Spherical metrics with conical singularities on 2-spheres
aa r X i v : . [ m a t h . DG ] M a y Spherical metrics with conical singularities on 2-spheres
Subhadip Dey
Abstract
Suppose that θ , θ , . . . , θ n are positive numbers and n ≥
3. We want to know whether thereexists a spherical metric on S with n conical singularities of angles 2 πθ , πθ , . . . , πθ n . Asufficient condition was obtained by Gabriele Mondello and Dmitri Panov [6]. We show thattheir condition is also necessary when we assume that θ , θ , . . . , θ n N . A classical question in the theory of Riemann surfaces asks which real functions f on a Riemannsurface S are equal to the curvature of a pointwise conformal metric. For simplicity, all the surfacesconsidered in this section are orientable closed 2-manifolds. If S has genus ≥
2, then Berger [1]showed that any smooth negative function is the curvature of a unique conformal metric. In thecase of a Riemann surface of genus 1, i.e. when S is a torus with a flat metric g , Kazdan andWarner [4] proved that a function f (
0) : S → R is the curvature of a metric in the conformalclass of g if and only if f changes sign and satisfies R S f dA <
0, where dA is the area form g .These results have been generalized by Marc Troyanov in the case of surfaces with singularitiesof a special type, called conical singularities which we shall define below. Let S be a surface. A real divisor βββ on S is a formal sum βββ = β x + · · · + β n x n , where x i ∈ S are pairwise distinct and β i are real numbers. For the pair ( S, βββ ), define χ ( S, βββ ) = n X i =1 β i + χ ( S ) , called the Euler characteristic of (
S, βββ ).Suppose that βββ = β x + · · · + β n x n is a real divisor on S such that β i > −
1. Let g be aRiemannian metric on S defined away from x , . . . , x n such that each point x i has a neighborhood U i in S with coordinate z i satisfying z i ( x i ) = 0 on which g has the following form, ds = e u i | z i | β i | dz i | . Here u i : U i → R is a continuous function such that u i | U i −{ x i } is differentiable (of class at least C ). The point x i is called a conical singularitiy of the metric g of angle 2 π ( β i + 1). We refer tothis type of metrics g as (Riemannian) metrics with conical singularities . We say that the metric g (with conical singularities) represents the divisor βββ .In this terminology, Troyanov [8] proved the following two theorems.1 heorem 1 ([8]) . Let S be a Riemann surface with a real divisor βββ = β x + · · · + β n x n such that β i > − , i = 1 , . . . , n . If χ ( S, βββ ) < , then any smooth negative function on S is the curvature ofa unique conformal metric which represents βββ . Theorem 2 ([8]) . Let S be a Riemann surface with a real divisor βββ = β x + · · · + β n x n such that β i > − , i = 1 , . . . , n . If χ ( S, βββ ) = 0 , then any function f (
0) : S → R is the curvature of aconformal metric representing βββ if and only if f changes sign and satisfies R S f dA < , where dA is the area element of a conformally flat metric on S with singularities. Theorem 1 generalizes Berger’s result, and Theorem 2 generalizes Kazdan and Warner’s result.Restricting our attention only to constant curvature metrics with conical singularities, we formulatethe following question.
Question 1.
Let S be a Riemann surface with a real divisor βββ = β x + · · · + β n x n such that β i > − , i = 1 , . . . , n . Can βββ be represented by a conformal metric of constant curvature? The case when βββ = 0 is completely understood due to the classical uniformization theorems. If χ ( S ) ≥
0, then any conformal class has a representative of constant curvature. When χ ( S ) < βββ = 0, the Theorems 1 and 2 give complete understanding in the case when χ ( S, βββ ) ≤ χ ( S, βββ ) < S is the Riemann sphere, which has been leastunderstood. Given a real divisor βββ = β x + · · · + β n x n with β i > − i = 1 , . . . , n , does there exista conformal metric g with conical singularities on S which represents βββ ? If we further assume that g has constant curvature 1, then Gauss-Bonnet Theorem gives a restriction on βββ , namely χ ( S, βββ ) > . But further restrictions on βββ were found in addition to this in the case n = 2 ,
3. When n = 2, wesimply need to require that β = β by the work of Troyanov [7]. When n = 3, by Eremenko’sresult in [2], a conformal metric with conical singularities exists if and only if some inequalities onthe numbers β , β , β are satisfied. Moreover, in this case, there exists a spherical triangle withangles π ( β + 1) , π ( β + 1) , π ( β + 1). Here a spherical triangle means a topological closed disk witha metric of constant curvature 1 such that the boundary is a piecewise geodesic loop with threesingular points. Examples include geodesic triangles immersed in ordinary sphere S . A spherewith three conical singularities is the double of such a triangle. When n >
3, the situation becomesmore complicated.Therefore, before answering the Question 1 for higher n ’s, perhaps one needs to have a completelist real divisors βββ which may be represented by a spherical metric g on a sphere with conicalsingularities. Here spherical means that the metric has constant curvature 1. Note that the divisor βββ can be represented by a spherical metric with conical singularities if and only if there exists aspherical metric with n conical singularities of angles 2 π ( β + 1) , . . . , π ( β n + 1).Let R n + be the set of all points in R n with positive coordinates. A point θθθ = ( θ , . . . , θ n ) ∈ R n + iscalled admissible if there exists a sphere S with a spherical metric g with n conical points x , . . . , x n of angles 2 πθ , . . . , πθ n , respectively. In this case, the real divisor βββ = ( θ − x + · · · + ( θ n − x n is represented by g . By abuse of notation, we write θθθ −
111 = βββ , where 111 = (1 , . . . , uestion 2.
Which points θθθ ∈ R n + are admissible? A major progress was done to answer this question by Mondello and Panov [6], as stated in thenext two theorems.
Theorem 3 ([6]) . If θθθ ∈ R n + is admissible, then χ ( S, θθθ − > , (P) d ( Z no , θθθ − ≥ , (H) where d is the standard ℓ distance on R n and Z no is the set of all points vvv ∈ R n with integercoordinates such that d ( vvv, is odd. Moreover, if the equality in (H) is attained, then the holonomyof the a metric which corresponds to θθθ is coaxial. Here we clarify what we mean by saying that the holonomy of a metric with conical sigularitiesis coaxial. Note that a metric g on S with conical singularities is actually defined as a Riemannianmetric only on S − { conical sigularities } . Thus we have a holonomy representation of the metric g which is a homomorphism φ : π ( S − { conical sigularities } ) → SO (3) from the fundamental groupof S − { conical sigularities } to the group of rotations of the standard sphere S . The metric g is saidto have coaxial holonomy if the image of φ is contained in an one-parameter subgroup of SO (3).The following is a partial converse to Theorem 3. Theorem 4 ([6]) . If θθθ ∈ R n + satisfies the positivity constraints (P) and the holonomy constraints (H)strictly, then θθθ is admissible. Moreover, each metric corresponding to θθθ has non-coaxial holonomy. Theorems 3 and 4 classified all the admissible points which do not satisfy (H’), where (H’) isthe equality in (H), d ( Z no , θθθ − , (H’)but did not provide answer in the complementary case. We formulate this case in the followingquestion. Question 3.
For n ≥ , which points in R n + satisfying (P) and (H’) together are admissible? In Theorem 5, by restricting to the “non-integral” case, we give an answer to this question. Weshow that when θ , . . . , θ n N , the sufficient conditions in Theorem 4 are actually necessary.The “integral” case has been analyzed by Kapovich [3] who showed that a spherical metric withconical singularities of angles 2 πθ , . . . , πθ n , where θ , . . . , θ n ∈ N , exists if and only if the integers θ , . . . , θ n satisfy (H’) together with the polygon inequality( θ i − ≤ n X j =1 ( θ j − , for all 0 ≤ i ≤ n. A well-formulated collection of necessary and sufficient conditions on the complement of thesetwo cases (the “mixed” case) is unknown to the author.
Acknowledgement
I thank my advisor, Prof. Michael Kapovich, for introducing this problem to me. I am gratefulfor his constant encouragement, helpful comments and numerous discussions on this project. I alsothank an anonymous referee for commenting on an earlier version of this paper which helped toimprove the proof of the main result. 3
Main Result
The aim of this paper is to establish the following theorem.
Theorem 5.
Suppose that θθθ = ( θ , . . . , θ n ) ∈ R n + satisfies (P) and (H’) and n ≥ . If θ i N forall ≤ i ≤ n then θθθ is not admissible. The case n = 3 has already been treated in [2].By a singular spherical surface , we mean a surface, possibly with boundary, with a sphericalmetric g with a set X of conical singularities. The points in X are called singular points andthe points in S − ( ∂S ∪ X ) are called regular points . We further assume that the boundary, ifnonempty, is a union of smooth curves. A path γ in S is called a geodesic arc if it’s restriction tothe complement of X is a connected geodesic arc in the Riemannian sense. We allow geodesic arcsto have singular endpoints. If a geodesic arc has same initial and final point, we call it a geodesicloop . A composition of piecewise geodesic arcs is called a peicewise geodesic path . The metric g induces a distance function d S defined by d S ( x ′ , x ′′ ) = infimum over the length of piecewise geodesic paths connecting x ′ and x ′′ . We prove,
Theorem 6.
Suppose that S is a singular spherical surface without boundary with a discrete set ofconical singularities X . Suppose that the metric d S is complete. For x ∈ X , if S has no geodesicloop based at x of length shorter than π , then min x = x ′ ∈ X { d S ( x, x ′ ) } ≤ π. If the equality occurs, then S is compact and | X | ≤ . Combining Theorem 5 with Theorems 3 and 4, we get necessary and sufficient conditions on( θ , . . . , θ n ), provided θ i N , for which there exists a sphere S with n ≥ πθ , . . . , πθ n . The analogous case when n ≤ Theorem 7.
Suppose that θθθ = ( θ , . . . , θ n ) ∈ R n + where n ≥ . If θ i N , ≤ i ≤ n , then θθθ isadmissible if and only if χ ( S, θθθ − > ,d ( Z no , θθθ − > , where d is the standard ℓ distance on R n and Z no is the set of all points vvv ∈ R n with integercoordinates such that d ( vvv, is odd. Let Σ be a closed, genus zero, singular spherical surface with two conical singularities y and y ′ of angle θ . Such a surface has a shape of a football, which we call a “Troyanov’s football”. Wedenote the closed r -neighborhood of y by Σ r . For small r , Σ r is a model neighborhood of a conicalsingularity of angle θ . Note that when r ≥ π , Σ r = Σ.4 efinition 8. Let f : S → S be a map between two singular spherical surfaces. Such a map f is called a locally isometric map (or a local isometry ) between these surfaces if the followingconditions are satisfied.1. f maps the singular (resp. regular) points of S to the singular (resp. regular) points of S .2. f is a local isometry on S − ( ∂S ∪ X ) in the Riemannian sense.3. For each boundary point s ∈ ∂S , there exists a neighborhood U of s , a neighborhood V of s , isometries φ : U → S , φ : V → S and ˆ f ∈ Isom( S ) such that the following squarecommutes: U V S S fφ φ ˆ f f is called an isometric embedding if it is an injective locally isometric map.Let S be a singular spherical surface without boundary with a discrete set X of conical singu-larities such that the underlying metric space structure of S is complete. Let x ∈ X be a singularpoint. We define l x = inf { l ′ > | x can be connected to a point in X by a geodesic arc of length l ′ } . Note that in the definition of l x , we also allow geodesic loops based at x . If S has has only onesingular point x and no geodesic loops based at x , i.e. when the definition of l x given above becomesvoid, then we make the convention l x = π .Let Σ be a Troyanov’s football with singular points y and y ′ of angles equal to the angle of S at x . Our goal is the following: We show that, for any r < l x , there exists a local isometry f : Σ r → S which sends y to x . When l x ≥ π , we argue that S can have at most two conical singular points. Proposition 9.
There exists a family of local isometries f r : Σ r → S , f r ( y ) = x , indexed by r ∈ [0 , min { l x , π } ) , such that, for s > r , f s | Σ r = f r .Proof. Let L = min { l x , π } . We start by showing that the set R = { r ∈ [0 , L ) | ∃ a local isometry f : Σ r → S such that f ( y ) = x } is equal to [0 , L ).We prove that R is a closed subset of [0 , L ). Let ( r k ) k ∈ N be an increasing sequence in R whichconverges to some number r in [0 , L ). Let f r k : Σ r k → S , f r k ( y ) = x , be a sequence of localisometries indexed by k ∈ N . We can extend the domain of f r k to Σ r by keeping f r k constantalong the geodesics in Σ r − int (Σ r k ) orthogonal to the boundary ∂ Σ r k . Therefore, we have anequicontinuous family of maps f r k : Σ r → S , for k ∈ N . Using Arzel´a-Ascoli Theorem, by passingto a subsequence we can assume that the sequence ( f r k ) converges to a limit f : Σ r → S . In thefollowing, we prove that f is a local isometry which in turn proves that R is closed.We first prove that f is a local isometry in the interior of Σ r . We take any point p = y inthe interior of Σ r . Let q = lim k →∞ f r k ( p ) ∈ S . Since 0 < d S ( x, q ) < L , q is a regular point of S ,and hence, there exists π/ > δ > δ -neighborhood V of q in S is a spherical disk5f radius δ . We can choose k ∈ N and an δ/ U of p in Σ such that, for k ≥ k , U ⊂ Σ r k and d S ( q, f r k ( p )) < δ/
2. We may also assume that U is isometric a spherical disk. It isclear that, for k ≥ k , U maps into V under f r k . Since V has radius < π/ V does not containany closed geodesic. Moreover, U is geodesically convex. Therefore, f r k | U is an isometry onto it’simage. As a result, f | U is an isometric embedding into S .We now show that f is a local isometry at any boundary point c ∈ ∂ Σ r . The boundary C r = ∂ Σ r is a round circle in Σ. Let c ∈ C and U be an δ -neighborhood of c in Σ, for some δ < min { r, L − r } .Note that f ( c ) is a regular point because we assumed that a geodesic from x i to any singular pointhas length at least l x > r . Let V be the δ -neighborhood of f ( c ) in S . We can also assume that U and V are spherical disks with centers c and f ( c ) respectively. From above, we know that f maps U = U ∩ Σ r − ∂ Σ r locally isometrically into V . We show that f | U is an isometric embedding.If U is geodesically convex, which happens precisely when r < π/
2, this follows as before. Butthis argument fails when r > π/
2. In this case, let f ( t ) = f ( t ), for some t , t in U . We canfind a third point t in U such that, for i = 1 , t i and t can be connected by an (unnormalized)geodesic segment γ i : [0 , → U , γ i (0) = t , γ i (1) = t i . The geodesics f ◦ γ and f ◦ γ in V bothhave same initial and final points. Since df is injective at all points of U , γ ′ (0) = γ ′ (0) which thenimplies, after suitable renormalization, that either γ ⊂ γ or γ ⊃ γ . Therefore, t and t must bejoined by a geodesic segment in U , say γ . But then f ◦ γ is a closed geodesic in S which forces t = t because V has radius at most π/
2. As a result, f | U is injective i.e. an embedding. In thiscase, f | U can be extended to an embedding of U in V i.e. f is a local isometry at c . Thus R isclosed in [0 , L ).Next we show that R is also an open subset of [0 , L ). It is clear that if r ∈ R , then the interval[0 , r ] is also a subset of R . If for all r ∈ R there exists some s > r such that s ∈ R , then r is aninterior point of R . This shows that R is an open subset of [0 , L ). Lemma 10.
Let f : Σ r → S be a local isometry such that f ( y ) = x . Then, there exists some s > r and a local isometry ˜ f : Σ s → S such that ˜ f | Σ r = f .Proof. Let c ∈ C r be any point in the boundary C r = ∂ Σ r . Since f is a locally isometry, thereexists a neighborhood B of c in Σ and an embedding f : B → S such that f | B ∩ Σ r = f | B ∩ Σ r . Themaps f and f patch together to extend f on the larger domain Σ r ∪ B and such an extension isunique because our maps are analytic.Let c , . . . , c p be points on C r with the following properties: (i) Each point c j has a neighborhood B j in Σ such that f can be extended to a local isometry ˜ f j : B j ∪ Σ r → S . (ii) S pj =1 B j coversthe circle C r . (iii) ( B j ∩ B i ) ∪ Σ r is connected. For example, we could choose B j to be a smalldisk centered at c j . Clearly, (ii) implies that Σ r ∪ S pj =1 B j contains Σ s , for some s > r . Moreover,the uniqueness of the extensions ˜ f j and, (iii) ensure that they can be patched together to form anextension ˜ f : Σ r ∪ S pj =1 B j → S of f . Then ˜ f | Σ s is an extension of f .The lemma combined with the fact that R is closed in [0 , L ) shows that any given local isometry f r : Σ r → S , sending y to x , can be extended to a local isometry f s : Σ r → S , for any L > s > r .Therefore, given a local isometry f r : Σ r → S sending y to x , for 0 < r < L , we have a sequenceof local isometries f r : Σ r → S , f r ( y ) = x , indexed by r ∈ [0 , L ) such that f s | Σ r = f r , for s ≥ r .This completes the proof of the proposition. Remark . The family of local isometries in the proposition may not be replaced by a family ofisometries, as shown in the following example: 6 xample 12.
Let D be the closed upper hemisphere of the standard sphere S . The boundary of D is a geodesic. Let x , x , x , x be evenly distributed points on the boundary as shown in thefigure below. By identifying the directed edges a with c and b with d , we obtain a torus T with a x x x x abcd D spherical metric having only one conical singularity x of angle 4 π , and in this case, l x = π/ x has length π/
2. Note that a local isometry f r : Σ r → S failsto be an injective map when r > π/ l x ≥ π . Let ˙Σ denote the punctured sphere Σ − { y } . UsingProposition 9, we have a family of local isometries f r : Σ r → S , f r ( y ) = x , indexed by r ∈ [0 , min { l x , π } ), such that, for s > r , f s | Σ r = f r . We construct a map˙ F : ˙Σ → S by setting ˙ F ( z ) = f r ( z ), where r = d Σ ( y, z ). It is clear that ˙ F is a local isometry, and hence, aLipschitz continuous function with Lipschitz constant 1. Since the codomain of ˙ F is a completemetric space, ˙ F can be extended to a map F : Σ → S . Proposition 13. F : Σ → S constructed above is a surjective map.Proof. Since F | Σ −{ y,y ′ } is a local isometry, image S ′ of Σ − { y, y ′ } under F is open in S . Adjoiningthe points x = F ( y ) and F ( y ′ ) compactifies S ′ . F ( y ) = x is an interior point of the image F (Σ).We prove that F ( y ′ ) is also an interior point of the image F (Σ). Let D be a neighborhood of F ( y ′ ),homeomorphic to R . D ∩ S ′ is a nonempty open set in D which becomes a closed subset whenwe adjoin F ( y ′ ). So, D ∩ S ′ and complement of ( D ∩ S ′ ) ∪ { F ( y ′ ) } are disjoint open subsets whoseunion is D − { F ( y ′ ) } . Since D − { F ( y ′ ) } is connected, the complement of ( D ∩ S ′ ) ∪ { F ( y ′ ) } mustbe empty, i.e. D ∩ S ′ ⊂ S ′ . This means that F (Σ) contains D .Hence, the image F (Σ) is open in S . Since S is connected, F (Σ) = S .We close this section by completing the proof of Theorem 6. Proof of Theorem 6.
Suppose that S has no geodesic loop based at x of length ≤ π . If min x = x ′ ∈ X { d S ( x, x ′ ) }≥ π , then l x ≥ π . The image F ( ˙Σ) constructed above is a regular open subset of S , i.e. it containsno conical singular point of S . From Proposition 13, this image misses at most two points of S ,namely, F ( x ) and F ( x ′ ). Therefore, S can have at most two singular points. Compactness of S follows from the surjectivity of F . 7 Proof of Theorem 5
Let S be a closed, genus zero surface with a spherical metric g with n conical singularities x , . . . , x n of angles 2 πθ , . . . , πθ n respectively, where all θ i ’s are non-integers and the tuple ( θ , . . . , θ n )satisfies (H’). An important feature of the metric g on S is that the holonomy representation iscoaxial, which follows from Theorem 3. Proposition 14.
Let γ be a geodesic arc in S with singular endpoints. Then the length of γ is anintegral multiple of π . In particular, for i ≥ , d S ( x , x i ) ≥ π and S has no geodesic loop based at x of length shorter than π .Proof. We shall use Lemma 2.10 of [6] which states the following.
Lemma 15.
Let S ′ be a sphere with a spherical metric with conical singularities having coaxialholonomy. Suppose that γ ′ is a geodesic arc from x ′ to x ′ , where x ′ and x ′ are distinct conicalsingularities of angles π N . Then the length of γ ′ is an integral multiple of π . The lemma shows that the length of γ is an integral multiple of π if γ has distinct endpoints.So, we can assume that γ is a geodesic loop based at x . The complement of γ in S is comprisedof connected components among which there are exactly two components, say U and U , whoseboundary contains the point x . For i = 1 ,
2, let y i ∈ U i be a regular point. Let ρ : ˜ S → S bethe two-fold branched covering, branched at the points y and y , where ˜ S is a closed, genus zerosurface. By pulling back the metric of S , ˜ S gets a natural spherical metric with a set of conicalsingularities ρ − ( X ∪ { y , y } ) where X = { x , . . . , x n } . Moreover, the holonomy of ˜ S is coaxialwhich follows from holonomy representation π ( ˜ S − singularities) π ( S − X ∪ { y , y } ) SO (3) , ρ ∗ φ where φ : π ( S − X ∪ { y , y } ) → SO (3) is the holonomy representation associated to S withconical singularities y , y , x , x . . . , x n . Here we treat the regular points y and y of S as singularpoints of angle 2 π . The loop γ can be lifted to a geodesic path ˜ γ with distinct endpoints which areprecisely the pre-images of x . We havelength of γ = length of ˜ γ, and the later is an integral multiple of π which follows from the lemma. Proof of Theorem 5.
Suppose that, for n ≥ θθθ = ( θ , . . . , θ n ) ∈ R n + − Z n satisfies (P) and (H’). If θθθ is admissible, then there is a sphere with a spherical metric with conical singularities x , . . . , x n such that the angle at x i of g is 2 πθ i . By Proposition 14, d S ( x , x i ) ≥ π , for i ≥
1, and there is nogeodesic loop based at x of length shorter than π . Using Theorem 6, we get | X | ≤
2. This is acontradiction.
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Department of MathematicsUniversity of California, Davis1 Shields Ave, Davis, CA 95616
E-mail address : [email protected]@math.ucdavis.edu