Structure Theorem for Riemannian surfaces with arbitrary curvature
aa r X i v : . [ m a t h . DG ] M a y STRUCTURE THEOREM FOR RIEMANNIAN SURFACES WITH ARBITRARYCURVATURE
ANA PORTILLA (1)(2)(3) , JOSE M. RODRIGUEZ (1)(2)(3)
AND EVA TOURIS (1)(2)(3)
Abstract.
In this paper we prove that any Riemannian surface, with no restriction of curvature at all, canbe decomposed into blocks belonging just to some of these types: generalized Y-pieces, generalized funnelsand halfplanes.
Key words and phrases:
Decomposition of surfaces; arbitrary curvature. Introduction.
The Classification Theorem of compact surfaces says that every orientable compact topological surface ishomeomorphic either to a sphere or to a “torus” of genus g ≥ topological Y-piece . A Y-piececan be visualized as a tubing with the shape of the letter Y. A cylinder is a bordered topological surfacehomeomorphic to SS × [0 , ∞ ), where SS is the unit circle.We refer to the next section for precise definitions and background.The Classification Theorem of compact surfaces says, in other words, that every orientable compacttopological surface except for the sphere and the torus (of genus 1) can be obtained by gluing topologicalY-pieces along their boundaries.In [1], the Classification Theorem is generalized to noncompact surfaces in the following way: Theorem 1.1. ([1, Theorem 1.1])
Every complete orientable topological surface which is homeomorphicneither to the sphere nor to the plane nor to the torus is the union (with pairwise disjoint interiors) oftopological Y-pieces and cylinders.
The following result is the most important in [1] and is a geometric version of the theorem above forcomplete surfaces with constant negative curvature. In this case we have more information about the basicblocks of the surface: the surface can be decomposed in such a way that the boundary of the blocks is theunion of at most three simple closed geodesics. Since the Riemannian structure is more restrictive than thetopological one, an additional piece is necessary in order to achieve the decomposition: the halfplane.We state now this result for Riemannian surfaces.
Theorem 1.2. ([1, Theorem 1.2])
Every complete orientable Riemannian surface with constant curvature K = − , which is not the punctured disk, is the union (with pairwise disjoint interiors) of generalizedY-pieces, funnels and halfplanes. In the applications of Theorem 1.2, it is a crucial fact that the boundaries of the generalized Y-pieces aresimple closed geodesics. There is a clear reason for this: it is very easy to cut and paste surfaces along suchkind of curves.
Date : February 28, 2007.2000 AMS Subject Classification: 41A10, 46E35, 46G10.(1) Research partially supported by a grant from DGI (BFM 2006-11976), Spain.(2) Research partially supported by a grant from DGI (BFM 2006-13000-C03-02), Spain.(3) Research partially supported by a grant from MEC (MTM 2006-26627-E), Spain. (1)(2)(3) , JOSE M. RODRIGUEZ (1)(2)(3)
AND EVA TOURIS (1)(2)(3)
Furthermore, closed geodesics in a Riemannian surface S are geometrical objects interesting by themselves.Since they are the periodic orbits of the dynamical system associated to S on its unit tangent bundle, theyprovide tools to study the geodesic flow, just like the fixed points of an automorphism helps to study it.Lastly, the closed geodesics are becoming more and more important in the study of heat and wave equations,and of the spectrum of S . The lengths of all closed geodesics determine largely the spectrum. Conversely,the spectrum determines completely the lengths of the closed geodesics (see [4], [8], [5]).In this paper we prove the conclusion of Theorem 1.2 with no restriction of curvature at all (see Theorem4.3 forward). In our context, we require that the boundaries of the generalized Y-pieces are minimizingsimple closed geodesics (in its free homotopy class). Although if K = − K ≤ − c < K = − Theorem 1.3. ([7, Theorem 1])
Let S be a complete orientable Riemannian surface with curvature K = − .There are three possibilities: ( i ) S has finite area. Then for every p ∈ S there is exactly a countable collection of directions in E ( p ) . ( ii ) S is transient. Then for every p ∈ S , E ( p ) has full measure. ( iii ) S is recurrent and of infinite area. Then for every p ∈ S , E ( p ) has zero length but its Hausdorffdimension is . A surface is said to be transient (respectively, recurrent ) if Brownian motion of S is transient (respectively,recurrent). Also, we define E ( p ) as the set of unitary directions v in the tangent plane of S at p such thatthe unit speed geodesic emanating from p in the direction of v , escapes to infinity.Just like Theorem 1.2 played an important role in the proof of Theorem 1.3, we are sure that Theorems4.3 and 4.8 will be crucial in order to generalize this latest result to surfaces with curvature K ≤ − c < every standard fact used in the proof of Theorem 1.2 is false when there is no restriction of curvature.Hence, it was unavoidable both to state definitions for the new objects appearing in our current context andto prove alternate results valid for arbitrary curvature. This work has provided some results with intrinsicinterest, as Theorems 3.7 and 3.11.One can think that in the decomposition of Theorem 4.3 we might not need halfplanes. There is anexample in [1] which shows that, even with curvature K = −
1, we do need them. The necessity of halfplanesis, in fact, one of the most difficult parts in the proof of this theorem.The outline of the paper is as follows. Section 2 presents some definitions and technical results whichwe will need. We prove some additional technical results in Section 3. Section 4 is dedicated to the mainresults.
Notations.
We denote by L M ( γ ) the length of a curve γ in a Riemannian manifold M . If there is nopossible confusion, we usually write L ( γ ). Acknowledgements.
We would like to thank Professor Jes´us Gonzalo his proof of Theorem 3.11.2.
Background in Riemannian manifolds.
Definition 2.1.
Any divergent curve σ : [0 , ∞ ) −→ Y , where Y is a noncompact Hausdorff space, determinesan end E of Y . Given a compact set F of Y , one defines E ( F ) to be the arc component of Y \ F that containsa terminal segment σ ([ a, ∞ )) of σ . A set U ⊂ Y is a neighborhood of an end E if U contains E ( F ) forsome compact set F of Y . An end E in a surface S is collared if E has a neighborhood homeomorphic to (0 , ∞ ) × SS . A neighborhood U of E will be called Riemannian collared if there exists a C diffeomorphism X : (0 , ∞ ) × SS −→ U such that the metric in U relative to the coordinate system X is written ds = dr + G ( r, θ ) dθ , where G is a positive continuous function. A sequence of curves { C n } converges to E if TRUCTURE THEOREM FOR RIEMANNIAN SURFACES WITH ARBITRARY CURVATURE 3 for any neighborhood U of E we have C n ⊂ U for sufficiently large n . We say that a closed curve γ bounds a collared end E in S if some arc component of S \ γ is a neighborhood U of E . It follows directly from the metric expression ds = dr + G ( r, θ ) dθ of a Riemannian collared parametriza-tion that the r -parameter curves have unit speed and minimize the distance between any two of their points.Consequently the r -parameter curves are geodesics of S . If the curvature K satisfies K ≤
0, then G is a C ∞ function of r for each fixed θ and satisfies the Jacobi equation ∂ G∂r ( r, θ ) + K ( r, θ ) G ( r, θ ) = 0 , where K ( r, θ ) is the curvature of S at X ( r, θ ) ([6, p. 17]).Every manifold is connected, C ∞ and satisfy the second axiom of countability (has a countable basis forits topology). In a Riemannian surface we always assume that the Riemannian metric is C ∞ unless perhapsin some simple closed geodesics, each of them bounding a collared end, where we allow the metric to be C and piecewise C ∞ , with the “singularities” along these geodesics. Then the curvature is a (possiblydiscontinuous) function along these geodesics.Geodesic always means local geodesic (unless we say explicitly something else). Definition 2.2.
Given a Riemannian surface S , a geodesic γ in S , and a continuous unit vector field ξ along γ , orthogonal to γ , we define the Fermi coordinates based on γ as the map Y ( r, θ ) := exp γ ( θ ) rξ ( θ ) . It is well known that the Riemannian metric can be expressed in Fermi coordinates as ds = dr + G ( r, θ ) dθ , where G ( r, θ ) is the solution of the scalar equation ∂ G∂r ( r, θ ) + K ( r, θ ) G ( r, θ ) = 0 , G (0 , θ ) = 1 , ∂G∂r (0 , θ ) = 0 , (see e.g. [3, p. 247]).We will need the following three results. Theorem 2.3. ([6, Theorem 4.2])
Let S be a complete Riemannian surface with K ≤ and E an end of S .Then the following are equivalent: (1) E is a collared end. (2) E is a Riemannian collared end. (3) There exists a sequence { C n } of continuous piecewise smooth closed curves converging to E such that { C n } belongs to a single nontrivial free homotopy class. It is clear that (2) can be deduced from (1) since S verifies K ≤
0. However, since (1) and (3) aretopological conditions, we have the following result without conditions on K . Theorem 2.4.
Let S be a complete Riemannian surface and E an end of S . Then E is a collared end ifand only if there exists a sequence { C n } of continuous piecewise smooth closed curves converging to E suchthat { C n } belongs to a single nontrivial free homotopy class. Theorem 2.5. ([2, Theorem (5.16)])
Giving a sequence of rectifiable curves { α k } contained in a compactset of a Riemannian manifold M with { L ( α k ) } a bounded sequence, there exists a subsequence of curves(which we also call { α k } for simplicity), a rectifiable curve α , and parametrizations x k : [0 , −→ X of α k and x : [0 , −→ X of α , such that { x k } converges uniformly to x in [0 , and L ( α ) ≤ lim inf k →∞ L ( α k ) . In fact, Theorem (5.16) in [2] is stronger than Theorem 2.5, but this statement is general enough for ourpurposes.
Definition 2.6.
Given a n -dimensional Riemannian manifold M and a closed curve α in M , we define the length of the freely homotopy class [ α ] as L ([ α ]) := inf (cid:8) L ( σ ) : σ ∈ [ α ] (cid:9) . The curve α is minimizing if L ( α ) = L ([ α ]) .A minimizing sequence for α is a sequence of closed curves { α k } ⊂ [ α ] such that lim k →∞ L ( α k ) = L ([ α ]) . ANA PORTILLA (1)(2)(3) , JOSE M. RODRIGUEZ (1)(2)(3)
AND EVA TOURIS (1)(2)(3)
Definition 2.7. A halfplane is a bordered Riemannian surface which is simply connected and whose borderis a unique nonclosed simple geodesic.A generalized funnel is a bordered Riemannian surface which is a neighborhood of a collared end andwhose border is a minimizing simple closed geodesic. A funnel is a generalized funnel such that there doesnot exist another simple closed geodesic freely homotopic to the border of the funnel. (a) Generalized funnel (b) Generalized funnel (c) Funnel A generalized puncture is a collared end whose fundamental group is generated by a simple closed curve σ and there is no minimizing closed geodesic γ ∈ [ σ ] . A puncture is a generalized puncture such that L ([ σ ]) = 0 and there is no closed geodesic in [ σ ] . (d)Generalized puncture (e) Generalized puncture (f) Puncture A bordered or nonbordered surface is doubly connected if its fundamental group is isomorphic to Z . Everygeneralized funnel and every generalized puncture are doubly connected surfaces.A geodesic domain G is a bordered Riemannian surface (which is neither simply nor doubly connected)with finitely generated fundamental group and such that ∂G consists of finitely many minimizing simpleclosed geodesics, and it may contain generalized punctures but not generalized funnels.A Y-piece is a compact bordered Riemannian surface which is topologically a sphere without three opendisks and whose boundary curves are minimizing simple closed geodesics. They are a standard tool to con-struct Riemannian surfaces. A clear description of these Y-pieces and their use is given in [3, chapterX.3] .A generalized Y-piece is a bordered or nonbordered Riemannian surface which is topologically a spherewithout three open disks, such that there exist integers n, m ≥ with n + m = 3 , so that the border are n minimizing simple closed geodesics and there are m generalized punctures.Notice that a generalized Y-piece is topologically the union of a Y-piece and m cylinders, with ≤ m ≤ .It is clear that every generalized Y-piece is a geodesic domain (unless m = 3 , in which case it has no border).Furthermore, every geodesic domain is a finite union (with pairwise disjoint interiors) of generalized Y-pieces(see Proposition 4.1 below). TRUCTURE THEOREM FOR RIEMANNIAN SURFACES WITH ARBITRARY CURVATURE 5
We say that the set A is exhausted by { A n } if A n ⊆ A n +1 for every n and A = ∪ n A n .We say that a bordered Riemannian surface S is simple if the border of S is a (finite or infinite) unionof pairwise disjoint simple closed geodesics. Technical results.
Lemma 3.1.
Let us consider a n -dimensional complete Riemannian manifold M and an homotopicallynontrivial closed curve α in M . If there exists a minimizing sequence { α k } for α contained in a compactset, then there exists a minimizing closed geodesic γ ∈ [ α ] .Proof. Since { L ( α k ) } is convergent, it is a bounded sequence. By Theorem 2.5, there exists a subsequence ofcurves (which we also call { α k } for simplicity), a rectifiable curve γ , and parametrizations x k : [0 , −→ M of α k and x : [0 , −→ M of γ , such that { x k } converges uniformly to x in [0 ,
1] and L ( γ ) ≤ lim inf k →∞ L ( α k ) = L ([ α ]) . The curve γ is closed since each α k is a closed curve and x (0) = lim k →∞ x k (0) = lim k →∞ x k (1) = x (1).Then, in order to finish the proof of the lemma, it is enough to show that γ ∈ [ α ], since then γ attains theminimum length in its homotopy class, and it must be a geodesic.We can assume that x k and x are 1-periodic functions in R . For each t ∈ [0 ,
1] let us consider r t > B ( x ( t ) , r t ) in M is simply connected. For each t ∈ [0 , J t the connected component of γ ∩ B ( x ( t ) , r t ) which contains x ( t ). Since γ is a compact topologicalspace and { J t } t ∈ [0 , is an open covering of γ , there exist 0 ≤ t < t < · · · < t m − < t m ≤ γ ⊂ ∪ mj =1 J t j . Choosing a subset of { t , t , . . . , t m } if it is necessary, without loss of generality we can assumethat the subcovering { J t j } mj =1 is minimal in the following sense: each y ∈ γ belongs at most to two sets of { J t j } mj =1 .Let us consider 0 ≤ s < · · · < s m − < s m ∈ ( s m − , s + 1) such that x ( s ) ∈ J t ∩ J t , . . . , x ( s m − ) ∈ J t m − ∩ J t m , x ( s m ) ∈ J t m ∩ J t . Hence, x ( s m ) , x ( s ) ∈ B ( x ( t ) , r t ) , x ( s ) , x ( s ) ∈ B ( x ( t ) , r t ) , . . . , x ( s m − ) , x ( s m ) ∈ B ( x ( t m ) , r t m ) . Since { x k } converges uniformly to x in R , there exists k such that x k ([ s m , s ]) ⊂ B ( x ( t ) , r t ) , x k ([ s , s ]) ⊂ B ( x ( t ) , r t ) , . . . , x k ([ s m − , s m ]) ⊂ B ( x ( t m ) , r t m ) , for every k ≥ k .This proves that γ ∈ [ α k ] for every k ≥ k (since each ball B ( x ( t ) , r t ) is simply connected), and then γ ∈ [ α ]. This finishes the proof of the lemma. (cid:3) Proposition 3.2.
Let us consider a complete Riemannian surface S and an homotopically nontrivial closedcurve α in S . Then, one and only one of the two following possibilities holds: (1) There exists a minimizing closed geodesic γ ∈ [ α ] . (2) The curve α bounds a generalized puncture E in S . Furthermore, if S is not doubly connected, thenany minimizing sequence for α converges to E .Proof. If there exists a minimizing sequence { α k } for α contained in a compact set, then Lemma 3.1 gives(1).Otherwise, every minimizing sequence { α k } for α escapes from any compact set. Hence, there not existsany minimizing closed geodesic in [ α ].If S is doubly connected, then α bounds a collared end E (in fact, α bounds exactly two collared ends).This collared end E is a generalized puncture since there not exists any minimizing closed geodesic in [ α ].If S is not doubly connected, then any minimizing sequence { α k } for α converges to an end E . Since thecurves { α k } belong to a single nontrivial free homotopy class, Theorem 2.4 gives that E is a collared end in S . Hence, α bounds a collared end in S , which must be a generalized puncture since there not exists anyminimizing closed geodesic in [ α ]. (cid:3) ANA PORTILLA (1)(2)(3) , JOSE M. RODRIGUEZ (1)(2)(3)
AND EVA TOURIS (1)(2)(3)
In order to deal with bordered surfaces, we need the following results.
Lemma 3.3.
Any simple complete bordered Riemannian surface S is a subset of a complete Riemanniansurface R , which can be obtained by attaching a neighborhood of a collared end to each simple closed geodesic γ ⊆ ∂S , with the following properties: (1) If σ is a closed curve in R which is not contained in S , then there exists σ ⊆ ( S ∩ σ ) ∪ ∂S ⊂ S with σ ∈ [ σ ] and L ( σ ) < L ( σ ) . (2) A closed geodesic is minimizing in R if and only if it is minimizing in S (in particular, it is containedin S ). (3) If σ is a closed curve in S , then L S ([ σ ]) = L R ([ σ ]) . (4) If σ is a closed curve in S , and { σ k } is a minimizing sequence for σ verifying { σ k } ⊆ ( R \ S ) ∪ K ,with K a compact subset of S , then there exists a minimizing closed geodesic in [ σ ] . (5) The curvature satisfies K = − in R \ S . (6) The fundamental group of R is isomorphic to the fundamental group of S . (7) If S is not doubly connected, then there exists a minimizing simple closed geodesic in [ γ ] for eachsimple closed geodesic γ ⊆ ∂S .Proof. The border of S is a (finite or infinite) union of pairwise disjoint simple closed geodesics. Let us fix aclosed geodesic γ ⊆ ∂S with length l . We can consider the Fermi coordinates based on γ . The Riemannianmetric can be expressed in Fermi coordinates as ds = dr + G ( r, θ ) dθ , with G ( r, θ ) a l -periodic functionin θ defined in [ − r , × R , for some r >
0. We have G (0 , θ ) = 1 and ∂G/∂r (0 , θ ) = 0 for every θ ∈ R . If wedefine G ( r, θ ) := cosh r in (0 , ∞ ) × R , then it is C (and even piecewise C ∞ ) in [ − r , ∞ ) × R , and l -periodicin θ . These coordinates ( r, θ ) ∈ [ − r , ∞ ) × R , with the Riemannian metric ds = dr + G ( r, θ ) dθ , attacha neighborhood of a collared end F to γ ; by this way we get a C ∞ surface. We have that K ( r, θ ) = − , ∞ ) × R . We also have the following properties:( a ) Any homotopically nontrivial closed curve σ in F verifies L ( γ ) < L ( σ ):Without loss of generality we can assume that σ can be parametrized in Fermi coordinates based on γ as σ ( θ ) = ( r ( θ ) , θ ), with θ ∈ [0 , l ]. Then, L ( σ ) = Z l p r ′ ( θ ) + cosh r ( θ ) dθ ≥ Z l cosh r ( θ ) dθ > Z l dθ = l = L ( γ ) . ( b ) Given any closed curve σ intersecting S and the interior of F , there exists σ ∈ [ σ ] contained in S verifying L ( σ ) < L ( σ ):We can construct this curve in the following way: given any subarc a of σ contained in F and joining twopoints p, q ∈ γ , we replace it by the subarc of γ joining p, q, which is homotopic to a . The argument abovegives L ( σ ) < L ( σ ).We define R as the surface obtained by attaching this neighborhood of a collared end to each closedgeodesic in ∂S .Properties ( a ) and ( b ) give that if σ is a closed curve in R which is not contained in S , then there exists σ ⊆ ( S ∩ σ ) ∪ ∂S ⊂ S with σ ∈ [ σ ] and L ( σ ) < L ( σ ). This finishes the proof of (1), (5) and (6).Now, the statements (2) and (3) are direct consequences of (1).We prove now (4). If σ is a closed curve in S , and { σ k } is a minimizing sequence for σ verifying { σ k } ⊆ ( R \ S ) ∪ K , with K a compact subset of S , by (1) there exists { σ k } ⊆ K with σ k ∈ [ σ ] and L ( σ k ) ≤ L ( σ k ). Then { σ k } is a minimizing sequence for σ contained in a compact set and Lemma 3.1 givesthat there exists a minimizing closed geodesic in [ σ ].In order to prove (7), fix a simple closed geodesic γ ⊆ ∂S . Let us call F the neighborhood of a collaredend in R with ∂F = γ . Seeking for a contradiction, assume that there not exists a minimizing closedgeodesic in [ γ ]. Since R is not doubly connected, by Proposition 3.2 γ bounds a generalized puncture E in R and any minimizing sequence for α converges to E . Since R is not doubly connected, F is a neighborhoodof E , and for any minimizing sequence { α k } for [ γ ] there exists N with α k ⊂ F for every k ≥ N . By ( a )we have L ( γ ) < L ( α k ) for every k ≥ N , which is the required contradiction. (cid:3) Using Lemma 3.3, Proposition 3.2 can be generalized to simple bordered Riemannian surfaces.
TRUCTURE THEOREM FOR RIEMANNIAN SURFACES WITH ARBITRARY CURVATURE 7
Proposition 3.4.
Let us consider a simple complete bordered Riemannian surface S and an homotopicallynontrivial closed curve α in S . Then, one and only one of the two following possibilities holds: (1) There exists a minimizing closed geodesic γ ∈ [ α ] . (2) The curve α bounds a generalized puncture E in S and any minimizing sequence for α converges to E .Proof. By Lemma 3.3, S is a subset of a complete Riemannian surface R .Assume first that S is not doubly connected (then R is not doubly connected).By Lemma 3.3 (2), if there exists a minimizing closed geodesic γ ∈ [ α ] in R , then γ ∈ S .If there not exist such minimizing geodesic, by Proposition 3.2 the curve α bounds a generalized puncture E in R and any minimizing sequence for α converges to E . By Lemma 3.3 (7), α can not be freely homotopicto any closed geodesic in ∂S , and therefore E ⊂ S .Assume now that S is doubly connected (then R is also doubly connected). The curve α bounds exactlytwo collared ends in R .Since S is doubly connected, ∂S can be either a simple closed geodesic or two simple closed geodesics.Assume first that ∂S is a simple closed geodesic γ . Then α bounds the unique collared end E in S .Consider a minimizing sequence { α n } for [ α ] in R . If (2) does not hold, then either α does not bound ageneralized puncture (and then (1) holds) or there exists a neighborhood U of E and a subsequence { α n k } with α n k * U for every k . Since { L ( α n ) } is a bounded sequence, without loss of generality we can assumethat α n k ∩ U = ∅ for every k . Since S is doubly connected, R = S ∪ F with ∂F = ∂S = γ , and thecomplement of U in R is F ∪ K , where K is a compact set in S . Therefore, α n k ⊂ F ∪ K for every k andby Lemma 3.3 (4) there exists a minimizing closed geodesic in [ α ]. Then (1) also holds.Assume now that ∂S is the union of two simple closed geodesics γ , γ . Then, R = S ∪ F ∪ F with ∂F j = γ j ( j = 1 ,
2) and S is compact. Consider a minimizing sequence { α n } for [ α ] in R . By Lemma 3.3(1), there exists a minimizing sequence { α n } ⊂ S for [ α ]. Since S is compact, Lemma 3.1 gives that thereexists a minimizing closed geodesic γ ∈ [ α ]. (cid:3) Lemma 3.5.
Let us consider a positive constant c and two functions y , y, satisfying respectively y ′′ = c y , y ( t ) > , y ′ ( t ) > , and y ′′ ( t ) ≥ c y ( t ) > , if t ≥ t ,y ( t ) = y ( t ) ,y ′ ( t ) ≥ y ′ ( t ) . Then, y ( t ) ≥ y ( t ) for every t ≥ t .Proof. Since y ′′ ( t ) > t ≥ t , then y ′ is an increasing function in [ t , ∞ ). This fact and y ′ ( t ) ≥ y ′ ( t ) > y ′ ( t ) ≥ y ′ ( t ) > t ≥ t . Then, for every t ≥ t , we can deduce y ′′ ( t ) ≥ c y ( t ) ,y ′′ ( t ) y ′ ( t ) ≥ c y ( t ) y ′ ( t ) ,y ′ ( t ) − y ′ ( t ) ≥ c (cid:0) y ( t ) − y ( t ) (cid:1) ,y ′ ( t ) ≥ q c (cid:0) y ( t ) − y ( t ) (cid:1) + y ′ ( t ) . For each fixed ε ∈ (0 , y ′ ( t )), we define the function y ε as the unique solution of y ′′ ε ( t ) = c y ε ( t ) , if t ≥ t ,y ε ( t ) = y ( t ) ,y ′ ε ( t ) = y ′ ( t ) − ε > . Then y ′ ( t ) > y ′ ε ( t ) >
0. Using the same argument above in the case of y ε (with equality instead ofinequality) we obtain that y ′ ε ( t ) ≥ y ′ ε ( t ) > t ≥ t and y ′ ε ( t ) = q c (cid:0) y ε ( t ) − y ε ( t ) (cid:1) + y ′ ε ( t ) . We prove now that y ( t ) ≥ y ε ( t ) for every t ≥ t . Seeking for a contradiction, suppose that y ( t ) < y ε ( t )for some t > t . Then, we can define t := min { t > t : y ( t ) = y ε ( t ) } ; this minimum is attained since ANA PORTILLA (1)(2)(3) , JOSE M. RODRIGUEZ (1)(2)(3)
AND EVA TOURIS (1)(2)(3) y ( t ) = y ε ( t ) and y ′ ( t ) > y ′ ε ( t ); consequently, y ( t ) > y ε ( t ) > t ∈ ( t , t ), and y ε ( t ) − y ε ( t ) = Z t t y ′ ε ( t ) dt = Z t t q c (cid:0) y ε ( t ) − y ε ( t ) (cid:1) + y ′ ε ( t ) dt< Z t t q c (cid:0) y ( t ) − y ( t ) (cid:1) + y ′ ( t ) dt ≤ Z t t y ′ ( t ) dt = y ( t ) − y ( t ) = y ε ( t ) − y ε ( t ) . This is a contradiction and we have proved that y ( t ) ≥ y ε ( t ) for every t ≥ t . It is easy to check that y ε ( t ) = y ( t ) cosh c ( t − t ) + y ′ ( t ) − εc sinh c ( t − t ) , for every t, ε ∈ R . Hence y ( t ) ≥ y ( t ) cosh c ( t − t ) + y ′ ( t ) − εc sinh c ( t − t ) , for every t ≥ t and ε ∈ (0 , y ′ ( t )). If ε →
0, we obtain y ( t ) ≥ y ( t ) cosh c ( t − t ) + y ′ ( t ) c sinh c ( t − t ) = y ( t ) , for every t ≥ t . This finishes the proof of the lemma. (cid:3) Lemma 3.5 has the following direct consequence.
Corollary 3.6.
Let us consider a positive constant c and a function y satisfying y ′′ ( t ) ≥ c y ( t ) > and y ′ ( t ) > . Then y ( t ) ≥ y ( t ) cosh c ( t − t ) , for every t ≥ t .Proof. Let us consider the function y with y ′′ ( t ) = c y ( t ) , if t ≥ t ,y ( t ) = y ( t ) ,y ′ ( t ) = y ′ ( t ) . The first inequality in the following expression is obtained by applying Lemma 3.5 and the first equality bysolving the above differential equation: y ( t ) ≥ y ( t ) = y ( t ) cosh c ( t − t ) + y ′ ( t ) c sinh c ( t − t ) ≥ y ( t ) cosh c ( t − t ) = y ( t ) cosh c ( t − t ) , for every t ≥ t . (cid:3) The following result assures that if K ≤ − c <
0, there always exists a closed geodesic in every freehomotopy class, except for punctures, in which is impossible to have one.
Theorem 3.7.
Let us consider a Riemannian surface S , which can be either simple bordered or withoutborder. Besides, S must be complete and with curvature K ≤ − c < . Fix an homotopically nontrivialclosed curve α in S . Then there exists a minimizing closed geodesic γ ∈ [ α ] if and only if L ([ α ]) > . Remark 3.8.
The conclusion of this Theorem does not hold if we replace the hypothesis K ≤ − c < bythe weaker one K < , as shows the revolution surface of the graph of f ( x ) = 1 + e x around the horizontalaxis (with the standard metric induced by the Euclidean metric in R ).Proof. We deal first with nonbordered surfaces S .If L ([ α ]) = 0, it is clear that there does not exist a closed geodesic γ ∈ [ α ] with L ( γ ) = L ([ α ]) = 0, since α is an homotopically nontrivial closed curve in S .Let us assume now that L ([ α ]) > S is not doubly connected. Seeking for a contradiction, suppose that there not exist aclosed geodesic γ ∈ [ α ]. Then, by Proposition 3.2, the curve α bounds a generalized puncture E in S . Sincethe curvature satisfies K ≤ − c <
0, this end E is a Riemannian collared end, by Theorem 2.3. TRUCTURE THEOREM FOR RIEMANNIAN SURFACES WITH ARBITRARY CURVATURE 9
For each r we define g r as the closed curve { r = r } . It is easy to check that l ( r ) := L ( g r ) = R π G ( r, θ ) dθ satisfies l ′′ ( r ) ≥ c l ( r ): ∂ G∂r ( r, θ ) = − K ( r, θ ) G ( r, θ ) ≥ c G ( r, θ ) > l ′′ ( r ) = Z π ∂ G∂r ( r, θ ) dθ ≥ Z π c G ( r, θ ) dθ = c l ( r ) > . Since L ([ α ]) >
0, there exist positive constants c , r with l ( r ) ≥ c for every r ≥ r . Hence, for every r ≥ r , l ′ ( r ) = l ′ ( r ) + Z rr l ′′ ( t ) dt ≥ l ′ ( r ) + Z rr c c dt = l ′ ( r ) + c c ( r − r ) , and consequently lim r →∞ l ′ ( r ) = ∞ . Sincelim r →∞ Z π ∂G∂r ( r, θ ) dθ = lim r →∞ l ′ ( r ) = ∞ , there exist r ≥ r and a set A ⊂ [0 , π ] with positive Lebesgue measure such that ∂G/∂r ( r , θ ) > θ ∈ A . Since ∂ G/∂r ( r, θ ) >
0, the function ∂G/∂r ( r, θ ) increases in r ≥ r for each fixed θ ∈ A , andconsequently ∂G/∂r ( r, θ ) ≥ ∂G/∂r ( r , θ ) > θ ∈ A and r ≥ r . Hence, G ( r, θ ) increases in r ≥ r for each fixed θ ∈ A . By Corollary 3.6, G ( r, θ ) ≥ G ( r , θ ) cosh c ( r − r ) for every θ ∈ A and r ≥ r .Let us consider a curve σ parametrized in the Riemannian collared end as σ ( θ ) = ( r ( θ ) , θ ), with θ ∈ [0 , π ]and r ( θ ) ≥ R ≥ r . Then L ( σ ) = Z π p r ′ ( θ ) + G ( r ( θ ) , θ ) dθ ≥ Z π G ( r ( θ ) , θ ) dθ ≥ Z A G ( r ( θ ) , θ ) dθ ≥ Z A G ( R, θ ) dθ ≥ cosh c ( R − r ) Z A G ( r , θ ) dθ . Since R A G ( r , θ ) dθ is a positive constant independent of R , there exists r > r withcosh c ( r − r ) Z A G ( r , θ ) dθ > L ( α ) . Hence, given any curve σ parametrized in the Riemannian collared end as σ ( θ ) = ( r ( θ ) , θ ), with θ ∈ [0 , π ]and r ( θ ) ≥ r , we have L ( σ ) > L ( α ). Consequently, any curve σ ∈ [ α ] contained in the region { r ≥ r } verifies L ( σ ) > L ( α ). Then a minimizing sequence for α can not converge to E . This fact contradictsProposition 3.2.If S is doubly connected, the argument is similar except for the fact that α bounds two collared ends.Therefore, a minimizing sequence might not converge to an end in S ; but we can always extract a subsequenceconverging to some end in S .Assume now that S is simple bordered. By Lemma 3.3, S is a subset of a complete Riemannian surface R . The previous argument gives the desired result in R . Then, Lemma 3.3 implies the result in S . (cid:3) The following lemma is a well known result, but we include a direct proof by the sake of completeness.
Lemma 3.9.
Let us consider a Riemannian surface S , which can be either simple bordered or without border.Besides, S must be complete and with curvature K < . Then in each free homotopy class there exists atmost a closed geodesic, and if there exists, then it is minimizing. Consequently, every generalized funnel isa funnel.Proof. By Lemma 3.3, without loss of generality we can assume that S is nonbordered.Seeking for a contradiction, suppose that there exist two freely homotopic closed geodesics γ , γ .If γ and γ intersect at some point, they can not be tangent at this point, since they are geodesics. Then, γ and γ intersect at least at another point, since they are freely homotopic. Therefore, some segment of (1)(2)(3) , JOSE M. RODRIGUEZ (1)(2)(3) AND EVA TOURIS (1)(2)(3) γ and some segment of γ determine a geodesic “bigon” B (a polygon with two sides) with interior angles α, β >
0. Gauss-Bonnet Formula gives
Z Z B K dA = α + β > , which is a contradiction with K < γ and γ do not intersect. We consider the geodesic segment σ joining x ∈ γ with x ∈ γ , whichgives the minimum distance between γ and γ ; then σ meets orthogonally to γ and to γ . Let us considera universal covering map π : ˜ S −→ S . Fix a lift ˜ γ of γ starting in ˜ x , a lift ˜ σ of σ starting in ˜ x , andfinishing in ˜ x , and a lift ˜ γ of γ starting in ˜ x . Then ˜ σ meets orthogonally to ˜ γ and to ˜ γ . If we denoteby y and y , respectively, the endpoints of ˜ γ and ˜ γ , there exists a covering isometry T : ˜ S −→ ˜ S with T (˜ x ) = y and T (˜ x ) = y . We also have that with T (˜ σ ) joins y and y , and meets orthogonally to ˜ γ and to ˜ γ . Consequently, ˜ γ , ˜ γ , ˜ σ and T (˜ σ ) bound a geodesical quadrilateral Q in ˜ S with four right angles.Gauss-Bonnet Formula gives − Z Z Q K dA = 2 π − π − π − π − π , which is a contradiction with K < γ . We need to prove that L ( γ ) = L ([ γ ]). Let us define l := L ( γ ).The Riemannian metric can be expressed in Fermi coordinates based on γ as ds = dr + G ( r, θ ) dθ ,where G ( r, θ ) satisfies ∂ G∂r ( r, θ ) + K ( r, θ ) G ( r, θ ) = 0 , G (0 , θ ) = 1 , ∂G∂r (0 , θ ) = 0 . Since ∂ G/∂r ( r, θ ) = − K ( r, θ ) G ( r, θ ) >
0, it follows that G ( r, θ ) is a convex function on r for each fixed θ ∈ [0 , l ]; since ∂G/∂r (0 , θ ) = 0, we deduce that for each fixed θ ∈ [0 , l ], G ( r, θ ) attains its minimum value 1at r = 0.We prove now that any curve σ ∈ [ γ ] verifies L ( σ ) ≥ L ( γ ). Let us consider a fixed curve σ ∈ [ γ ]. Withoutloss of generality we can assume that σ can be parametrized in Fermi coordinates as σ ( θ ) = ( r ( θ ) , θ ), with θ ∈ [0 , l ]. Then, L ( σ ) = Z l p r ′ ( θ ) + G ( r ( θ ) , θ ) dθ ≥ Z l G ( r ( θ ) , θ ) dθ ≥ Z l G (0 , θ ) dθ = Z l dθ = l = L ( γ ) . This shows that L ( γ ) = L ([ γ ]) and hence γ is minimizing.In order to prove the last part of the lemma, consider now a generalized funnel F in S . We have provedthat there does not exist another closed geodesic freely homotopic to the boundary of the generalized funnel.Hence, the generalized funnel is a funnel. (cid:3) Lemma 3.10.
Let us consider a Riemannian surface S , which can be either simple bordered or withoutborder. Besides, S must be complete and with curvature K ≤ − c < . Then, every generalized puncture isa puncture.Proof. By Lemma 3.3, without loss of generality we can assume that S is nonbordered.Seeking for a contradiction, consider a generalized puncture which is not a puncture. Then, its funda-mental group is generated by a simple closed curve σ , there is no minimizing simple closed geodesic γ ∈ [ σ ],and we have either:( i ) L ([ σ ]) >
0; then by Theorem 3.7 there exists a minimizing simple closed geodesic γ ∈ [ σ ], which is acontradiction,or ( ii ) L ([ σ ]) = 0 and there exists a simple closed geodesic γ ∈ [ σ ]; then by Lemma 3.9 γ is minimizing,which is a contradiction. (cid:3) TRUCTURE THEOREM FOR RIEMANNIAN SURFACES WITH ARBITRARY CURVATURE 11
Theorem 3.11.
Let us consider a complete Riemannian surface S and two disjoint nontrivial piecewisesmooth simple closed curves a and b in S , which are not freely homotopic. If α ∈ [ a ] and β ∈ [ b ] are anychoice of minimizing closed geodesics, then they are disjoint as well.Furthermore, if α = β are freely homotopic minimizing simple closed geodesics in S , then they are disjoint. Remark 3.12.
We have some examples that show that the conclusion of the previous Theorem does nothold if either α or β are not minimizing geodesics. See for example the figure below:PSfrag replacements α β Proof.
First, let us assume that the curves a and b are not freely homotopic. Without loss of generalitywe can assume that a and b are disjoint nontrivial smooth simple closed curves in S , since in other casewe can modify them slightly in order to obtain all these facts. Since α ∈ [ a ] and S is a surface, by Baer’sTheorem (see e.g. [11] or [10]) there exists an isotopy, that is to say, a continuous family of diffeomorphisms f t : S −→ S , such that f is the identity, and f ( a ) = α . Let us define b := f ( b ), which is a simple closedcurve freely homotopic to b . As f is bijective and a and b are disjoint curves, then b and α are disjoint too.Seeking for a contradiction, let us assume that α ∩ β = ∅ . If they do intersect each other tangentiallythen they should coincide, and this is not possible since they are not freely homotopic. Therefore, they mustintersect transversally. Since b ∈ [ β ], there exists an smooth homotopy F : A −→ S , where A is the annulus { z ∈ C : 1 ≤ | z | ≤ } , such that F ( e iθ ) = b ( θ ) and F (2 e iθ ) = β ( θ ) , with θ ∈ [0 , π ].No connected component γ of F − ( α ) can be a closed curve in A , since it should be either trivial or freelyhomotopic to F − ( β ). In this case, F ( γ ) = α would also be either trivial or homotopic to F ( F − ( β )) = β ,and this is contradiction with our hypothesis. Therefore, F − ( α ) must contain an arc σ joining two points z and z in { z ∈ C : | z | = 2 } = F − ( β ). As A is a planar domain, one of the two arcs joining z and z in F − ( β ) (let us denote such arc by η ), is homotopic to σ . This fact implies that the geodesics α and β intersect in F ( z ) and F ( z ) and that F ( σ ) and F ( η ) are homotopic. Since α and β are minimizing, L ( F ( σ )) = L ( F ( η )) and so, from α we can construct a new curve ˜ α ∈ [ α ] with the same length by replacingthe arc F ( σ ) by the arc F ( η ). By means of a smooth modification in small neighborhoods of F ( z ) and F ( z ) we can obtain a shorter curve freely homotopic to α , which is contradiction with the fact that α isminimizing.Now, we will deal with the second part of the Theorem. Once again we are going to seek for a contradiction:let us assume that α ∩ β = ∅ . If they do intersect in a single point, as they are in the same homotopy class,they must intersect each other tangentially and therefore α = β . If they do intersect in several points, thenthey must intersect transversally. The argument in the previous case allows to obtain a shorter curve freelyhomotopic to α , which is contradiction with the fact that α is minimizing. (cid:3) The main results.
Proposition 4.1.
Every geodesic domain in any complete orientable Riemannian surface is a finite union(with pairwise disjoint interiors) of generalized Y-pieces.
Remark 4.2.
The argument in the proof of Proposition 4.1 also proves the following: (1)(2)(3) , JOSE M. RODRIGUEZ (1)(2)(3)
AND EVA TOURIS (1)(2)(3)
Every complete orientable Riemannian surface without generalized funnels and with finitely generatedfundamental group, which is neither simply nor doubly connected nor homeomorphic to a torus, is a finiteunion (with pairwise disjoint interiors) of generalized Y-pieces.Proof.
Let us fix a geodesic domain G in a complete orientable Riemannian surface S . We denote by γ , γ , . . . , γ k the minimizing simple closed geodesics in ∂G . Since G is a simple complete bordered Rie-mannian surface, by Lemma 3.3 it is a subset of a complete Riemannian surface R .In particular, R is a complete orientable topological surface. Since R contains a geodesic domain, R isneither simply nor doubly connected nor homeomorphic to a torus. Then, by Theorem 1.1, R is the union oftopological Y-pieces { Y n } and cylinders { C n } . The fundamental group of R is isomorphic to the fundamentalgroup of G , and therefore it is finitely generated; then there are only a finite number of topological Y-piecesand cylinders. We denote by { η m } ⊂ R the set of pairwise disjoint simple closed curves in ∪ n ∂Y n . Withoutloss of generality we can assume that the curves are numbered such that η j ∈ [ γ j ] for each 1 ≤ j ≤ k .We want to change the curves η j by minimizing simple closed geodesics. For each 1 ≤ m ≤ k , we replace η m by γ m (Lemma 3.3 gives that γ m are also minimizing simple closed geodesic in R ). For each m > k , letus choose a minimizing simple closed geodesic γ m ∈ [ η m ], if it exists. In other case, by Proposition 3.2, thecurve η m bounds a generalized puncture in S and we define γ m := ∅ . By Theorem 3.11, the minimizingsimple closed geodesics { γ m } ⊂ G are pairwise disjoint; then, they split G in the required finite union ofgeneralized Y-pieces (if for some m we have γ m = ∅ , the corresponding Y -piece in R is a generalized Y -piecein G ). (cid:3) The following theorem is the main result of this paper. It generalizes an already known result for constantnegatively curved surfaces to arbitrary surfaces with no restricition of curvature at all.
Theorem 4.3.
Every complete orientable Riemannian surface which is neither simply nor doubly connectednor homeomorphic to a torus is the union (with pairwise disjoint interiors) of generalized Y-pieces, general-ized funnels and halfplanes.
Remark 4.4.
If there are several freely homotopic minimizing simple closed geodesics which bound a gen-eralized funnel, we will see in the proof that any of them can be chosen as border of this generalized funnel.Proof.
We assume first that the fundamental group of S is finitely generated. If S has not generalizedfunnels, then Remark 4.2 gives the result. If S has generalized funnels { F j } , then the closure of S \ ∪ j F j isa geodesic domain; Proposition 4.1 gives the result in this case.Let us consider a surface S with infinitely generated fundamental group, and fix a point p ∈ S . Next, wewill take an increasing sequence of positive numbers { r n } so that lim n →∞ r n = ∞ . For each r n we intendto associate a geodesic domain G n to the ball B ( r n ) centered in p with radius r n .The boundary of B ( r ) is a finite union of pairwise disjoint simple closed curves except for r ∈ A with A a countable set. Since S is of infinite type, we can always find a positive number r / ∈ A such that thefundamental group of the ball B ( r ) has, at least, two generators. We choose r n / ∈ A with r n > max { r n − , n } .As r n > r , the fundamental group of B ( r n ) has, at least, two generators as well. Since r n / ∈ A , the boundaryof B ( r n ) is a finite union of pairwise disjoint simple closed curves { η ni } i ∈ I n . In order to construct its geodesicdomain G n , our goal is to relate a minimizing geodesic γ ni to each curve η ni ⊆ ∂B ( r n ), and we do it inductivelyas it follows. There are two possibilities:(1) There not exists any minimizing simple closed geodesic in [ η ni ]. In this case γ ni := ∅ .(2) There exists at least one minimizing simple closed geodesic in [ η ni ]. If n > j ∈ I n − such that γ n − j ∈ [ η ni ], then γ ni := γ n − j . Otherwise (notice that this situation includes the case n = 1), choose γ ni as any of the minimizing simple closed geodesics in [ η ni ]. G n is the geodesic domain limited by all these geodesics { γ ni } i ∈ I n . By construction, G n ⊆ G n +1 .Before going on with the proof, we need the following lemma: Lemma 4.5.
If there exists some positive number N such that γ is a minimizing simple closed geodesiccontained in ∂G n for every n > N , then γ is the border of a generalized funnel in S . TRUCTURE THEOREM FOR RIEMANNIAN SURFACES WITH ARBITRARY CURVATURE 13
Proof.
For n > N , let us consider the simple closed curve η n ⊆ ∂B ( r n ) which is freely homotopic to γ . Sincelim n →∞ dist( p, η n ) = lim n →∞ r n = ∞ , and η n belongs to a single nontrivial freely homotopy class for every n > N , Theorem 2.4 gives that { η n } converges to a collared end F ; since its border γ is a minimizing simpleclosed geodesic, F must be a generalized funnel. (cid:3) Now, let us continue with the proof of Theorem 4.3. We can take a subsequence of radii (by simplicity ofthe notation we will denote this subsequence just like the whole sequence) such that G n ⊂ G n +1 and besides, ∂G n ∩ ∂G n +1 is either the empty set or a union of minimizing simple closed geodesics, each of them is theborder of a generalized funnel.Let us define H n as the closed set obtained as the union of G n and the generalized funnels whose bordersbelong to ∂G n , and H := ∪ n H n . Notice that due to the properties of G n , each H n is contained in theinterior of H n +1 . If S = H , then there is nothing else to prove. Otherwise, S \ H is a closed non-empty set.By Proposition 4.1, each connected component of the closure of G n +1 \ G n is a finite union (with pairwisedisjoint interiors) of generalized Y-pieces. Therefore, in order to finish the proof, we just have to see thatevery connected component J of S \ H is a halfplane, that is to say: J is a simply connected set and ∂J ⊆ ∂H is a unique nonclosed simple geodesic. From now on, by simplicity in the notation and as there is no possibleconfusion, we will denote γ ni ⊂ ∂H n by γ n .Next, we state a lemma that we will need along the proof: Lemma 4.6.
Let σ be a nontrivial simple closed curve in S . If there is a minimizing simple closed geodesic γ ∈ [ σ ] contained in B ( r n ) , then either γ is contained in G n or it is freely homotopic to some geodesic in ∂G n .Proof. Let us assume that γ is not contained in G n . Then, there are two possibilities: either γ ∩ G n = ∅ or γ ∩ ∂G n = ∅ . In the first case, γ is contained in a doubly connected set whose borders are γ n ⊂ ∂G n and η n ⊂ ∂B ( r n ) and therefore γ is freely homotopic to both of them.We finish the proof by showing that γ cannot intersect ∂G n . Seeking for a contradiction, assume that γ ∩ ∂G n = ∅ ; then, γ ∩ γ n = ∅ for some minimizing closed geodesic γ n ⊂ ∂G n . Since γ and γ n are minimizingsimple closed geodesics, Theorem 3.11 implies that γ ∩ η n = ∅ , where η n is the closed curve in ∂B ( r n ) with γ n ∈ [ η n ]. This is the required contradiction, since γ ⊂ B ( r n ). (cid:3) Going on with proof of Theorem 4.3 we will see that J is simply connected. In order to do so, let us provethat every simple closed curve contained in J must be trivial, since every topological obstacle must be in H :Let us consider a nontrivial simple closed curve σ in J . By Proposition 3.2, there are two possibilities:(1) There exists a minimizing simple closed geodesic γ ∈ [ σ ]. Consider r n with γ ⊂ B ( r n ); we provenow that γ ⊂ H n +1 . If γ is not contained in G n , then by Lemma 4.6 it is freely homotopic tosome geodesic in ∂G n . If γ bounds a generalized funnel, then γ ⊂ H n . If a curve in ∂G n is freelyhomotopic to some curve in ∂G n +1 then it must bound a generalized funnel. Since γ does not bounda generalized funnel, it is not freely homotopic to any geodesic in ∂G n +1 ; hence, Lemma 4.6 gives γ ⊂ G n +1 , since γ ⊂ B ( r n ) ⊂ B ( r n +1 ). Hence, in any case, γ ⊂ H n +1 , and it is not freely homotopicto any curve in ∂H n +1 . Therefore, σ must intersect H n +1 , which is a contradiction with σ ⊂ J .(2) The curve σ bounds a generalized puncture. Then, there exists some neighborhood of this collaredend contained in some H n . Since σ is not freely homotopic to any geodesic in ∂H n , σ must intersect H n , which is again a contradiction with σ ⊂ J .Next, we will prove that ∂J is a geodesic. Let us fix a point q ∈ ∂J ; we want to prove that q belongsto a geodesic arc γ contained in ∂J : There exist points q n ∈ γ n ⊆ ∂H n converging to q . Let us considernow the sequence { v n } of tangent vectors to γ n in q n . Notice that this latest sequence must converge to acertain vector v , since otherwise the geodesics γ n would intersect. As geodesics are solutions of a system ofordinary differential equations and the initial data { q n , v n } converge to { q, v } , then the geodesics γ n convergeuniformly in some neighborhood U of q to the geodesic γ whose tangent vector in q is v .Now, let us prove that γ ∩ U is contained in ∂J . Choose a point q ′ ∈ γ ∩ U . On the one hand, q ′ / ∈ ext H ,since there exists a sequence of points in γ n ⊂ H n converging to q ′ . On the other hand, q ′ does not belongto H either, since if it did, there would exist some H m containing q ′ for some positive integer m , and this isa contradiction with H n contained in the interior of H n +1 for every n . (1)(2)(3) , JOSE M. RODRIGUEZ (1)(2)(3) AND EVA TOURIS (1)(2)(3)
From the previous argument we also deduce that this geodesic can be prolonged to infinity at both sides:if it had an endpoint p , it is obvious that p ∈ ∂J , but as we have just seen, for every point in the boundaryof J there exists a neighborhood U such that U ∩ ∂J is a geodesic arc containing p .In order to see that every connected component γ ⊆ ∂J is simple, we will prove that it is not closed andit does not intersect itself transversally: If γ were a simple closed geodesic, it would be compact and as γ n locally uniformly converge to γ then there would exist a positive integer N and a collar C for γ such that γ n ⊂ C for every n ≥ N , and therefore γ n ∈ [ γ ], which is a contradiction. If γ intersected itself transversally,there would exist some positive integer N such that each γ n would intersect itself as well for every n ≥ N ,and this is not possible since they are all simple. This same argument also proves that two different geodesicscontained in ∂H must be simple and pairwise disjoint.To finish, there is only one fact to prove: ∂J consists of just one geodesic. Let us assume that thereexist two simple geodesics σ , σ ⊂ ∂J . Let us consider two points q ∈ σ , q ∈ σ , two simple connectedneighborhoods V , V of q and q respectively, two simple closed geodesics γ n ⊂ ∂H n , γ n ⊂ ∂H n , with γ n ∩ V = ∅ , γ n ∩ V = ∅ and n = n , and curves η ⊂ V , η ⊂ V joining, respectively, γ n with q and q with γ n . As J is path-connected, it is possible to construct the three following curves: η ⊂ J joining q and q , η := η + η + η and the closed curve β := η + γ n − η + γ n . Since β cannot bound a generalizedpuncture, every closed geodesic γ ∈ [ β ] verifies γ ∩ σ = ∅ and γ ∩ σ = ∅ ; in particular, this means thatevery minimizing simple closed geodesic do intersect ∂J . But, by Lemma 4.6, there must exist a minimizingclosed geodesic in [ β ] entirely contained in H , which is a contradiction. (cid:3) In fact, the proof of Theorem 4.3 gives the following result.
Theorem 4.7.
Every complete orientable Riemannian surface which is neither simply nor doubly connectednor homeomorphic to a torus is the union (with pairwise disjoint interiors) of generalized funnels, halfplanesand a set G which can be exhausted by geodesic domains. The curvature of a Riemannian surface homeomorphic to a torus can not verify
K <
0; then, Theorem4.3, Lemma 3.9 and Lemma 3.10 give directly the following result.
Theorem 4.8.
Every complete orientable Riemannian surface with curvature K ≤ − c < , which is neithersimply nor doubly connected is the union (with pairwise disjoint interiors) of generalized Y-pieces, funnelsand halfplanes. Furthermore, every generalized puncture is a puncture. In order to deal with bordered surfaces, we need a last definition.
Definition 4.9. A finite cylinder is a bordered Riemannian surface which is homeomorphic to SS × [0 , ,whose border is the union of two simple closed geodesics, and at least one of them is minimizing. Theorem 4.10.
Every simple complete orientable bordered Riemannian surface which is neither simplynor doubly connected is the union (with pairwise disjoint interiors) of generalized Y-pieces, finite cylinders,generalized funnels and halfplanes.Furthermore, there is a bijection between finite cylinders in the decomposition and nonminimizing simpleclosed geodesics in the border.Proof.
Let S be a simple complete orientable bordered Riemannian surface which is not simply nor doublyconnected, whose border is the union of simple closed geodesics { γ i } i ∈ I . By applying Lemma 3.3 we canconstruct another complete Riemannian surface R by gluing a neighborhood F i of a collared end to each γ i ,such that R = ∪ i ∈ I F i ∪ S . It is obvious that R is not homeomorphic to a torus, since it is not compact.By Theorem 4.3 we know that R is the union (with pairwise disjoint interiors) of generalized Y-pieces,generalized funnels and halfplanes. Furthermore, by Remark 4.4, if γ i is a minimizing simple closed geodesicin S , for some i ∈ I , we can choose the decomposition in such a way that γ i belongs to the border of ageneralized funnel.If γ i is a nonminimizing simple closed geodesic in S for some i ∈ I , Lemma 3.3 ((7) and (2)) guaranteesboth that there exists the minimizing simple closed geodesic γ i ∈ [ γ i ] and that it is contained in S . Then,the funnel F i in this decomposition intersects S in a finite cylinder whose border is γ i ∪ γ i .Consequently, we obtain the desired decomposition in S if we restrict to S this decomposition in R . (cid:3) TRUCTURE THEOREM FOR RIEMANNIAN SURFACES WITH ARBITRARY CURVATURE 15
Theorem 4.11.
Every simple complete orientable bordered Riemannian surface with curvature K ≤ − c < , is the union (with pairwise disjoint interiors) of generalized Y-pieces, funnels and halfplanes. Furthermore,every generalized puncture is a puncture.Proof. The proof follows the argument of the proof of Theorem 4.10, using Theorem 4.8, instead of Theorem4.3.Lemma 3.9 gives that in each free homotopy class there exists at most a closed geodesic. Consequently,there are not finite cylinders in the decomposition.We only need to study the simple complete orientable bordered Riemannian surfaces with curvature K ≤ − c < S be such a surface. S can not be simply connected: Seeking for a contradiction, suppose that S is simply connected; then ∂S can be considered a geodesic triangle with three angles equal to π , andGauss-Bonnet Formula gives − Z Z S K dA = π − π − π − π = − π , which is a contradiction with K < S is doubly connected, then Lemma 3.9 gives that ∂S is a single simple closed geodesic and besides itis minimizing. Then, S is a funnel. (cid:3) References [1] Alvarez, V., Rodr´ıguez, J. M., Structure theorems for Riemann and topological surfaces,
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Ana Portilla, Jos´e M. Rodr´ıguez, Eva Tour´ısDepartamento de Matem´aticasEscuela Polit´ecnica SuperiorUniversidad Carlos III de MadridAvenida de la Universidad, 3028911 Legan´es (Madrid)SPAIN
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