmapping theorem there exists an (unique) one-to-one holomorphic map f : ∆ → ˜∆ whichsends 0, ω , ω to e , e , e respectively. From the reflection principle, f can be extendedto an elliptic function whose period lattice is Λ := Z (2 ω ) ⊕ Z (2 ω ) ⊂ C . The set ofcritical points of f is Z ω + Z ω ⊂ C , and the critical values are e , e , e , e . We have For the upper bound, see also Gromov [5, p. 396, (c)] and Tsukamoto [12, Remark 1.6].
MASAKI TSUKAMOTO deg( f : C / Λ → C P ) = 2. f satisfies( f ′ ) = K ( f − e )( f − e )( f − e ) = K ( f − / √ K . ω can be derived from K by ω = 1 √ K Z ∞ / √ dx q x − / √ / √ K Z ∞ dx √ x − . The spherical derivative | df | ( z ) defined in (1) is given by | df | = 1 π | f ′ | (1 + | f | ) = Kπ | f − / √ | (1 + | f | ) . Some calculation shows sup z ∈ C | z − / √ | (1 + | z | ) = 1 / √ . Therefore sup z ∈ C | df | ( z ) = Kπ √ . We choose ω so that K = π √
8. Then sup z ∈ C | df | ( z ) = 1 and f becomes an elliptic Brodycurve. Since the volume of the fundamental domain of Λ in C is given by | C / Λ | = 2 √ ω ,we have e ( f ) = 2 | C / Λ | = 2 π √ (cid:18)Z ∞ dx √ x − (cid:19) − = 0 . · · · . From Theorem 1.2,dim( M ( C P ) : C ) ≥ π √ (cid:18)Z ∞ dx √ x − (cid:19) − = 2 . · · · . This inequality might be an equality.1.3.
Remark on residual dimension.
We want to remark about the “residual dimen-sion” introduced by Gromov (see [5, p. 330 and p. 346]). This subsection is logicallyindependent of the proof of Theorem 1.1, and readers can skip it. (But the idea of thissubsection is implicitly used in Section 3.) Let Λ ⊂ C be a lattice and M ( C P N ) Λ be theset of Brody curves f satisfying f ( z + λ ) = f ( z ) for all λ ∈ Λ. M ( C P N ) Λ is the set offixed-points of the natural action of Λ on M ( C P N ). In other words, M ( C P N ) Λ is thespace of holomorphic maps f : C / Λ → C P N satisfying | df | ≤
1. The usual deformationtheory gives (cf. Section 3)1 | C / Λ | dim M ( C P N ) Λ ≤ N + 1) sup f ∈M ( C P N ) Λ e ( f ) ≤ e ( C P N ) ell ( N + 1) . In particular, Theorem 1.1 givesresdim( M ( C P N ) : { n Λ } n ≥ ) := lim inf n →∞ | C /n Λ | dim M ( C P N ) n Λ , ≤ e ( C P N ) ell ( N + 1) ≤ dim( M ( C P N ) : C ) . (5) EFORMATION OF BRODY CURVES 5
Moreover some consideration showssup Λ ⊂ C resdim( M ( C P N ) : { n Λ } n ≥ ) = 2 e ( C P N ) ell ( N + 1) , where Λ runs over all lattices in C . Remark 1.5.
In (5) the residual dimension is not bigger than the mean dimension. Butin general residual dimension can be bigger than mean dimension; Consider the naturalaction of Z on [0 , Z . For n ≥ F n ⊂ [0 , /n ] Z be the set of fixed-points of the actionof n Z on [0 , /n ] Z . Set X := S n ≥ F n . X becomes a Z -invariant closed set in [0 , Z . Let X n ( n ≥
1) be the set of fixed-points of the action of n Z on X . Since F n ⊂ X n , we havedim X n = n . Therefore, resdim( X : { n Z } n ≥ ) := lim inf n →∞ dim X n /n = 1. On the otherhand, it is not difficult to see dim( X : Z ) = 0.1.4. Remark: twisted-elliptic Brody curves.
For a Brody curve f : C → C P N , wecall f a twisted-elliptic Brody curve if there exist a lattice Λ ⊂ C and a homomorphism(of groups) φ : Λ → P U ( N + 1) such that f ( z + λ ) = φ ( λ ) f ( z ) for all z ∈ C and λ ∈ Λ . Note that the projective unitary group
P U ( N + 1) is the holomorphic-isometry group of C P N . Perhaps it might be able to apply the methods in this paper to twisted-ellipticBrody curves also. I think this is a natural generalization. But I don’t know whether thisimproves the estimate of the mean dimension or not. So I don’t study this case in thispaper. If there is a reader who has an interest in this case, please pursue it.1.5. Organization of the paper.
In Section 2 we review the definition and basic prop-erties of mean dimension. In Section 3 we prove Theorem 1.1, assuming an analytic resultabout the “deformation theory of Brody curves” proved in Section 5. Section 4 is a prepa-ration for Section 5. In Section 5 we develop the deformation theory of Brody curves andcomplete the proof of Theorem 1.1. We give a remark about Gromov’s conjecture onrational curves and mean dimension in Section 6.1.6.
Acknowledgement.
I wish to thank Professors Minoru Murata and Yoshio Tsut-sumi. They gave me several helpful advices on elliptic partial differential equations.Especially I learned the basic idea of the proof of Proposition 4.2 from Professor Mi-noru Murata. I also wish to thank Professor Katsutoshi Yamanoi for various valuablediscussions. He gave me an important suggestion about Conjecture 1.4.2.
Review of mean dimension
We review the definitions of mean dimension. For the detail, see Gromov [5] orLindenstrauss-Weiss [8]. Let (
X, d ) be a compact metric space, Y a topological space. For ε >
0, a continuous map f : X → Y is called an ε -embedding if we have Diam f − ( y ) ≤ ε MASAKI TSUKAMOTO for all y ∈ Y . Let Widim ε ( X, d ) be the minimum number n ≥ n -dimensional polyhedron K and an ε -embedding from X to K . The following is provedin Gromov [5, p. 333]. (This is a basic result for us. So we will give its proof in Appendix.) Proposition 2.1.
Let ( V, ||·|| ) be an n -dimensional normed linear space (over R ). Let B ⊂ V be the closed ball of radius r > with the distance d ( x, y ) := || x − y || . Then Widim ε ( B, d ) = n for all ε < r. Suppose the Lie group C continuously acts on the compact metric space X . For anypositive number R , we define the distance d R ( · , · ) on X by d R ( p, q ) := sup z ∈ C , | z |≤ R d ( z.p, z.q ) for p, q ∈ X. Set Widim ε ( X : C ) := lim R →∞ πR Widim ε ( X, d R ) . This limit always exists (see Gromov [5, pp. 335-338] and Lindenstrauss-Weiss [8, Ap-pendix]). We define the mean dimension dim( X : C ) by settingdim( X : C ) := lim ε → Widim ε ( X : C ) . dim( X : C ) is a topological invariant, i.e., it does not depend on the given distance d .Let Λ = Z ω ⊕ Z ω ⊂ C be a lattice ( ω , ω ∈ C ). Then Λ also acts on X and we candefine the mean dimension dim( X : Λ) as follows: For any positive integer n we set(6) Ω n := { xω + yω ∈ Λ | x, y ∈ Z , ≤ x, y ≤ n − } We define the distance d Ω n ( · , · ) on X by(7) d Ω n ( p, q ) := max z ∈ Ω n d ( z.p, z.q ) for p, q ∈ X. Set (the following limit always exists)Widim ε ( X : Λ) := lim n →∞ n Widim ε ( X, d Ω n ) . We define the mean dimension dim( X : Λ) bydim( X : Λ) := lim ε → Widim ε ( X : Λ) . The following gives the relation between dim( X : C ) and dim( X : Λ). (This is given inGromov [5, p.329] and Lindenstrauss-Weiss [8, Proposition 2.7]. For its proof, see alsoTsukamoto [12, Proposition 4.5].) Proposition 2.2. dim( X : Λ) = | C / Λ | dim( X : C ) , where | C / Λ | denotes the volume the fundamental domain of Λ in C . EFORMATION OF BRODY CURVES 7 Proof of Theorem 1.1
Let Λ ⊂ C be a lattice and π : C → C / Λ be the natural projection. Let ϕ : C / Λ → C P N be a non-constant holomorphic map satisfying | dϕ | <
1, and set ˜ ϕ := ϕ ◦ π : C → C P N .We have e ( ˜ ϕ ) = deg ϕ/ | C / Λ | , where deg ϕ = h c ( ϕ ∗ O (1)) , [ C / Λ] i . Let T ′ C P N be the holomorphic tangent bundle of C P N and consider its pull-back E := ˜ ϕ ∗ T ′ C P N over C . E is equipped with the Hermitianmetric induced by the Fubini-Study metric. We define a Banach space V as the space ofbounded holomorphic sections of E with the sup-norm ||·|| ∞ :(8) V := { u : C → E | u is a holomorphic section and satisfies || u || ∞ := sup z ∈ C | u ( z ) | < ∞} . The following result is the keystone of the proof of Theorem 1.1.
Proposition 3.1.
There are positive numbers δ and C such that for any u ∈ V with || u || ∞ ≤ δ there exists a Brody curve f u : C → C P n satisfying the following:(i) f = ˜ ϕ .(ii) The map B δ ∋ u f u ∈ M ( C P N ) is Λ -equivariant. Here B δ = { u ∈ V | || u || ∞ ≤ δ } and we have considered the natural Λ -action on E and V .(iii) For any u, v ∈ V with || u || ∞ , || v || ∞ ≤ δ , we have C − || u − v || ∞ ≤ sup z ∈ C d ( f u ( z ) , f v ( z )) ≤ C || u − v || ∞ , where d ( · , · ) denotes the distance on C P N defined by the Fubini-Study metric. We will prove this proposition in Section 5 by constructing a “deformation theory”of ˜ ϕ . (Each f u is a “small deformation” of ˜ ϕ .) Here we prove Theorem 1.1, assumingProposition 3.1. Proof of Theorem 1.1.
To begin with, we define the distance d ( · , · ) on M ( C P N ) by d ( f, g ) := X n ≥ − n sup | z |≤ n d ( f ( z ) , g ( z )) for f, g ∈ M ( C P N ) . Let Λ = Z ω ⊕ Z ω ⊂ C be a lattice in C ( ω , ω ∈ C ). For any positive integer n we set(9) K n := { xω + yω ∈ C | x, y ∈ R , ≤ x, y ≤ n } .K n is a fundamental domain of n Λ in C . There is a positive constant C = C (Λ) suchthat sup z ∈ K d ( f ( z ) , g ( z )) ≤ C d ( f, g ) for f, g ∈ M ( C P N ) . Then for any n > z ∈ K n d ( f ( z ) , g ( z )) ≤ C d Ω n ( f, g ) for f, g ∈ M ( C P N ) , where Ω n and d Ω n ( · , · ) are defined by (6) and (7). MASAKI TSUKAMOTO
Let ϕ : C / Λ → C P N be a non-constant holomorphic map satisfying | dϕ | <
1. Wedefine ˜ ϕ , E and V as before. For any positive integer n , let π n : C /n Λ → C / Λ be thenatural n -fold covering map, and set ϕ n := ϕ ◦ π n : C /n Λ → C P N . Consider V n := H ( C /n Λ , O ( ϕ ∗ n T ′ C P N )) .V n is the space of holomorphic sections of ϕ ∗ n T ′ C P N over C /n Λ, and it can be identifiedwith the subspace of V consisting of n Λ-invariant holomorphic sections of E . From theRiemann-Roch formula and the vanishing of H (cf. Section 5), we havedim V n = 2 dim C V n = 2 h ϕ ∗ n c ( C P N ) , [ C /n Λ] i = 2 n ( N + 1) deg ϕ. (Actually we need only the inequality dim V n ≥ n ( N + 1) deg ϕ in this proof. Hencewe don’t need H = 0.) Let δ, C be the positive constants in Proposition 3.1. Set B δ ( V n ) := { u ∈ V n | || u || ∞ ≤ δ } . For any u ∈ B δ ( V n ) there exists a Brody curve f u . Fromthe Λ-equivariance in Proposition 3.1 (ii), f u is n Λ-invariant (i.e., it can be considered asa holomorphic map from C /n Λ to C P N ). Then from Proposition 3.1 (iii) and (10), forany u, v ∈ B δ ( V n ) || u − v || ∞ ≤ C sup z ∈ C d ( f u ( z ) , f v ( z )) = C sup z ∈ K n d ( f u ( z ) , f v ( z )) ≤ CC d Ω n ( f u , f v ) . Moreover Proposition 3.1 shows that the map B δ ( V n ) → M ( C P N ), u f u , is continuous.Therefore for any ε > ε ( M ( C P N ) , d Ω n ) ≥ Widim CC ε ( B δ ( V n ) , ||·|| ∞ ) , where B δ ( V n ) is equipped with the distance || u − v || ∞ . Then Proposition 2.1 implies, for ε < δ/CC , Widim ε ( M ( C P N ) , d Ω n ) ≥ dim V n = 2 n ( N + 1) deg ϕ. Note that δ/CC is independent of n (this is the crucial point). HenceWidim ε ( M ( C P N ) : Λ) = lim n →∞ n Widim( M ( C P N ) , d Ω n ) ≥ N + 1) deg ϕ, for any ε < δ/CC . Thus dim( M ( C P N ) : Λ) ≥ N + 1) deg ϕ. Using Proposition 2.2, we getdim( M ( C P N ) : C ) = 1 | C / Λ | dim( M ( C P N ) : Λ) , ≥ N + 1) deg ϕ/ | C / Λ | = 2( N + 1) e ( ˜ ϕ ) . (11)Then we can prove Theorem 1.1. Let f ∈ M ( C P N ) be any elliptic Brody curve. Take apositive number c < g ( z ) := f ( cz ). Then g is an elliptic Brody curve satisfying | dg | <
1, and we can apply (11) to g :dim( M ( C P N ) : C ) ≥ N + 1) e ( g ) = 2 c ( N + 1) e ( f ) . EFORMATION OF BRODY CURVES 9
Let c →
1. Then dim( M ( C P N ) : C ) ≥ N + 1) e ( f ) . This shows Theorem 1.1. (cid:3)
Remark 3.2.
In the above proof, each B δ ( V n ) describes a small deformation of ϕ n : C /n Λ → C P N . The small deformations of each ϕ n can be constructed by the usualdeformation theory. The point of Proposition 3.1 is that we can construct the deformationsof all ϕ n with the estimates independent of n ; This is essential in the above proof.4. Analytic preliminaries
This section is a preparation for the proof of Proposition 3.1.4.1.
Helmholtz equation.
We will need some elementary facts about the Helmholtzequation on the plane R :(12) ( − ∆ + λ ) w = 0 , where λ > ∂ ∂x + ∂ ∂y . Set(13) w λ ( z ) := 12 π Z π exp √ λ ( x cos θ + y sin θ ) dθ.w λ satisfies (12) and w λ >
0. The following fact can be easily checked:
Lemma 4.1.
The minimum value of w λ is w λ (0) = 1 , and w λ ( z ) → + ∞ as | z | → ∞ . L ∞ -estimate. Let F be a holomorphic vector bundle over the complex plane C witha Hermitian metric h . Let ¯ ∂ : Ω ( F ) → Ω , ( F ) be the Dolbeault operator, and ∇ thecanonical connection on ( F, h ). We denote the formal adjoint of ¯ ∂ and ∇ by ¯ ∂ ∗ and ∇ ∗ .We have the following Weintzenb¨ock formula: for any ξ ∈ Ω , ( F )(14) ¯ ∂ ¯ ∂ ∗ ξ = 12 ∇ ∗ ∇ ξ + Rξ, where Rξ = [ ∇ ∂/∂z , ∇ ∂/∂ ¯ z ] ξ . Note that for ξ = u ⊗ d ¯ z ( u ∈ Γ( F )) we have ∇ ∂/∂z ξ = ( ∇ ∂/∂z u ) ⊗ d ¯ z, ∇ ∂/∂ ¯ z ξ = ( ∇ ∂/∂ ¯ z u ) ⊗ d ¯ z. For ξ = u ⊗ d ¯ z and η = v ⊗ d ¯ z ( u, v ∈ Γ( F )), we set h ξ, η i := 2 h ( u, v ). We suppose that F is “positive” in the following sense: there exists a positive number a such that for any ξ ∈ Ω , ( F )(15) h Rξ, ξ i ≥ a | ξ | . Proposition 4.2.
Let ξ ∈ Ω , ( F ) be a F -valued (0 , -form of class C , and set η := ¯ ∂ ¯ ∂ ∗ ξ .If || ξ || ∞ , || η || ∞ < ∞ , then || ξ || ∞ ≤ a || η || ∞ . Proof.
There is a point z ∈ C satisfying | ξ ( z ) | ≥ || ξ || ∞ /
2. We suppose z = 0 forsimplicity. We have ∆ | ξ | = − h∇ ∗ ∇ ξ, ξ i + 2 |∇ ξ | . Using the Weintzenb¨ock formula (14) and η = ¯ ∂ ¯ ∂ ∗ ξ , we have∆ | ξ | = − h η, ξ i + 4 h Rξ, ξ i + 2 |∇ ξ | , ≥ − h η, ξ i + 4 a | ξ | . Set M := 4 || ξ || ∞ || η || ∞ . We have ( − ∆ + 4 a ) | ξ | ≤ M .Set w ( z ) := M w a ( z ) / a , where w a is a function defined in (13). w ( z ) satisfies( − ∆ + 2 a ) w = 0 , w ≥ M/ a. Then ( − ∆ + 4 a ) w = 2 aw ≥ M . Therefore( − ∆ + 4 a )( w − | ξ | ) ≥ . Since || ξ || ∞ < ∞ and w ( z ) → ∞ ( | z | → ∞ ), we have w ( z ) − | ξ | > | z | ≫ w (0) − | ξ (0) | ≥ . Therefore || ξ || ∞ / ≤ | ξ (0) | ≤ w (0) = M/ a = 2 || ξ || ∞ || η || ∞ /a. Thus || ξ || ∞ ≤ || η || ∞ /a . (cid:3) Perturbation of a Hermitian metric.
We briefly discuss a perturbation techniqueof a Hermitian metric. M. Gromov also discuss it in [5, p. 399]. Let Λ ⊂ C be a lattice and ϕ : C / Λ → C P N a non-constant holomorphic map. Let ϕ ∗ T ′ C P N → C / Λ be the pull-back of the holomorphic tangent bundle T ′ C P N with the Hermitian metric h induced bythe Fubini-Study metric. Since the holomorphic bisectional curvature of the Fubini-Studymetric is positive, there is c > u ∈ Γ( ϕ ∗ T ′ C P N )(16) h ( Ru, u ) ≥ c | dϕ | | u | , where R is the curvature defined by Ru := [ ∇ ∂/∂z , ∇ ∂/∂ ¯ z ] u ( ∇ is the canonical connection). Lemma 4.3.
There is a Hermitian metric h ′ on ϕ ∗ T ′ C P N satisfying the following: Thereexists a > such that for any u ∈ Γ( ϕ ∗ T ′ C P N ) h ′ ( R ′ u, u ) ≥ a | u | , where R ′ is the curvature of h ′ . EFORMATION OF BRODY CURVES 11
Proof.
Set h ′ = e − f h where f is a real valued function defined later. Then for any u ∈ Γ( ϕ ∗ T ′ C P N ) R ′ u = 14 (∆ f ) u + Ru and h ′ ( R ′ u, u ) = e − f {
14 (∆ f ) | u | + h ( Ru, u ) } . Set { p ∈ C / Λ | dϕ ( p ) = 0 } =: { p , · · · , p n } . Let δ > A := ` i B δ ( p i ) ⊂ C / Λ ( B δ ( p i ) is the closed ball of radius δ centered at p i ). From (16)there is c ′ > h ( Ru, u ) ≥ c ′ | u | for u ∈ ( ϕ ∗ T ′ C P N ) p at p ∈ A c = ( C / Λ) \ A. Let g be a real valued function on C / Λ satisfying( i ) g > A, ( ii ) g ≥ − c ′ / A c , ( iii ) Z C / Λ g dxdy = 0 . From the condition (iii), there exists f satisfying ∆ f / g . Therefore h ′ ( R ′ u, u ) = e − f ( g | u | + h ( Ru, u )) . From the conditions (i) and (ii), it is easy to see that there exists a > h ′ ( R ′ u, u ) ≥ a | u | for all sections u . (cid:3) Deformation theory
In this section we prove Proposition 3.1 by constructing “deformation theory”.
Remark 5.1.
M. Gromov gives a certain “deformation” argument different from ours in[5, pp. 399-400].5.1.
Deformation and the proof of Proposition 3.1.
Let Λ ⊂ C be a lattice and π : C → C / Λ the natural projection. Let ϕ : C / Λ → C P N be a non-constant holomorphicmap satisfying | dϕ | < ϕ := ϕ ◦ π . Let E := ˜ ϕ ∗ T ′ C P N be the pull-back of theholomorphic tangent bundle T ′ C P N . E is equipped with the Hermitian metric h inducedby the Fubini-Study metric. E admits the natural Λ-action.Let k be a non-negative integer and α a real number satisfying 0 < α <
1. We want todefine the H¨older spaces C k,α ( E ) and C k,α (Ω , ( E )). Let { U n } mn =1 , { U ′ n } mn =1 and { U ′′ n } mn =1 be open coverings of C / Λ satisfying the following (i), (ii), (iii).(i) ¯ U n ⊂ U ′ n and ¯ U ′ n ⊂ U ′′ n , and all U n , U ′ n , U ′′ n are smooth regions i.e., their boundariesare smooth.(ii) The covering map π : C → C / Λ can be trivialized on each U ′′ n , i.e., there is a disjointunion π − ( U ′′ n ) = ` λ ∈ Λ U ′′ n,λ such that each U ′′ n,λ is a connected component of π − ( U ′′ n ) and π | U ′′ n,λ : U ′′ n,λ → U ′′ n is biholomorphic. Set U n,λ := π − ( U n ) ∩ U ′′ n,λ and U ′ n,λ := π − ( U ′ n ) ∩ U ′′ n,λ ,then π | U n,λ : U n,λ → U n and π | U ′ n,λ : U ′ n,λ → U ′ n are biholomorphic and we have disjointunions π − ( U n ) = ` λ ∈ Λ U n,λ and π − ( U ′ n ) = ` λ ∈ Λ U ′ n,λ . (iii) A bundle trivialization of ϕ ∗ T ′ C P N is given on each U ′′ n , i.e., we have a holomorphicbundle isomorphism ϕ ∗ T ′ C P N | U ′′ n → U ′′ n × C N . Then we also have a trivialization of E over each U ′′ n,λ through the isomorphisms U ′′ n,λ → U ′′ n .Let u be a section of E (not necessarily holomorphic). From (iii) in the above, u | U ′′ n,λ canbe seen as a vector-valued function on U ′′ n,λ . Hence we can define its C k,α -norm || u || C k,α ( ¯ U n,λ ) over ¯ U n,λ as a vector-valued function (see Gilbarg-Trudinger [3, Chapter 4]). We definethe C k,α ( E )-norm of u by || u || C k,α ( E ) := sup n,λ || u || C k,α ( ¯ U n,λ ) . We define the H¨older space C k,α ( E ) as the space of sections of E whose C k,α ( E )-normsare finite. For ξ = u ⊗ d ¯ z ∈ Ω , ( E ) ( u ∈ Γ( E )) we define its C k,α (Ω , ( E ))-norm by || ξ || C k,α (Ω , ( E )) := √ || u || C k,α ( E ) , and we define C k,α (Ω , ( E )) := C k,α ( E ) ⊗ d ¯ z . Then C k,α ( E ) and C k,α (Ω , ( E )) becomeBanach spaces. (In the above definition of the H¨older spaces we have not used the opensets U ′ n,λ . They will be used in the next subsection.)The holomorphic tangent bundle T ′ C P N is the eigenspace of the complex structure J on T C P N ⊗ R C of eigenvalue √−
1. We naturally identify T ′ C P N with the tangent bundle T C P N by T C P N ∋ u ←→ u − √− J u ∈ T ′ C P N . So E can be identified with ˜ ϕ ∗ T C P N .Consider (cf. McDuff-Salamon [9, Chapter 3])Φ : C ,α ( E ) → C ,α (Ω , ( E )) u P u ( ¯ ∂ exp u ) ⊗ d ¯ z. Here, exp : T C P N → C P N is the exponential map defined by the Fubini-Study metric,and P u : T exp u C P N → T ˜ ϕ C P N is the parallel transport along the geodesic exp tu (0 ≤ t ≤ ∂ exp u ∈ T exp u C P N is defined by¯ ∂ exp u := 12 (cid:18) ∂∂x exp u + J ∂∂y exp u (cid:19) . Φ is a smooth map between the Banach spaces, and it is Λ-equivariant. The map C ∋ z exp u ( z ) ∈ C P N becomes a holomorphic curve if and only if Φ( u ) = 0. The derivative ofΦ at the origin is the Dolbeault operator:(17) ( d Φ) = ¯ ∂ : C ,α ( E ) → C ,α (Ω , ( E )) . Proposition 5.2.
The small deformation of ˜ ϕ is unobstructed, i.e., there exists a Λ -equivariant bounded linear operator Q : C ,α (Ω , ( E )) → C ,α ( E ) satisfying ¯ ∂ ◦ Q = 1 . EFORMATION OF BRODY CURVES 13
This proposition will be proved later. Let V := ker ¯ ∂ be the kernel of (17) (this definitioncoincides with (8)). Note that V is a complement of the image of Q in C ,α ( E ) and thatit is Λ-invariant. From the elliptic regularity (cf. Subsection 5.2), we have || u || ∞ ≤ const · || u || C ,α ( E ) ≤ const ′ · || u || ∞ for any u ∈ V , where const and const ′ are independent of u .For r > B r := { u ∈ V | || u || ∞ ≤ r } . From Proposition 5.2 and the implicit functiontheorem, there are δ > g : B δ → Image( Q ) satisfying(18) ( i ) g (0) = 0 , ( ii ) Φ( u + g ( u )) = 0 for all u ∈ B δ , ( iii ) ( dg ) = 0 . Set f u := exp( u + g ( u )) : C → C P N for u ∈ B δ . We want to show that these f u satisfy the conditions in Proposition 3.1. From (i) and (ii) in (18), f = ˜ ϕ and each f u is aholomorphic curve. Since | dϕ | <
1, if we choose δ sufficiently small, all f u ( u ∈ B δ ) becomeBrody curves, i.e., | df u | ≤
1. Since g is Λ-equivariant, the map B δ ∋ u f u ∈ M ( C P N )is also Λ-equivariant.If we choose δ > K > u, v ∈ B δ K − || u + g ( u ) − v − g ( v ) || ∞ ≤ sup z ∈ C d ( f u ( z ) , f v ( z )) ≤ K || u + g ( u ) − v − g ( v ) || ∞ (this is a standard property of the exponential map) and we have || g ( u ) − g ( v ) || ∞ ≤ || u − v || ∞ . Here we have used the condition (iii) in (18). Hence12 K − || u − v || ∞ ≤ sup z ∈ C d ( f u ( z ) , f v ( z )) ≤ K || u − v || ∞ . Then all the conditions in Proposition 3.1 have been proved (assuming Proposition 5.2).5.2.
Proof of Proposition 5.2.
To begin with, we consider a perturbation of the Her-mitian metric on E . E has the Hermitian metric h induced by the Fubini-Study metric.From Proposition 4.3, ϕ ∗ T ′ C P N admits a Hermitian metric which is “positive” in thesense of Proposition 4.3. Then, pulling back this metric to E , E admits a Λ-invariantHermitian metric h ′ satisfying (15) for some a >
0. In this subsection we use this h ′ asthe Hermitian metric on E . Note that the definitions of the H¨older spaces C k,α ( E ) and C k,α (Ω , ( E )) does not use the Hermitian metric. So they are independent of the choiceof the Hermitian metric. (The sup-norm ||·|| ∞ depends on the Hermitian metric, but thesup-norms defined by h and h ′ are equivalent to each other.)We prove Proposition 5.2 by showing that(19) ¯ ∂ ¯ ∂ ∗ : C ,α (Ω , ( E )) → C ,α (Ω , ( E )) is an isomorphism. (Note that the Dolbeault operator ¯ ∂ is independent of the Hermitianmetric h ′ , but its formal adjoint ¯ ∂ ∗ depends on h ′ .) Then Q := ¯ ∂ ∗ ( ¯ ∂ ¯ ∂ ∗ ) − gives a Λ-equivariant right inverse of ¯ ∂ . The injectivity of (19) directly follows from the L ∞ -estimatein Proposition 4.2. So the problem is its surjectivity. Lemma 5.3. If η ∈ C ,α (Ω , ( E )) has a compact support, then there exists ξ ∈ C ,α (Ω , ( E )) satisfying ¯ ∂ ¯ ∂ ∗ ξ = η .Proof. We set L (Ω , ( E )) := { ξ ∈ L (Ω , ( E )) | ∇ ξ ∈ L } , where ∇ ξ is the distributional derivative of ξ . (The L -norm and the L -space are definedby using the Hermitian metric h ′ .) Let ξ ∈ Ω , ( E ) be a compact-supported smoothsection. From the Weitzenb¨ock formula (14), (cid:12)(cid:12)(cid:12)(cid:12) ¯ ∂ ∗ ξ (cid:12)(cid:12)(cid:12)(cid:12) L = ( ¯ ∂ ¯ ∂ ∗ ξ, ξ ) L = ( 12 ∇ ∗ ∇ ξ + Rξ, ξ ) L , ≥ ||∇ ξ || L + a || ξ || L . Therefore for any ξ ∈ L (Ω , ( E )) (cid:12)(cid:12)(cid:12)(cid:12) ¯ ∂ ∗ ξ (cid:12)(cid:12)(cid:12)(cid:12) L ≥ ||∇ ξ || L + a || ξ || L . This means that the inner-product ( ¯ ∂ ∗ ξ , ¯ ∂ ∗ ξ ) L ( ξ , ξ ∈ L (Ω , ( E ))) is equivalent tothe natural inner-product ( ξ , ξ ) L := ( ∇ ξ , ∇ ξ ) L + ( ξ , ξ ) L on L (Ω , ( E )). η defines a bounded functional ( · , η ) L : L (Ω , ( E )) → C . From the Riesz represen-tation theorem, there (uniquely) exists ξ ∈ L (Ω , ( E )) satisfying ( ¯ ∂ ∗ φ, ¯ ∂ ∗ ξ ) L = ( φ, η ) L for all φ ∈ L (Ω , ( E )) and(20) || ξ || L := ( ξ, ξ ) / L ≤ const · || η || L . In particular, ¯ ∂ ¯ ∂ ∗ ξ = η in the sense of distribution. (The above is a standard argumentin the “ L -theory”.)We want to show ξ ∈ C ,α (Ω , ( E )). Remember the open covering C = S n,λ U n,λ = S n,λ U ′ n,λ = S n,λ U ′′ n,λ ( n = 1 , · · · , m, λ ∈ Λ) used in the definition of the H¨older spaces.Each ξ | U ′′ n,λ can be seen as a vector-valued function. From the Sobolev embedding L ֒ →C , the elliptic regularity (see Gilbarg-Trudinger [3, Chapter 8]) and (20), (cid:12)(cid:12)(cid:12)(cid:12) ξ | U n,λ (cid:12)(cid:12)(cid:12)(cid:12) ∞ ≤ const n · (cid:12)(cid:12)(cid:12)(cid:12) ξ | U n,λ (cid:12)(cid:12)(cid:12)(cid:12) L ≤ const ′ n (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ξ | U ′ n,λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) η | U ′ n,λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L (cid:19) ≤ const ′′ n · || η || L , where const n , const ′ n and const ′′ n are positive constants which depend on n = 1 , · · · , m .The point is that they are independent of λ ∈ Λ; this is due to the Λ-symmetry of theequation. Then || ξ || ∞ ≤ const · || η || L . EFORMATION OF BRODY CURVES 15
From the Schauder interior estimate (see Gilbarg-Trudinger [3, Chapter 6]), || ξ || C ,α ( ¯ U n,λ ) ≤ const n (cid:16) || ξ || ∞ + || η || C ,α ( ¯ U ′ n,λ ) (cid:17) ≤ const ( || η || L + || η || C ,α (Ω , ( E )) ) . Here we have used the following fact (which can be easily checked):(21) sup n,λ || η || C ,α ( ¯ U ′ n,λ ) ≤ const · || η || C ,α (Ω , ( E )) (= const · sup n,λ || η || C ,α ( ¯ U n,λ ) )Thus || ξ || C ,α (Ω , ( E )) < ∞ and ξ ∈ C ,α (Ω , ( E )). (cid:3) Then we can prove that (19) is surjective (and hence isomorphic). Take an arbitrary η ∈ C ,α (Ω , ( E )). Let φ k ( k ≥
1) be cut-off functions on the plane C such that 0 ≤ φ k ≤ φ k ( z ) = 1 for | z | ≤ k and φ k ( z ) = 0 for | z | ≥ k + 1. Set η k := φ k η . From Lemma 5.3, thereexists ξ k ∈ C ,α (Ω , ( E )) satisfying ¯ ∂ ¯ ∂ ∗ ξ k = η k . From the L ∞ -estimate in Proposition 4.2,(22) || ξ k || ∞ ≤ const · || η k || ∞ ≤ const · || η || ∞ . Using the Schauder interior estimate on each U ′ n,λ , we get || ξ k || C ,α ( ¯ U n,λ ) ≤ const n (cid:16) || η k || C ,α ( ¯ U ′ n,λ ) + || ξ k || ∞ (cid:17) ≤ const ′ n (cid:16) || η k || C ,α ( ¯ U ′ n,λ ) + || η || ∞ (cid:17) . Since η k | U ′ n,λ = η | U ′ n,λ for k ≫ n, λ )), { ξ k | U n,λ } k ≥ is a bounded sequencein C ,α ( ¯ U n,λ ). Hence, if we choose a subsequence, { ξ k | U n,λ } k ≥ becomes a convergentsequence in C ( ¯ U n,λ ) (by Arzela-Ascoli’s theorem). Therefore (by using the diagonalargument) there exists ξ ∈ Ω , ( E ) of class C such that { ξ k | U n,λ } k ≥ converges to ξ | U n,λ in C ( ¯ U n,λ ) for each ( n, λ ). Since ¯ ∂ ¯ ∂ ∗ ξ k = η k , we have ¯ ∂ ¯ ∂ ∗ ξ = η . From (22), || ξ || ∞ ≤ const · || η || ∞ < ∞ . Using the Schauder interior estimate, we get || ξ || C ,α ( ¯ U n,λ ) ≤ const n (cid:16) || η || C ,α ( ¯ U ′ n,λ ) + || ξ || ∞ (cid:17) ≤ const ′ n (cid:16) || η || C ,α ( ¯ U ′ n,λ ) + || η || ∞ (cid:17) . From (21), || ξ || C ,α (Ω , ( E )) ≤ const · || η || C ,α (Ω , ( E )) < ∞ . Therefore we get ξ ∈ C ,α (Ω , ( E )) satisfying ¯ ∂ ¯ ∂ ∗ ξ = η . Then (19) is an isomorphism, andthe proof of Proposition 5.2 is finished (and hence the proof of Theorem 1.1 is completed).6. Remark on Gromov’s conjecture on rational curves and meandimension
Gromov gives the following (very beautiful) conjecture in [5, p. 329].
Conjecture 6.1.
Let X ⊂ C P N be a projective manifold, and M ( X ) the space of Brodycurves in X . Then dim( M ( X ) : C ) > if and only if X contains a rational curve. The “if” part is easy and the problem is the “only if” part. The purpose of this sectionis to show the following proposition:
Proposition 6.2.
There exists a compact Hermitian manifold X such that X contains norational curve and satisfies dim( M ( X ) : C ) > . Here M ( X ) is the space of holomorphicmaps f : C → X satisfying sup z ∈ C | df | ( z ) := sup z ∈ C √ | df ( ∂/∂z ) | ≤ . This shows that the projectivity (or the K¨ahler condition) is essential in Conjecture6.1. (Actually I feel that the following argument suggests that the true conjecture mightbe something like the following; if dim( M ( X ) : C ) > X (cf. Gromov [5, p. 330, EXAMPLE]).) We prove Proposition 6.2 by usingan argument similar to that of Section 3. (But this case is much easier than the proofof Theorem 1.1; We don’t need a serious analytic argument. In particular we don’t usethe results in Section 4,5. Perhaps we can also prove Proposition 6.2 by applying theargument in Gromov [5, pp. 385-388] to the following construction.)The compact Hermitian manifold X constructed below is actually known as a counter-example of “bend-and-break” technique for general complex manifolds (see Koll´ar-Mori[6, Example 1.8]). We follow the description of [6, Example 1.8].Let C / Λ be an elliptic curve (Λ is a lattice in C ). Let L be a holomorphic line bundleof deg ≥ C / Λ such that there exists two holomorphic sections s, t of L satisfying { z ∈ C / Λ | s ( z ) = t ( z ) = 0 } = ∅ . Set F := L ⊕ L . The vector bundle F has the followingfour sections: ( s, t ) , ( √− s, −√− t ) , ( t, − s ) , ( √− t, √− s ) . These are R -linearly independent all over C / Λ. (Therefore F becomes a product bundleas a real vector bundle.) Hence we can define a lattice bundle Γ ⊂ F byΓ := { x ( s, t ) + x ( √− s, −√− t ) + x ( t, − s ) + x ( √− t, √− s ) | x , x , x , x ∈ Z } . We define a compact complex threefold X by X := F/ Γ. (Topologically X = T × T = T .) Obviously X contains no rational curve. But X can contain lots of Brody curves aswe will see below.We give a Hermitian metric (of a complex vector bundle) to F and a Hermitian metric(of a complex manifold) to X . Let π : C → C / Λ be the natural projection and E := π ∗ F the pull-back of F by π . E is equipped with the Λ-invariant Hermitian metric induced bythe metric on F . Let V be the (Banach) space of bounded holomorphic sections of E withthe sup-norm ||·|| ∞ , and set B δ ( V ) := { u ∈ V | || u || ∞ ≤ δ } for δ >
0. Let p : F → X be thenatural projection. If we choose δ sufficiently small and consider some scale-change of theHermitian metric of X , then, for any u ∈ B δ ( V ), p ◦ u : C → X belongs to M ( X ) andthe map Φ : B δ ( V ) ∋ u p ◦ u ∈ M ( X ) becomes injective. (Here we consider u ∈ B δ ( V )as a map from C to F .) We define a distance d ( · , · ) on B δ ( V ) by d ( u, v ) := X n ≥ − n sup | z |≤ n | u ( z ) − v ( z ) | for any u, v ∈ B δ ( V ) . EFORMATION OF BRODY CURVES 17
We consider the topology defined by this distance on B δ ( V ). Then B δ ( V ) becomes com-pact, and Φ : B δ ( V ) → M ( X ) becomes a Λ-equivariant continuous embedding (here weconsider the compact-open topology on M ( X )). Hence dim( M ( X ) : Λ) ≥ dim( B δ ( V ) :Λ). Let π n : C /n Λ → C / Λ be the natural n -fold covering. Then the argument in Section3 shows dim( B δ ( V ) : Λ) ≥ lim n →∞ n dim H ( C /n Λ : O ( π ∗ n F )) = 2 deg( F ) > . (This is an inequality of the type “mean-dimension ≥ residual-dimension”.) Thereforedim( M ( X ) : C ) = dim( M ( X ) : Λ) / | C / Λ | > . Remark 6.3.
The above X does not admit a K¨ahler metric. In fact the space of holo-morphic one-forms in X is (complex) one-dimensional. Since the first Betti number of X = T is 6, the Hodge theory implies that there is no K¨ahler metric on X . Appendix A. Proof of Proposition 2.1
Gromov [5, p. 333] proved Proposition 2.1 by using the notion “filling radius”. (Fillingradius is a notion introduced in his celebrated paper [4].) Our following proof is a variantof the argument of Lindenstrauss-Weiss [8, Lemma 3.2].It is enough to prove that for the unit ball B := { x ∈ V | || x || ≤ } we haveWidim ε ( B, d ) = dim V = n for ε < . Suppose there exists ε < ε ( B, d ) ≤ n −
1. Then there is a finite opencovering { U i } i ∈ I of B such that δ := max i ∈ I Diam U i < ≤ n −
1, i.e., U i ∩ U i ∩ · · · ∩ U i n +1 = ∅ for distinct i , i , · · · , i n +1 ∈ I .Let { φ i } i ∈ I be a partition of unity on B satisfying supp φ i ⊂ U i . Take an arbitrarypoint p i in U i . We define a map f : B → B by f ( x ) := − P i ∈ I φ i ( x ) · p i . For any x ∈ B we have(23) || f ( x ) + x || = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i ∈ I φ i ( x )( x − p i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ δ X i ∈ I φ i = δ. For each x ∈ B we have ♯ { i ∈ I | φ i ( x ) = 0 } ≤ n . Therefore f ( B ) is contained in a unionof at most n − f ( B ) does not contain an innerpoint. Hence there exists a ∈ B such that a / ∈ f ( B ) and || a || ≤ − δ . Then we can define g : B → ∂B by g ( x ) := ( f ( x ) − a ) / || f ( x ) − a || . g does not have a fixed point. In fact if g ( x ) = x , then x ∈ ∂B and f ( x ) − a = x || f ( x ) − a || . Then f ( x )+ x − a = x (1+ || f ( x ) − a || )and || f ( x ) + x − a || = 1 + || f ( x ) − a || >
1. From (23),1 < || f ( x ) + x − a || ≤ || f ( x ) + x || + || a || ≤ δ + || a || ≤ . This is a contradiction. Therefore g does not have a fixed point, and this contradicts theBrouwer fixed-point theorem. References [1] M. Bonk, A. Eremenko, Covering properties of meromorphic functions, negative curvature and spher-ical geometry, Ann. of Math. (2000) 551-592[2] R. Brody, Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc. (1978) 213-219[3] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Reprint of the1998 edition, Classics in Mathematics, Springer-Verlag, Berlin (2001)[4] M. Gromov, Filling Riemannian manifolds, J. Differential Geom. (1983) 1-147[5] M. Gromov, Topological invariants of dynamical systems and spaces of holomorphic maps: I, Math.Phys. Anal. Geom. (1999) 323-415[6] J. Koll´ar, S. Mori, Birational geometry of algebraic varieties, With the collaboration of C.H. Clemensand A. Corti, Cambridge Tracts in Mathematics, 134, Cambridge University Press, Cambridge (1998)[7] E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes´Etudes Sci. Publ. Math. (1999) 227-262[8] E. Lindenstrauss, B. Weiss, Mean topological dimension, Israel J. Math. (2000) 1-24[9] D. McDuff, D. Salamon, J -holomorphic curves and quantum cohomology, University Lecture Series,6, American Mathematical Society, Providence (1994)[10] M. Tsukamoto, A packing problem for holomorphic curves, preprint, arXiv: math.CV/0605353[11] M. Tsukamoto, Mean dimension of the unit ball in ℓ p Masaki TsukamotoDepartment of Mathematics, Faculty of ScienceKyoto UniversityKyoto 606-8502Japan
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