A Cartan-type result for invariant distances and one-dimensional holomorphic retracts

Publ. Mat. 45 (2001), 387–397 A CARTAN-TYPE RESULT FOR INVARIANT DISTANCES AND ONE-DIMENSIONAL HOLOMORPHIC RETRACTS Colum Watt Abstract We derive conditions under which a holomorphic mapping of a taut Riemann surface must be an automorphism. This is an analogue involving invariant distances of a result of H. Cartan. Using similar methods we prove an existence result for 1-dimensional holomorphic retracts in a taut complex manifold. 1. Introduction In what follows, Wdenotes a connected Riemann surface, Hol(W, W ) denotes the set of holomorphic self-maps of Wand Aut(W) denotes the set of biholomorphic maps of Wonto itself. A complex manifold M is taut if and only if for each complex manifold N, each sequence of holomorphic mappings from Nto Mcontains a subsequence which either converges uniformly on compact subsets of Nor is uniformly divergent to infinity (in the one point compactification of M) on compact subsets of N. Definition 1.1. We call a distance function don a Riemann surface W invariant if d(f(w),f(z)) ≤d(w, z)∀w, z ∈W, ∀f∈Hol(W, W ). A Hermitian metric hon Wis called invariant provided h(f∗(u),f ∗(u)) ≤h(u, u)∀u∈O wW, ∀w∈W, ∀f∈Hol(W, W ). We say that a distance function don Wis Ck(k≥1) if dis the integrated distance function associated to a Ck−1Hermitian metric hon W. 2000 Mathematics Subject Classification. 32F45, 32Q40, 53C60. Key words. Taut complex manifold, invariant distance, automorphism, holomorphic retract. 388 C. Watt Remark 1.A standard example of an invariant metric is the square of the Kobayashi metric on a taut Riemann surface. For the unit disc in the complex plane, this metric is usually referred to as the Poincar´e metric. In his paper [4], J. P. Vigu´e proved the following result. Theorem. Let Xand Wbe connected, taut complex manifolds with W 1-dimensional. Assume that his an invariant Hermitian metric on W and choose w∈W,x∈Xand v∈O xX\{0}. Then there exists a holomorphic retraction ρ:X→Xsuch that ρ(w)=x,ρ∗(OwW)=Cv and ρ(X)is biholomorphic to Wif and only if E(x, v)=F(x, v)(where E,F :OX→R+are invariant Finsler metrics which are defined in terms of w,W,h,xand Xand which generalise the usual Carath´eodory and Kobayashi metrics). Key to his proof of this is the following result of H. Cartan [1]. Cartan’s Theorem. Let Wbe a taut Riemann surface. If f∈Hol(W,W ) fixes some point w∈Wand has unimodular derivative at wthen f∈ Aut(W). Note that the hypothesis of Cartan’s result is equivalent to the requirement that ffixes wand that its derivative at wis unitary with respect to every (in particular every invariant) Hermitian metric on W.Ina remark in [4], J. P. Vigu´e defined the invariant pseudodistances cW,w X,x and kW,w X,x in terms of an invariant distance on W(see Section 3 below). He would have liked to use these to investigate the existence of 1-dimensional holomorphic retracts through two given points of X.To do so, Vigu´e would have needed (but did not possess) an analogue of Cartan’s result which uses an invariant distance in place of an invariant Hermitian metric. In Section 2 we prove such an analogue (Theorem 2.3) of Cartan’s theorem under the assumption that the invariant distance arises from a continuous Hermitian metric on W. Then in the final section we use the invariant pseudodistances cW,w X,x and kW,w X,x (which generalise the usual Carath´eodory distance and Kobayashi function on a complex manifold) and apply Theorem 2.3 to investigate the existence of holomorphic retractions of a complex manifold onto a 1-dimensional submanifold through two given points. 2. Automorphisms of Riemann surfaces First we recall some standard notions from differential geometry. Let hbe a continuous Hermitian metric on a connected Riemann surface W. Thus hdetermines a sesquilinear, positive definite inner product hw Invariant Distances and Holomorphic Retracts 389 on each tangent space OwWand hwvaries continuously with w. The associated norm on OwWis denoted |·| w(to simplify notation, the subscript wis often omitted). If f∈Hol(W, W ) we denote its derivative at wby f∗w:OwW→O f(w)W. The operator norm of f∗w(with respect to |·| wand |·| f(w)) is denoted ||f∗w|| (or by ||f∗|| when wis clear from the context). As OwWis onedimensional it follows that |f∗w(u)|f(w)=||f∗w||·|u|w∀u∈O wW. Continuity of himplies that the map w→ ||f∗w|| is continuous. A piecewise C1path in Wis a mapping γ:[a, b]→Wfor which there exists a finite set of points a=t0<t 1<···<t n=bsuch that γ|[ti,ti+1] is C1and has nowhere vanishing tangent for each i=0,... ,n−1. The length of such a path γis defined by l(γ)=b a h(γ(t),γ(t))1 2dt =b a |γ(t)|dt. For any two points wand zin Wthe distance d(w, z) is defined by d(w,z) = inf{l(γ):γis a piecewise C1path from wto z}. This function is clearly symmetric, positive and satisfies the triangle inequality. It is a standard result that d(w,z)>0 when w=zand that dgenerates the given topology on W(for example, see [2]). The open ball B(w, r)⊂Wwith centre wand radius r>0 is given by B(w,r)={z∈W:d(w, z)<r}. If dis the distance arising from a hermitian metric h, it is easy to show that invariance of himplies invariance of d. In this first proposition we prove a converse result. Proposition 2.1. Let dbe the integrated distance associated to a continuous Hermitian metric hon a Riemann surface W.Ifdis invariant then hmust also be invariant. Proof: Assume that there exists f∈Hol(W, W ) such that ||f∗w|| >1 for some w∈W. We will show that dcannot be invariant. As ||f∗|| =0atw,fmaps some neighbourhood Uof Wbiholomorphically onto an open neighbourhood of f(w). Shrinking Uif necessary and using continuity, we may assume that ||f∗|| ≥ 1+ on Ufor some >0. 390 C. Watt Choose δ>0 such that B(f(w),δ)⊂f(U). Let y∈B(f(w),δ) and let γbe a path from f(w)toysuch that d(f(w),y)≤l(γ)<δ. As any path which leaves B(f(w),δ) will have length at least δ, the image of γmust be contained in B(f(w),δ). Thus we may write γ=fσ where σ=(f|U)−1γlies in Uand starts at w. Denote the endpoint (f|U)−1(y) of σby z. Then l(γ)=|γ(t)|dt =|(fσ)(t)|dt =||f∗σ(t)||·|σ(t)|dt ≥(1 + )|σ(t)|dt ≥(1 + )d(w,z). Taking the infimum over paths joining f(w)toy=f(z), it follows that d(f(w),f(z)) ≥(1 + )d(w, z). Hence dcannot be invariant. Proposition 2.2. Let dbe an invariant C1distance on a Riemann surface W.Letf∈Hol(W, W )and assume that there exists a sequence of paths γn:[0,a n]→Wwhich all start at wand for which lim n→∞ l(γn) = lim n→∞ l(fγn)=a>0. Then ||f∗w|| =1. Proof: Invariance of dimplies that ||f∗|| ≤ 1 everywhere. Assume that ||f∗w|| =r<1. As dis C1,w→ ||f∗w|| is continuous and hence there exists >0 such that ||f∗|| <1+r 2on B(w, ). For each ndefine tn∈ (0,a n]by tn=anif γn([0,a n]) ⊂B(w, ) sup{t:γn([0,t]) ⊂B(w, )}otherwise. Invariant Distances and Holomorphic Retracts 391 Then lfγn|[0,tn]=tn 0 |(fγn)(t)|dt =tn 0 ||f∗γn(t)||·|γ n(t)|dt <1+r 2tn 0 |γ n(t)|dt =1+r 2lγn|[0,tn] and hence l(fγn)=lfγn|[0,tn]+lfγn|[tn,an] <1+r 2lγn|[0,tn]+lγn|[tn,an] =l(γn)−1−r 2lγn|[0,tn] ≤l(γn)−1−r 2min( ,l(γn)) ∀n. Thus lim n→∞ l(fγn)≤a−1−r 2min( ,a)<a since a>0. As this contradicts the hypothesis that l(fγn) converges to a, our assumption that ||f∗w|| <1 must have been false. We combine these two propositions to prove the following theorem. Theorem 2.3. Let dbe a C1invariant distance on a taut Riemann surface W. Assume that f∈Hol(W, W )and that there are distinct points wand zin Wsatisfying f(w)=wand d(w,z)=d(w,f(z)). Then f∈Aut(W). Proof: Let γnbe a sequence of paths from wto zwhose lengths converge to d(w, z). Taking the limit as n→∞in the inequality l(γn)≥l(fγn)≥d(f(w),f(z)) = d(w, z) we deduce that lim n→∞ l(γn) = lim n→∞ l(fγn)=d(w,z)>0. 392 C. Watt Proposition 2.2 now implies that ||f∗w|| = 1. Since wis fixed by f and Ow(W) is one dimensional, f∗wmust be given by multiplication by a unimodular complex number. Cartan’s theorem now implies that f∈Aut(W). Corollary 2.4. Let Wbe a taut Riemann surface and suppose f∈ Hol(W, W )fixes two distinct points of W. Then f∈Aut(W). Proof: The map fpreserves the Kobayashi distance between the two fixed points. As the Kobayashi distance is C∞(for any taut Riemann surface) the preceding theorem implies that f∈Aut(W). Corollary 2.5. If f∈Hol(W, W )fixes two distinct points w,z ∈W which can be joined by a unique path γ:[0,a]→Wsatisfying d(w,z)=l(γ)and l(γ|[0,t])=t∀t then fis the identity map. Proof: The path fγ also joins wto z. For any t∈[0,a]wehave d(w,z)=l(γ|[0,t])+l(γ|[t,a]) ≥l(fγ|[0,t])+l(fγ|[t,a]) by invariance ≥d(f(w),f(z)) =d(w,z). It follows that we must have l(γ|[0,t])=l(fγ|[0,t]) for all t. Our uniqueness hypothesis for γnow implies that f(γ(t)) = γ(t) for all t,sof fixes each point on γ([0,a]). The identity theorem for analytic functions implies that fis the identity map. Remark 2.If the distance dis C4then each point w∈Whas a neighbourhood Usuch that any two points of Ucan be joined by a unique path in Uwhich satisfies the hypothesis of γin the preceding corollary (such paths are usually called length minimising geodesics). For a taut Riemann surface W, the Kobayashi distance is C∞and hence any holomorphic map f∈Hol(W, W ) which fixes two sufficiently close points must be the identity mapping. Remark 3.The biholomorphism w→ 1 won the annulus A={w∈C: 1 2<|w|<2}fixes the two points 1 and −1. However there is more than one length minimising geodesic joining these two points in A(with respect to the Kobayashi metric). Invariant Distances and Holomorphic Retracts 393 3. One-dimensional holomorphic retracts In this section, following J. P. Vigu´e[4], we define analogues of the Carath´eodory pseudodistance and the Kobayashi function on a complex manifold. We use these to examine the existence of holomorphic retractions of a complex manifold onto a 1-dimensional complex submanifold through two given points. Let x1,x 2,... ,x nand y1,y 2,... ,y nbe ordered sequences of points in the complex manifolds Xand in Yrespectively. Then Hol(X, x1,... ,x n,Y,y 1,... ,y n) =f∈Hol(X, Y ):f(xi)=yi∀i=1,... ,n . Let Wbe a connected Riemann surface and dan invariant distance on W. Fix a point w∈Wand let Xbe a complex manifold with basepoint x∈X. Then we define a Carath´eodory type function on X×Xwith values in [0,∞]by cW,w X,x (x1,x 2) = supd(f(x1),f(x2)) : f∈Hol(X, x, W, w)∀x1,x 2∈X. The Kobayashi version kW,w X,x (x1,x 2) is defined as follows (i) If Hol(W, w, w1,w 2,X,x,x 1,x 2)=∅for all w1and w2in Wthen kW,w X,x (x1,x 2)=∞. (ii) Otherwise kW,w X,x (x1,x 2) is given by infd(w1,w 2):w1,w 2∈W, f ∈Hol(W, w, w1,w 2,X,x,x 1,x 2). It follows from the invariance of dthat cW,w X,x (x1,x 2)≤kW,w X,x (x1,x 2)∀x1,x 2. For the special case (X, x)=(W, w), the invariance of dalso implies cW,w W,w(w1,w 2)=kW,w W,w (w1,w 2)=d(w1,w 2)∀w1,w 2∈W. As in the cases of the usual Carath´eodory and Kobayashi functions it is straightforward to show that for all f∈Hol(X, x, Y, y) and x1,x 2∈X cW,w X,x (x1,x 2)≥cW,w Y,y (f(x1),f(x2)) and kW,w X,x (x1,x 2)≥kW,w Y,y (f(x1),f(x2)). If we take the usual Kobayashi distance on Was our invariant distance d, then it is easy to see that the resulting function kW,w X,x satisfies kW,w X,x ≤k 394 C. Watt where kdenotes the usual Kobayashi function given by k(x, x1) = inf tanh−1 z−w 1−wz :∃w,z ∈D with Hol(D,w,z,X,x,x 1)=∅ where Ddenotes the unit disc in the complex plane. We now use the functions cW,w X,x and kW,w X,x to give a criterion for deciding when there exists a holomorphic retraction of a complex manifold X onto a submanifold biholomorphic to Wwhich passes through two given points of X. First we recall the definition of a holomorphic retract. Definition 3.1. Aholomorphic retraction of Xis a holomorphic mapping ρ:X→Xsuch that ρ|ρ(X)is the identity map on ρ(X). The set ρ(X) is called a holomorphic retract of X. It is closed and analytic. Proposition 3.2. Let xand x1be distinct points in a complex manifold Xand let dbe an invariant distance on a Riemann surface W. Assume that there exists a holomorphic retraction ρ:X→Xsuch that x, x1∈ρ(X)and ρ(X)is biholomorphic to W. Then there exists some point w∈Wfor which 0<c W,w X,x (x, x1)=kW,w X,x (x, x1)<∞. Proof: Let i:W→Xbe a biholomorphism of Wonto ρ(X). Put w= i−1(x) and w1=i−1(x1). The three inequalities (i) d(w,w1)≥kW,w X,x (i(w),i(w1)) = kW,w X,x (x, x1) (ii) cW,w X,x (x, x1)≥d(i−1ρ(x),i −1ρ(x1)) = d(w,w1) (iii) cW,w X,x (x, x1)≤kW,w X,x (x, x1) combine to give 0<d(w, w1)≤cW,w X,x (x, x1)≤kW,w X,x (x, x1)≤d(w,w1)<∞ since w=w1. The result follows. By strengthening our hypotheses, we can prove the following converse to this proposition. Invariant Distances and Holomorphic Retracts 395 Theorem 3.3. Let xand x1be distinct points in a connected, taut complex manifold X.Letdbe a C1invariant distance on a taut Riemann surface W. If there is some point w∈Wfor which (a) cW,w X,x (x, x1)=kW,w X,x (x, x1)<∞and (b) the open ball B(w,r)has compact closure in W(where r=kW,w X,x (x,x1)), then there exists a holomorphic retraction ρ:X→Xsuch that x, x1∈ ρ(X)and ρ(X)is biholomorphic to W. Proof: Assume that there is a point w∈Wwhich satisfies the hypotheses (a) and (b). By tautness of Xwe can find maps f,f1,f 2,... in Hol(W, w, X, x) and a sequence of points zn∈Wsuch that (i) fn→funiformly on compact sets, (ii) fn(zn)=x1for each n, (iii) lim n→∞ d(w, zn)=kW,w X,x (x, x1). As B(w, r) is compact and Wis locally compact, there exists >0 such that B(w,r + ) is compact. By (iii), there exists Nsuch that zn∈ B(w,r + ) for all n≥N. Compactness of B(w,r + ) implies that zn has a convergent subsequence. Passing to this subsequence if necessary, we may assume that znconverges to z(say). Since dis continuous, we obtain d(w, z) = lim n→∞ d(w,zn)=kW,w X,x (x, x1).(1) As the set {z,z1,z 2,...}is compact, conditions (i) and (ii) imply that f(z) = lim n→∞ fn(zn)=x1. Note that zand ware distinct. Otherwise we would would have x= f(w)=f(z)=x1which contradicts the hypothesis that xand x1are distinct. Next we use the tautness of Wto construct a sequence gn∈ Hol(X, x, W, w) which converges uniformly on compact sets (to gsay) such that lim n→∞ d(gn(x),g n(x1)) = cW,w X,x (x, x1). Let w1=g(x1). As (gn(x),g n(x1)) converges to (g(x),g(x1))=(w, w1) and dis continuous, we obtain d(w,w1)=cW,w X,x (x, x1).(2)