A Riemannian Bieberbach estimate
aa r X i v : . [ m a t h . DG ] M a y A Riemannian Bieberbach estimate
F. Fontenele ∗ and F. Xavier Abstract . The Bieberbach estimate, a pivotal result in the classical theory ofunivalent functions, states that any injective holomorphic function f on the openunit disc D satisfies | f ′′ (0) | ≤ | f ′ (0) | . We generalize the Bieberbach estimate byproving a version of the inequality that applies to all injective smooth conformalimmersions f : D → R n , n ≥
2. The new estimate involves two correction terms.The first one is geometric, coming from the second fundamental form of the imagesurface f ( D ). The second term is of a dynamical nature, and involves certainRiemannian quantities associated to conformal attractors. Our results are partlymotivated by a conjecture in the theory of embedded minimal surfaces. A conformal orientation-preserving local diffeomorphism that is defined in the openunit disc D = { z ∈ C : | z | < } and takes values into R can be viewed as a holomorphicfunction f : D → C . Of special interest is the case when f is univalent, that is, injective.The class S of all holomorphic univalent functions in D satisfying f (0) = 0 and f ′ (0) = 1was the object of much study in the last century, culminating with the solution by deBranges ([4], [22]) of the celebrated Bieberbach conjecture: for any f ∈ S , the estimate | f ( k ) (0) | ≤ kk ! (1.1)holds for all k ≥
2. Equivalently, | f ( k ) (0) | ≤ kk ! | f ′ (0) | for any injective holomorphicfunction on D . The case k = 2, due to Bieberbach, yields the so-called distortion theoremswhich, in turn, imply the compactness of the class S ([19], [22]). Thus, the basic estimate | f ′′ (0) | ≤ f ∈ S , most commonly written in the form | a | ≤ f ( z ) = z + a z + · · · , already yields important qualitative information. In particular, it followsfrom the compactness of S that there are constants C k such that | f ( k ) (0) | ≤ C k | f ′ (0) | forevery k ≥ f on D . The Bieberbach conjecture (thede Branges theorem) asserts that one can take C k to be kk !.The aim of this paper is to establish a generalization of Bieberbach’s fundamentalestimate | f ′′ (0) | ≤ | f ′ (0) | that applies to all injective smooth conformal immersions f : D → R n , n ≥
2. The new estimate involves two correction terms. The first one is geo-metric, coming from the second fundamental form of the image surface f ( D ). The second ∗ Work partially supported by CNPq (Brazil) R n , we denote by ∇ and σ its connection and second fun-damental form, both extended by complex linearity. Recall also that a vector field ona Riemannian manifold is said to be conformal if it generates a local flow of conformalmaps. We use subscripts to denote differentiation with respect to z , where 2 ∂ z = ∂ x − i∂ y . Theorem 1.1.
Let f : ( D, → ( N, p ) be an injective smooth conformal immersion,where N ⊂ R n is a smooth embedded topological disc, n ≥ . Let X be the (non-empty)family of all normalized conformal attractors on ( N, p ) , namely those smooth vector fields X on N satisfyingi) X is conformal, X ( p ) = 0 , ( ∇ X ) p = − I .ii) Every positive orbit of X tends to p .Then sup X ∈ X (cid:13)(cid:13) f zz (0) − σ (cid:0) f z (0) , f z (0) (cid:1) + ( ∇ X ) p (cid:0) f z (0) , f z (0) (cid:1)(cid:13)(cid:13) ≤ k f z (0) k . (1.2)We expect to develop further the ideas in this paper and establish analogues of thehigher order estimates | f ( k ) (0) | ≤ kk ! | f ′ (0) | , k ≥
3, in the broader context of injectiveconformal immersions f : D → R n . One would then have, in all dimensions, a geometric-conformal version of the de Branges theorem. Example.
Taking N = R ⊂ R n as a totally geodesic submanifold and X ( w ) = − ( w − p ),we have ∇ X = − I and so ∇ X = 0. Since σ = 0, one sees that the original Bieberbachestimate | f ′′ (0) | ≤ | f ′ (0) | can be recovered from (1.2).It follows from classical results that for any injective smooth immersion g : D → R n there is a diffeomorphism h : Ω → D , where Ω is either D or C , such that f = g ◦ h is aninjective conformal immersion. Restricting h to D in the case Ω = C , one then sees thatthe above theorem applies to every injective immersion, after a suitable reparametrization.We observe that the original Bieberbach estimate does not carry over to the higherdimensional case. Indeed, if every injective conformal immersion f : D → R n were tosatisfy k f zz (0) k ≤ k f z (0) k , a contradiction could be reached as follows. Let g : C → R n be an injective conformal harmonic immersion which is not totally geodesic, i.e., g ( C ) isa parabolic simply-connected embedded minimal surface. A concrete example is providedby the helicoid given in coordinates z = x + iy by g ( x, y ) = (sinh x cos y, sinh x sin y, y ) (seethe Conjecture below). Applying the above estimate to f : D → R n , f ( z ) = g ( Rz ), andletting R → ∞ , one obtains g zz (0) = 14 (cid:16) g xx (0) − ig xy (0) − g yy (0) (cid:17) = 12 (cid:16) g xx (0) − ig xy (0) (cid:17) = 0 . Replacing g ( z ) by g ( z + z o ) in the above reasoning, we obtain g zz ≡
0, and so g xx , g yy and g xy are also identically zero, contradicting the fact that g ( C ) is not a plane.2t is tempting to believe that the contribution in (1.2) coming from the dynamic termcan be made to vanish, regardless of the embedding, in which case one would obtain anestimate that depends solely on the geometry of the surface f ( D ). To investigate thispossibility, suppose that inf X ∈ X (cid:13)(cid:13) ( ∇ X ) p (cid:0) f z (0) , f z (0) (cid:1)(cid:13)(cid:13) = 0for all conformal embeddings f : D → R n of the unit disc. Consider the map g above,parametrizing a helicoid. After applying (1.2) to f : D → R , f ( z ) = g ( Rz ), and reason-ing as above, one has g zz − σ (cid:0) g z , g z (cid:1) = 12 (cid:16) g Txx − ig Txy (cid:17) ≡ . In particular, the coordinates curves of g are geodesics of g ( D ). Since these curves areeasily seen to be asymptotic lines of g ( D ), one would conclude that the traces of thecoordinates curves of g are (segments of) straight lines in R . But this contradicts thefact that the coordinates curves y g ( x, y ) are helices. Hence, the dynamic term isessential for the validity of (1.2).As the reader may have suspected by now, part of our motivation for proving a versionof the Bieberbach estimate in the realm of conformal embeddings D → R n comes fromthe theory of minimal surfaces. We explain below how these two sets of ideas merge.The past few years have witnessed great advances in the study of simply-connectedembedded minimal surfaces (i.e., minimal surfaces without self-intersetions). From an an-alytic standpoint, these geometric objects correspond to conformal harmonic embeddingsof either D or C into R . In a series of groundbreaking papers, Colding and Minicozzi([5]-[9]) were able to give a very detailed description of the structure of embedded mini-mal discs. Using their theory, as well as other tools, Meeks and Rosenberg showed in alandmark paper [14] that helicoids and planes are the only properly embedded simply-connected minimal surfaces in R (subsequently, properness was weakened to mere com-pleteness).The classical link between minimal surfaces and complex analysis has been explored,with great success, to tackle other fundamental geometric problems. Given the historyof the subject, one is naturally inclined to look for a complex-analytic interpretation ofthe works of Colding-Minicozzi and Meeks-Rosenberg, with the hope that more could berevealed about the structure of embedded minimal discs. Although this effort is still inits infancy, one can already delineate the contours of a general programme. A centraltheme to be explored is the role of the conformal type in the embeddedness question forminimal surfaces. In particular, one would like to know, in the Meeks-Rosenberg theorem,if parabolicity alone suffices: Conjecture. If g : C → R is a conformal harmonic embedding, then g ( C ) is either aflat plane or a helicoid.There is a compelling analogy between the theory of conformal harmonic embeddings of the open unit disc D ⊂ C into R , and the very rich theory of holomorphic univalent D . It is an easy matter to use a scaling argument, as it was done above,together with the (classical) Bieberbach estimate to establish the scarcity of univalententire functions: they are all of the form f ( z ) = az + b , a = 0. One ought to regard thisstatement as the complex-analytic analogue of the above conjecture.A major hurdle in trying to use similar scaling arguments to settle the above conjectureis that one does not have an a priori control on the dynamic term in (1.2). Nevertheless, afirst step in the programme of using complex analysis towards studying embedded minimaldiscs has been taken in [12], where a new proof was given of the classical theorem ofCatalan, characterizing (pieces of) planes and helicoids as the only ruled minimal surfaces.The proof of Catalan’s theorem is reduced, after careful normalizations, to the uniquenessof solutions of certain holomorphic differential equations.The present work is in fact part of a much larger programme, going well beyondminimal surface theory, whose aim is to identify the analytic, geometric and topologicalmechanisms behind the phenomenon of global injectivity; see, for instance, [1]-[3], [11],[13], [15]-[18], [21], [23]. The reader can find in [24], p.17, a short description of some ofthe recent results in the area of global injectivity.In closing, we would like to point out that the main result in [23] – a rigidity theoremcharacterizing the identity map of C n among injective local biholomorphisms –, is alsobased on ideas suggested by the original Bieberbach estimate. Acknowledgements.
This paper was written during an extended visit of the first namedauthor to the University of Notre Dame. He would like to record his gratitude to themathematics department for the invitation, as well as for its hospitality.
The proof of Theorem 1.1 will be split into a series of lemmas, reflecting the dynamical,complex-analytic and Riemannian aspects of the argument.
Lemma 2.1.
Let e X be a smooth vector field defined in an open set U ⊂ R n . If e X ( p ) = 0 and ( d e X ) p = − I , then the local flow η t of e X satisfies (i) ( dη t ) p = e − t I, t > . (ii) lim t →∞ k ( dη t ) p k − ( d η t ) p ( v, w ) = ( d e X ) p ( v, w ) , v, w ∈ R n . Proof. (i)
For all x ∈ U and v ∈ R n , ddt ( dη t ) x ( v ) = ddt (cid:26) dds (cid:12)(cid:12)(cid:12) s =0 η t ( x + sv ) (cid:27) = dds (cid:12)(cid:12)(cid:12) s =0 (cid:18) ddt η t ( x + sv ) (cid:19) = dds (cid:12)(cid:12)(cid:12) s =0 e X ◦ η t ( x + sv ) = ( d e X ) η t ( x ) ◦ ( dη t ) x ( v ) . (2.1)4ince e X ( p ) = 0, η t ( p ) = p , for all t. Setting x = p in (2.1), one obtains ddt ( dη t ) p ( v ) = ( d e X ) p ◦ ( dη t ) p ( v ) = − ( dη t ) p ( v ) . (2.2)Since η ( x ) = x for all x , ( dη ) p = I and, by (2.2),( dη t ) p ( v ) = e − t ( dη ) p ( v ) = e − t v, v ∈ R n . (ii) For all x ∈ U and v, w ∈ R n ,( d η t ) x ( v, w ) = dds (cid:12)(cid:12)(cid:12) s =0 (cid:26) ddh (cid:12)(cid:12)(cid:12) h =0 η t ( x + sv + hw ) (cid:27) . (2.3)Thus, ddt ( d η t ) x ( v, w ) = dds (cid:12)(cid:12)(cid:12) s =0 (cid:26) ddh (cid:12)(cid:12)(cid:12) h =0 ddt η t ( x + sv + hw ) (cid:27) = dds (cid:12)(cid:12)(cid:12) s =0 (cid:26) ddh (cid:12)(cid:12)(cid:12) h =0 e X ◦ η t ( x + sv + hw ) (cid:27) = d ( e X ◦ η t ) x ( v, w ) . (2.4)It follows that ddt ( d η t ) x ( v, w ) = ( d e X ) η t ( x ) (cid:0) ( dη t ) x v, ( dη t ) x w (cid:1) + ( d e X ) η t ( x ) (cid:0) ( d η t ) x ( v, w ) (cid:1) . (2.5)Taking x = p in (2.5) and recalling that ( d e X ) p = − I , we have ddt ( d η t ) p ( v, w ) = ( d e X ) p (cid:0) ( dη t ) p v, ( dη t ) p w (cid:1) + ( d e X ) p (cid:0) ( d η t ) p ( v, w ) (cid:1) = ( d e X ) p (cid:0) ( dη t ) p v, ( dη t ) p w (cid:1) − ( d η t ) p ( v, w ) . (2.6)From (i) and (2.6), one has ddt ( d η t ) p ( v, w ) = e − t ( d ˜ X ) p ( v, w ) − ( d η t ) p ( v, w ) . (2.7)Letting V ( t ) = ( d η t ) p ( v, w ) and W = ( d ˜ X ) p ( v, w ), the last equation becomes ddt V ( t ) = − V ( t ) + e − t W, (2.8)so that ddt ( e t V ( t )) = e − t W, (2.9)5nd, by integration, e t V ( t ) − V (0) = W (1 − e − t ) . (2.10)Since η ( x ) = x, x ∈ U , it follows that V (0) = ( d η ) p ( v, w ) = 0. Using this in (2.10), wehave e t V ( t ) = W (1 − e − t ) . (2.11)Taking the limit as t → ∞ and using (i) one obtains (ii). Lemma 2.2.
Let f : D → R n and g : D → R n be two conformal embeddings of theunit disc into R n such that g ( D ) ⊂ f ( D ) and f (0) = g (0) . Assume that the orientationsinduced on g ( D ) by f and g are the same. Then there exists ζ ∈ C , | ζ | = 1 , such that (cid:13)(cid:13)(cid:13)(cid:13) g zz (0) k g z (0) k − ζ f zz (0) k f z (0) k (cid:13)(cid:13)(cid:13)(cid:13) ≤ k g z (0) k . (2.12) Proof.
The function ϕ = f − ◦ g : D → D is well defined, conformal and orientationpreserving. Hence ϕ is holomorphic, ϕ (0) = 0. Using g z (0) = f z (0) ϕ z (0) and g zz (0) = f zz (0) ϕ z (0) + f z (0) ϕ zz (0), one computes g zz (0) k g z (0) k = f zz (0) k f z (0) k ϕ z (0) | ϕ z (0) | + f z (0) ϕ zz (0) k f z (0) k | ϕ z (0) | . (2.13)But (cid:13)(cid:13)(cid:13)(cid:13) f z (0) ϕ zz (0) k f z (0) k | ϕ z (0) | (cid:13)(cid:13)(cid:13)(cid:13) ≤ k f z (0) k| ϕ z (0) | = 4 k g z (0) k , (2.14)by Bieberbach’s inequality | ψ ′′ (0) | ≤ | ψ ′ (0) | , valid for all univalent holomorphic functions ψ : D → C . Hence (cid:13)(cid:13)(cid:13)(cid:13) g zz (0) k g z (0) k − ϕ z (0) | ϕ z (0) | f zz (0) k f z (0) k (cid:13)(cid:13)(cid:13)(cid:13) ≤ k g z (0) k (2.15)and the lemma follows from (2.15) with ζ = ϕ z (0) | ϕ z (0) | .6 emma 2.3. Let N ⊂ R n , n ≥ , be a smooth surface and f : ( D, → ( N, p ) aninjective smooth conformal immersion. Let U be an open neighborhood of N in R n and e X a smooth vector field on U such that: (i) e X ( p ) = 0 , ( d e X ) p = − I . (ii) e X is tangent to N , e X | N is conformal and every positive orbit of e X | N converges to p .Then (cid:13)(cid:13) ( d e X ) p (cid:0) f z (0) , f z (0) (cid:1) + f zz (0) (cid:13)(cid:13) ≤ k f z (0) k . (2.16) Proof.
We may assume that f ( D ) has compact closure in N (the general case followsby replacing f ( z ) by f ( Rz ), with R <
1, and letting R → e X | N converges to p , there exists t > η t of e X satisfies η t ( f ( D )) ⊂ f ( D ) for all t ≥ t . Since e X | N is conformal, so is g ( t ) = g = η t ◦ f . It followseasily from Lemma 2.1 (i) that f and g induce the same orientations on g ( D ) ⊂ f ( D ), asrequired by Lemma 2.2. We have g z ( z ) = ( η t ◦ f ) z ( z ) = ( dη t ) f ( z ) ( f z ( z )) (2.17)and g zz (0) = ( d η t ) p ( f z (0) , f z (0)) + ( dη t ) p ( f zz (0)) . (2.18)By (2.12), k g zz (0) kk g z (0) k ≤ k g z (0) k + k f zz (0) kk f z (0) k . (2.19)From (2.17), (2.18) and (2.19), we then have k ( d η t ) p ( f z (0) , f z (0)) + ( dη t ) p ( f zz (0)) kk ( dη t ) p k k f z (0) k ≤ k ( dη t ) p k k f z (0) k + k f zz (0) kk f z (0) k (2.20)Multiplying the last equation by k ( dη t ) p kk f z (0) k , k ( d η t ) p ( f z (0) , f z (0)) + ( dη t ) p ( f zz (0)) kk ( dη t ) p k ≤ k f z (0) k + k ( dη t ) p k k f zz (0) k . (2.21)Taking the limit as t → ∞ and using Lemma 2.1, one obtains (2.16).In the next lemma we denote by ∇ and ∇ the Riemannian connections of R n and M ,respectively, and by σ the second fundamental form of M . Lemma 2.4.
Let M ⊂ R n be an m -dimensional submanifold and X a smooth vector fieldon M such that X ( p ) = 0 and ∇ v X = − v , for some p ∈ M and all v ∈ T p M . Then, forany extension e X of X to an open neighborhood of p in R n and all v, w ∈ T p M , one has ( d e X ) p ( v, w ) = ( ∇ X ) p ( v, w ) − ∇ σ ( v,w ) e X − σ ( v, w ) . (2.22)7 roof. Writing u , . . . , u n for the canonical basis of R n one has, for all v, w ∈ R n ,( d e X ) p ( v, w ) = n X i,j =1 ∂ e X∂x i ∂x j ( p ) h v, u i ih w, u j i , (2.23)so that the coordinates of this vector are given by n X i,j =1 ∂ e X k ∂x i ∂x j ( p ) h v, u i ih w, u j i = n X i,j =1 h v, u i ih w, u j i hess e X k ( p )( u i , u j )= hess e X k ( p )( v, w ) , k = 1 , . . . , n. (2.24)Choose an orthonormal frame field { e , . . . , e n } in an open neighborhood of p in such away that e , . . . , e m are tangent along M , e m +1 , . . . , e n are normal along M and ∇ e i e j ( p ) =0 , i, j = 1 , . . . , m . Hence, by (2.24), n X i,j =1 ∂ e X k ∂x i ∂x j ( p ) h v, u i ih w, u j i = n X i,j =1 v i w j hess e X k ( p )( e i , e j )= n X i,j =1 h∇ e i grad e X k ( p ) , e j ( p ) i v i w j , (2.25)where the v ′ i s and w ′ i s are the components of v and w in the basis { e , . . . , e n } . It followsthat n X i,j =1 ∂ e X k ∂x i ∂x j ( p ) h v, u i ih w, u j i = n X i,j =1 v i w j (cid:0) e i h grad e X k , e j i − h grad e X k , ∇ e i e j i (cid:1) ( p )= n X i,j =1 v i w j (cid:0) e i e j ( e X k ) − ∇ e i e j ( e X k ) (cid:1) ( p ) . (2.26)From (2.23) and (2.26) we obtain,( d e X ) p ( v, w ) = n X i,j =1 (cid:0) ∇ e i ∇ e j e X (cid:1) ( p ) v i w j − n X i,j =1 (cid:0) ∇ ∇ ei e j e X (cid:1) ( p ) v i w j . (2.27)Suppose now that v and w are tangent to M . Since the restriction to M of { e , . . . , e n } is an adapted frame satisfying ∇ e i e j ( p ) = 0 , i, j = 1 , . . . , m , and e X restricted to M is X ,it follows from (2.27) and the Gauss equation [10] ∇ Y X = ∇ Y X + σ ( X, Y ) (2.28)that ( d e X ) p ( v, w ) = m X i,j =1 (cid:0) ∇ e i ∇ e j X (cid:1) ( p ) v i w j − m X i,j =1 (cid:0) ∇ σ ( e i ,e j ) e X (cid:1) v i w j = m X i,j =1 (cid:0) ∇ e i ∇ e j X (cid:1) ( p ) v i w j − ∇ σ ( v,w ) e X. (2.29)8e will now determine the first term on the right hand side of (2.29). From ∇ e j X = ∇ e j X + σ ( e j , X ) , we have at p , for 1 ≤ i, j ≤ m , ∇ e i ∇ e j X = ∇ e i ∇ e j X + ∇ e i σ ( e j , X ) = ∇ e i ∇ e j X + σ ( e i , ∇ e j X ) + ∇ e i σ ( e j , X ) . Hence m X i,j =1 ( ∇ e i ∇ e j X )( p ) v i w j = m X i,j =1 ( ∇ e i ∇ e j X )( p ) v i w j + m X i,j =1 σ ( e i , ∇ e j X )( p ) v i w j + m X i,j =1 (cid:0) ∇ e i σ ( e j , X ) (cid:1) ( p ) v i w j . (2.30)The third term on the right hand side of (2.30) involves the quantity ∇ e i σ ( e j , X ) whichcan be computed using the normal connection ∇ ⊥ : ∇ e i σ ( e j , X ) = ∇ ⊥ e i σ ( e j , X ) + ( ∇ e i σ ( e j , X )) T = ( ∇ ⊥ e i σ )( e j , X ) + σ ( e j , ∇ e i X ) + m X k =1 (cid:10) ( ∇ e i σ ( e j , X )) T , e k (cid:11) e k = ( ∇ ⊥ e i σ )( e j , X ) + σ ( e j , ∇ e i X ) + m X k =1 (cid:10) ∇ e i σ ( e j , X ) , e k (cid:11) e k = ( ∇ ⊥ e i σ )( e j , X ) + σ ( e j , ∇ e i X ) − m X k =1 (cid:10) σ ( e j , X ) , ∇ e i e k (cid:11) e k = ( ∇ ⊥ e i σ )( e j , X ) + σ ( e j , ∇ e i X ) − m X k =1 (cid:10) σ ( e j , X ) , σ ( e i , e k ) (cid:11) e k . (2.31)From (2.30) and (2.31), m X i,j =1 ( ∇ e i ∇ e j X ) v i w j = m X i,j =1 ( ∇ e i ∇ e j X ) v i w j + σ ( v, ∇ w X ) + m X i,j =1 ( ∇ ⊥ e i σ )( e j , X ) v i w j + m X i,j =1 σ ( e j , ∇ e i X ) v i w j − m X i,j,k =1 (cid:16)(cid:10) σ ( e j , X ) , σ ( e i , e k ) (cid:11) e k (cid:17) v i w j = m X i,j =1 ( ∇ e i ∇ e j X ) v i w j + σ ( v, ∇ w X ) + ( ∇ ⊥ v σ )( w, X )+ σ ( w, ∇ v X ) − m X k =1 h σ ( w, X ) , σ ( v, e k ) i e k . (2.32)9ince, by assumption, X ( p ) = 0 and ( ∇ X ) p = − I , we have m X i,j =1 ( ∇ e i ∇ e j X )( p ) v i w j = m X i,j =1 ( ∇ e i ∇ e j X )( p ) v i w j − σ ( v, w ) . (2.33)It follows from (2.29) and (2.33) that( d e X ) p ( v, w ) = m X j =1 ( ∇ v ∇ e j X )( p ) w j − σ ( v, w ) − ∇ σ ( v,w ) e X. (2.34)Let V and W be arbitrary smooth extensions of v and w to an open neighborhood of p in M . Using again X ( p ) = 0 and ( ∇ X ) p = − I and recalling that ∇ e j e i ( p ) = 0, we have,from the definition of the curvature tensor R , m X j =1 ( ∇ v ∇ e j X )( p ) w j = m X j =1 (cid:0) R ( V, e j ) X + ∇ e j ∇ V X + ∇ [ V,e j ] X (cid:1) ( p ) w j = ∇ W ∇ V X ( p ) + ∇ W V ( p ) . (2.35)From (2.34) and (2.35) we obtain( d e X ) p ( v, w ) = ∇ W ∇ V X ( p ) + ∇ W V ( p ) − ∇ σ ( v,w ) e X − σ ( v, w ) . (2.36)On the other hand,( ∇ X ) p ( v, w ) = (cid:0) ∇ w ∇ X (cid:1) ( v ) = (cid:0) ∇ W ∇ X ( V ) (cid:1) ( p ) − ∇ X (cid:0) ∇ W V (cid:1) ( p )= ∇ W ∇ V X ( p ) − ∇ ∇ W V X ( p ) . (2.37)Since ( ∇ X ) p = − I , we thus obtain( ∇ X ) p ( v, w ) = ∇ W ∇ V X ( p ) + ∇ W V ( p ) . (2.38)Formula (2.22) now follows from (2.34), (2.35) and (2.38). Proof of Theorem 1.1.
We begin by showing that X = ∅ . Being simply-connected, theRiemannian surface N is globally conformally flat and so it is isometric to (Ω , e g ), where Ωis either D or C , e g = e ϕ g for some smooth function ϕ , and g is the standard flat metricon Ω. Writing e ∇ and ∇ for the Riemannian connections of e g and g , respectively, one has([20] p. 172): e ∇ Y X = ∇ Y X + Y ( ϕ ) X + X ( ϕ ) Y − h X, Y i∇ ϕ. (2.39)Being holomorphic, X ( z ) = − z generates a flow of conformal maps that clearly leaves Ωinvariant. It now follows from (2.39) that e ∇ Y X = ∇ Y X = − Y at 0, thus showing that X ∈ X . 10e now proceed to establish the estimate (1.2). Since N is contractible, one can choosea global orthonormal frame { ξ , . . . , ξ n − } of the normal bundle of N in R n . Extend X to an open neighborhood of N in R n by e X q + n − X i =1 s i ξ i ( q ) ! = X ( q ) − n − X i =1 s i ξ i ( q ) , | s i | < ǫ ( q ) , (2.40)with ǫ ( q ) sufficiently small, q ∈ N . For all v ∈ T p N we have( d e X ) p v = ∇ v e X = ∇ v X = ∇ v X + σ ( v, X ( p )) = − v, since X ( p ) = 0 and ( ∇ X ) p = − I . On the other hand, ∇ ξ i ( p ) e X = ( d e X ) p ξ i ( p ) = dds (cid:12)(cid:12)(cid:12) s =0 e X (cid:0) p + sξ i ( p ) (cid:1) = dds (cid:12)(cid:12)(cid:12) s =0 (cid:0) X ( p ) − sξ i ( p ) (cid:1) = − ξ i ( p ) . It follows from the last two equations that ( ∇ e X ) p = ( d e X ) p = − I . By Lemma 2.3, wehave (cid:13)(cid:13) ( d e X ) p (cid:0) f z (0) , f z (0) (cid:1) + f zz (0) (cid:13)(cid:13) ≤ k f z (0) k . (2.41)Also, from Lemma 2.4 and ( ∇ e X ) p = − I ,( d e X ) p ( v, w ) = ( ∇ X ) p ( v, w ) − ∇ σ ( v,w ) e X − σ ( v, w ) , = ( ∇ X ) p ( v, w ) − σ ( v, w ) , v, w ∈ T p N, (2.42)which, in particular, implies that ( ∇ X ) p is symmetric. From f z (0) = (cid:0) f x (0) − if y (0) (cid:1) and ( d e X ) p (cid:0) f z (0) , f z (0) (cid:1) = 14 ( d e X ) p (cid:0) f x (0) − if y (0) , f x (0) − if y (0) (cid:1) = 14 n ( d e X ) p (cid:0) f x (0) , f x (0) (cid:1) − ( d e X ) p (cid:0) f y (0) , f y (0) (cid:1) − i ( d e X ) p (cid:0) f x (0) , f y (0) (cid:1)o , (2.43)we obtain, applying (2.42) to each term in (2.43) and rearranging the terms,( d e X ) p (cid:0) f z (0) , f z (0) (cid:1) = ( ∇ X ) p (cid:0) f z (0) , f z (0) (cid:1) − σ (cid:0) f z (0) , f z (0) (cid:1) . (2.44)Formula (1.2) follows from (2.41) and (2.44).11 eferences [1] E. Andersen and L. Lempert, On the group of automorphisms of C n , Invent. Math., (1992) 371-388.[2] E. C. Balreira, Foliations and global inversion , to appear in Coment. Math. Helv.[3] H. Bass, E. Connell and D. Wright,
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