Metrics with four conic singularities and spherical quadrilaterals

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1 Metrics with four conic singulrities nd sphericl qudrilterls Alexndre Eremenko, Andrei Gbrielov nd Vitly Trsov July 7, 2015 Abstrct A sphericl qudrilterl is bordered surfce homeomorphic to closed disk, with four distinguished boundry points clled corners, equipped with Riemnnin metric of constnt curvture 1, except t the corners, nd such tht the boundry rcs between the corners re geodesic. We discuss the problem of clssifiction of these qudrilterls nd perform the clssifiction up to isometry in the cse tht two ngles t the corners re multiples of π. The problem is equivlent to clssifiction of Heun s equtions with rel prmeters nd unitry monodromy. MSC 2010: 30C20,34M03. Keywords: surfces of positive curvture, conic singulrities, Heun eqution, Schwrz eqution, ccessory prmeters, conforml mpping, circulr polygons 1 Introduction Let S be compct Riemnn surfce, nd 0,..., n 1 finite set of points on S. Let us consider conforml Riemnnin metric on S of constnt curvture K {0, 1, 1} with conic singulrities t the points j. This mens tht in Supported by NSF grnt DMS Supported by NSF grnt DMS

2 locl conforml coordinte z the length element of the metric is given by the formul ds = ρ(z) dz, where ρ is solution of the differentil eqution log ρ + Kρ 2 = 0 in S\{ 0,..., n 1 }, (1.1) nd ρ(z) z α j 1, for the locl coordinte z which is equl to 0 t j. Here α j > 0, nd 2πα j is the totl ngle round the singulrity j. The generl problem is how mny such metrics exist with prescribed j nd α j. This question goes bck to 19-s century, nd for the history we refer to [41, 16]. A complete nswer to this question is known when K 0, [34, 35, 36, 37, 22, 48], but very little is known on the cse K > 0. One necessry condition tht one hs to impose on these dt follows from the Guss Bonnet theorem: the quntity n 1 χ(s) + (α j 1) hs the sme sign s K. (1.2) j=0 Here χ is the Euler chrcteristic. Indeed, this quntity multiplied by 2π is equl to the integrl curvture of the smooth prt of the surfce. The result of Troynov [48] tht pplies to the cse K = 1 is the following: Let S be compct Riemnn surfce, 0,..., n 1 points on S, nd α 0,...,α n 1 positive numbers stisfying n 1 0 < χ(s) + (α j 1) < 2 min{1, min j}. 0 j n 1 (1.3) j=0 Then there exists conforml metric of positive curvture 1 on S with conic singulrities t j nd ngles 2πα j. F. Luo nd G. Tin [30] proved tht if the condition 0 < α j < 1 is stisfied, then (1.3) is necessry nd sufficient, nd the metric with given j nd α j is unique. In generl, the right hnd side inequlity in (1.3) is not necessry condition, nd the metric my not be unique. In this pper, we only consider the simplest cse when S is the sphere, so χ(s) = 2. For metrics on tori we refer to the recent work [6, 7]. 2

3 The problem of description nd clssifiction of conforml metrics of curvture 1 with conic singulrities on the sphere hs pplictions to the study of certin surfces of constnt men curvture [19, 9, 10], nd to severl other questions of geometry nd physics [39, 47]. The so-clled symmetric cse is interesting nd importnt. Suppose tht ll singulrities belong to circle on the sphere S, nd we only consider the metrics which re symmetric with respect to this circle. Then the circle splits S into two symmetric disks. Ech of them is sphericl polygon, (surfce) for which we stte forml definition: Definition 1.1 A sphericl n-gon is closed disk with n distinguished boundry points j clled the corners, equipped with conforml Riemnnin metric of constnt curvture 1 everywhere except the corners, nd such tht the sides (boundry rcs between the corners) re geodesic. The metric hs conic singulrities t the corners. In [50, 11], ll possibilities for sphericl tringles re completely described, see lso [19] where minor error in [11, Theorem 2] ws corrected. In the cse of tringles, the metric is uniquely determined by the ngles when none of the α j is n integer. The cse when ll α j re integers, nd n is rbitrry, is lso well understood. In this cse, the line element of the metric hs the globl representtion ds = 2 f dz 1 + f 2, where f is rtionl function. The singulr points j re the criticl points of f, α j 1 is the multiplicity of the criticl point j, nd α j is the locl degree of f t j. Thus the problem with ll integer α j is equivlent to describing rtionl functions with prescribed criticl points [21, 42, 12, 13, 14, 15, 18, 32]. Almost nothing is known in the cse when some of the α j re not integers, the number of singulrities is greter thn 3, nd the right-hnd side inequlity in (1.3) is violted. 1 In this pper we begin investigtion of the cse n = 4, with the emphsis on the symmetric cse. 1 After the first version of this work ws posted on the rxv, preprint [31] ppered, where the uthors find the generl necessry nd sufficient conditions on the ngles α j of metric on the sphere with some singulrities j. 3

4 In the symmetric cse, we my ssume without loss of generlity tht the circle is the rel line R { }. The rel line splits the sphere S into two symmetric sphericl n-gons with the rel corners j nd the ngles πα j t the corners. Thus we rrive t the problem of clssifiction of sphericl qudrilterls (surfces). We study this problem using two different methods. One of them is the geometric method of F. Klein [28] who clssified sphericl tringles. Klein clssified not only sphericl tringles with geodesic sides, but lso circulr tringles, whose sides hve constnt geodesic curvture (tht is, loclly they re rcs of circles). A modern pper which uses Klein s pproch to tringles is [52]. Clssifiction of tringles permitted Klein to obtin exct reltions for the numbers of zeros of hypergeometric functions on the intervls between the singulr points on the rel line. Vn Vleck [51] extended this pproch of Klein, using the sme geometric method, nd obtined exct inequlities for the numbers of zeros of hypergeometric functions in the upper nd lower hlf-plnes. Hurwitz [25] re-proved these results with different, nlytic method. We hope tht our results cn be used to obtin informtion bout solutions of Heun s eqution, in the sme wy s Klein obtined informtion bout solutions of the hypergeometric eqution. Klein s clssifiction of tringles ws prtilly extended to rbitrry circulr qudrilterls (not necessry geodesic ones) in the work of Schönflies [43, 44] nd Ihlenburg [26, 27]. They considered certin geometric reduction process of cutting circulr qudrilterl into simpler ones. Then they clssified the irreducible qudrilterls up to conforml utomorphisms of the sphere. Thus they obtined n lgorithm which permits to construct ll circulr qudrilterls. Using this lgorithm, Ihlenburg derived reltions between the ngles nd sides of circulr qudrilterl. However this lgorithm flls short of complete clssifiction. In prticulr, qudrilterl with prescribed ngles nd sides is not unique. Moreover, it seems difficult to single out geodesic qudrilterls in the construction of Schönflies nd Ihlenburg. We use somewht different pproch which consists in ssociting to every sphericl geodesic qudrilterl combintoril object which we cll net, thus reducing the clssifiction to combintorics. Then we solve this combintoril problem nd obtin clssifiction of sphericl qudrilterls up to isometry. Our pproch cn be lso pplied to generl (non-geodesic) 4

5 circulr qudrilterls. The boundry of sphericl polygon is closed curve on the sphere, consisting of geodesic pieces. The ngles of such curve t the corners mke sense only modulo 2π. All possible sequences of ngles hve been described by Bisws [3], see lso [5]. These inequlities on the ngles give necessry but not sufficient conditions tht the ngles of sphericl polygon (surfce) must stisfy. Our second method is direct study of Heun s eqution. Clssifiction of sphericl qudrilterls cn be stted in terms of specil eigenvlue problem for this eqution. This method leds to complete results when the eigenvlue problem cn be solved lgebriclly. The contents of the pper is the following. In section 2, we recll the connection of the problem with Heun s eqution, nd recll the results on Heun s eqution relted to our problem. In section 3 we begin the study of the cse when two of the α j re integers nd two others re not. (If three of the α j re integers then ll four must be integers.) Complete clssifiction for this cse is obtined in the remining sections. Sections 3-5 re bsed on direct study of the eigenvlue problem for Heun s eqution. The results re illustrted with numericl exmples in section 16. Sections 6-14 re bsed on geometric nd combintoril methods. The cse when three ngles re non-integers is treted in [17]. 2 Connection with liner differentil equtions Let (S,ds) be the Riemnn sphere equipped with metric with conic singulrities. Every smooth point of S hs neighborhood which is isometric to region on the stndrd unit sphere S; let f be such n isometry. Then f hs n nlytic continution long every pth in S\{ 0,..., n 1 }, nd we obtin multi-vlued function which is clled the developing mp. The monodromy of f consists of orienttion-preserving isometries (rottions) of S, so the Schwrzin derivtive is single vlued function. F(z) := f f 3 2 ( ) f 2 (2.1) f 5

6 Developing mp is completely chrcterized by the properties tht it hs n nlytic continution long ny curve in S\{ 1,..., n }, hs symptotics c(z j ) α j, z j, c 0, nd hs PSU(2) = SO(3) monodromy. It is possible tht two such mps with the sme j nd α j re relted by postcomposition with frctionl-liner trnsformtion. The metrics rising from such mps will be clled equivlent. Following [19], we sy tht the metric is reducible if its monodromy group is commuttive (which is equivlent to ll monodromy trnsformtions hving common fixed point). In the cse of irreducible metrics, ech equivlence clss contins only one metric. For reducible metrics, the equivlence clss is one-prmeter fmily when the monodromy is non-trivil nd two-prmetric fmily when monodromy is trivil. The symptotic behvior of f t the singulr points j implies tht the only singulrities of F on the sphere re double poles, so F is rtionl function, nd we obtin the Schwrz differentil eqution (2.1) for f. It is well-known tht the generl solution of the Schwrz differentil eqution is rtio of two linerly independent solutions of the liner differentil eqution y + Py + Qy = 0, f = y 1 /y 0, (2.2) where F = P P 2 /2 + 2Q. For exmple one cn tke P = 0, then Q = F/2. Another convenient choice is to mke ll poles but one of P nd Q simple. When n = 3, eqution (2.2) is equivlent to the hypergeometric eqution, nd when n = 4 to Heun s eqution [40]. The singulr points j of the metric re the singulr points of the eqution (2.2). These singulr points re regulr, nd to ech point correspond two exponents α j > α j, so tht α j = α j α j. If α j is n integer for some j, we hve n dditionl condition of the bsence of logrithms in the forml solution of (2.2) ner j. It is esy to write down the generl form of Fuchsin eqution with prescribed singulrities nd prescribed exponents t the singulrities. After normliztion, n 3 prmeters remin, the so-clled ccessory prmeters. To obtin conforml metric of curvture 1, one hs to choose these ccessory prmeters in such wy tht the monodromy group of the eqution is conjugte to subgroup of PSU(2). 6

7 By frctionl-liner chnge of the independent vrible, one cn plce one singulr point t. Then, mking chnges of the vrible y(z) y(z)(z j ) β j, one cn ssume tht the smller exponent t ech finite singulr point is 0, see [40]. For the cse of four singulrities 0,..., 3, where 3 =, we thus obtin Heun s eqution in the stndrd form where y + ( 2 j=0 1 α j z j ) A = α α, y + Az λ y = 0, (2.3) (z 0 )(z 1 )(z 2 ) 2 α j + α + α = 2. (2.4) j=0 Here the exponents t the singulr points re described by the Riemnn symbol P α ;z α 0 α 1 α 2 α. The first line lists the singulrities, the second the smller exponents, nd the third the lrger exponents. So the ngle t infinity is α 3 = α α. The ccessory prmeter is λ. Solving the second eqution (2.4) together with α α = α 3, we obtin α = 1 2 (2 + α 3 α 0 α 1 α 2 ) (2.5) nd α = 1 2 (2 α 3 α 0 α 1 α 2 ). The question of the existence of sphericl qudrilterl with given corners 0, 1, 2, nd given ngles πα j, 0 j 3, is equivlent to the following: when one cn choose rel λ so tht the monodromy group of Heun s eqution (2.3) is conjugte to subgroup of PSU(2)? The necessry condition (1.2) cn be restted for the eqution (2.3) s α < 0. (2.6) We lso hve A = α α = α (α 3 + α ) 7

8 by simple computtion. One cn write (2.3) in severl other forms. Assuming tht ll j re rel, we hve the Sturm-Liouville form: ( 2 ) d x j 1 α j y + dx j=0 ( 2 ) (Ax λ) sgn j=0 (x j) 2 j=0 x y = 0. (2.7) j α j Sometimes the Schrödinger form is more convenient: ( ) y λ αk 2 1 y ( k j ) 4 x 3 k j=0 (x = 0, (2.8) j) k=0 j k where ll four singulrities re in the finite prt of the plne. The exponents in the Schrödinger form re (1 ± α j )/2. The potentil in (2.8) is F/2, where F is the Schwrzin (2.1). A question similr to our problem ws investigted in [29, 23, 24, 45, 46]: when cn one choose the ccessory prmeter so tht the monodromy group of Heun s eqution preserves circle? All these uthors consider the problem under the ssumption 0 α j < 1, for 0 j 3. (2.9) The most comprehensive tretment of this problem is in Smirnov s thesis [45]. Smirnov proved tht for ll sets of dt stisfying (2.9), there exists sequence of vlues of the ccessory prmeter λ = λ k, k = 0, ±1, ±2,... such tht the monodromy group of the eqution hs n invrint circle. Ech of the two opposite sides of the corresponding qudrilterl covers circle k times, nd the other two sides re proper subsets of their corresponding circles. The problem of choosing the ccessory prmeter so tht the monodromy group is conjugte to subgroup in PSU(2) is discussed in [10]. However ll results of tht pper re lso proved only under the ssumption (2.9). Assumption (2.9) seems to be essentil for the methods of Klein [29], Hilb, Smirnov nd Dorfmeister. 8

9 3 The cse n = 4 with two integer corners: condition on the ngles In the rest of the pper, we study the cse n = 4 with two integer α j. We nswer the following questions: ) In the eqution (2.3), for which α j one cn choose λ so tht the monodromy group is conjugte to subgroup of PSU(2)? b) If α j stisfy ), how mny choices of λ re possible? c) If, in ddition, ll j re rel, how mny choices of rel λ re possible? One cnnot hve exctly one non-integer α j. Indeed, in this cse the developing mp f will hve just one brnching point on the sphere, which is impossible by the Monodromy Theorem. Let us consider the cse of two non-integer α j. In this section we obtin necessry nd sufficient condition on the ngles for this cse, tht is, solve the problem ). We plce the two singulrities corresponding to non-integer α t 0 = 0 nd 3 =, nd let the totl ngles t these points be 2πα 0 nd 2πα 3, where α 0 nd α 3 re not integers. Then the developing mp hs n nlytic continution in C from which we conclude tht the monodromy group must be cyclic group generted by rottion z ze 2πiα, with some α (0, 1). This mens tht f(z) is multiplied by e 2πiα when z describes simple loop round the origin. Thus g(z) = z α f(z) is single vlued function with t most power growth t 0 nd. Then we hve representtion f(z) = z α g(z), where g is rtionl function. Then α 0 = k+α, α 3 = j+α, where k nd j re integers, so either α 0 α 3 or α 0 + α 3 is n integer. The ngles 2πα 1 nd 2πα 2 t the other two singulr points 1 nd 2 of the metric re integer multiples of 2π, nd they re the criticl points of f other thn 0 nd. Let g = P/Q where P nd Q re polynomils without common zeros of degrees p nd q, respectively. Let p 0 nd q 0 be the multiplicities of zeros of P nd Q t 0. Then min{p 0,q 0 } = 0, becuse the frction P/Q is irreducible. The eqution for the criticl points of f is the following: z(p (z)q(z) P(z)Q (z)) + αp(z)q(z) = 0. (3.1) 9

10 Since α 1 nd α 2 re integers, we hve the following system of equtions: α 0 = p 0 q 0 + α, α 1 + α 2 2 = p + q mx{p 0,q 0 }, (3.2) α 3 = p q + α. The first nd the lst equtions follow immeditely from the representtion f(z) = z α P(z)/Q(z) of the developing mp. The second eqution holds becuse the left-hnd side of (3.1) is polynomil of degree exctly p + q, therefore the sum of the multiplicities of its zeros 1 nd 2 must be p + q mx{p 0,q 0 } Solving this system of equtions (3.2) in non-negtive integers stisfying min{p 0,q 0 } = 0, p 0 p, q 0 q, we obtin the necessry nd sufficient conditions the ngles should stisfy, which we stte s Theorem 3.1 Suppose tht four points 0,..., 3 on the Riemnn sphere nd numbers α j > 0, 0 j 3, re such tht α 1 nd α 2 re integers 2. The necessry nd sufficient conditions for the existence of metric of curvture 1 on the sphere, with conic singulrities t j nd ngles 2πα j re the following: ) If α 1 + α 2 + [α 0 ] + [α 3 ] is even, then α 0 α 3 is n integer, nd α 0 α α 1 + α 2. (3.3) b) If α 1 + α 2 + [α 0 ] + [α 3 ] is odd, then α 0 + α 3 is n integer, nd α 0 + α α 1 + α 2. (3.4) Sketch of the proof. For complete proof see [16]. Conditions ) nd b) re necessry nd sufficient for the existence of unique solution p,q,p 0,q 0,α of the system (3.2) stisfying min{p 0,q 0 } = 0, p 0 p, q 0 q, α (0, 1). Thus the necessity of these conditions follows from our rguments bove. We my ssume without loss of generlity tht 0 = 0, 3 =, 1 = 1 nd 2 = C. Then we set R(z) = z mx{p 0,q 0 } (z 1) α 1 (z ) α 2. The second eqution in (3.2) gives deg R = p + q. Now we consider the eqution z(p Q PQ ) + αpq = R. (3.5) 10

11 This eqution must be solved in polynomils P nd Q of degrees p nd q hving zeros of multiplicities p 0 nd q 0 t 0. Non-zero polynomils of degree t most p modulo proportionlity cn be identified with the points of the complex projective spce P p. The mp W α : P p P q P p+q, (P,Q) z(p Q PQ ) + αpq is well defined. It is finite mp between compct lgebric vrieties, nd it cn be represented s liner projection of the Veronese vriety. Its degree is equl to the degree ( ) p + q (3.6) p of the Veronese vriety. Thus the eqution (3.5) lwys hs complex solution (P,Q). The function f = z α P/Q is then developing mp with the required properties. So conditions ) nd b) re sufficient. This completes the proof. It will be convenient to introduce new prmeters insted of α j. Besides other dvntges, we eliminte the dditionl prmeter A, (see (2.3), (2.4)), nd the new prmeters llow us to tret the cses ) nd b) in Theorem 3.1 simultneously. We will rewrite (2.3) s z(z 1)(z ) ( y ( σ z + m z 1 + n z where κ is n integer, σ R is not n integer, ) y ) +κ(σ+1+m+n κ)zy = λy, (3.7) [σ] 1 in Cse ), nd [σ] < 1 in Cse b). (3.8) To chieve this we put m = min{α 1,α 2 } 1, nd n = mx{α 1,α 2 } 1. In cse ), we set σ = min{α 0,α 3 } 1 nd define κ by In cse b), we set nd define κ by 2κ = α 0 α 3 + α 1 + α 2 2. (3.9) σ = min{α 0,α 3 } 1 2κ = α 0 α 3 + α 1 + α 2 2. (3.10) 11

12 In both cses κ is n integer becuse the sums in the right-hnd sides of (3.9) nd (3.10) re even. Inequlities (3.3) nd (3.4) give tht κ 0 in both cses. Since m + n = α 1 + α 2 2, we lso get Furthermore in cse b), 2κ m + n. (3.11) σ + 1 κ (m + n)/2 (3.12) becuse α 0 + α 3 2 min{α 0,α 3 }. Notice tht in cse ), inequlity (3.12) holds trivilly becuse σ + 1 > 0. To summrize, the new prmeters re three integers m,n,κ, nd one rel non-integer number σ, subject to the conditions (3.12) nd Prmeters α j re recovered by the formuls 0 m n, 0 2κ m + n, (3.13) (α 0,α 3 ) = ( σ + 1, m + n + σ + 1 2κ ), (α 1,α 2 ) = (m + 1,n + 1), up to permuttion of α 1 nd α 2, nd permuttion of α 0 nd α 3, nd Heun s eqution is s (3.7). 4 Counting solutions In most cses, there is no uniqueness in Theorem 3.1. In this section we determine the number of equivlence clsses of metrics for given j nd α j, ssuming tht two of the α j re integers. As explined in the previous section, in this cse the Heun eqution hs polynomil solution, nd solution of the form z α P, where P is polynomil. We cll functions of this lst type qusipolynomils. Substituting forml power series H(z) = h(s)z s s Z+β to the eqution (3.7), we obtin recurrence reltions of the form c s 1 h(s 1) + s h(s) + b s h(s + 1) = 0, (4.1) 12

13 which cn be visulized s multipliction of the vector (h(s)) by Jcobi (three-digonl) mtrix c s 1 s b s c s s+1 b s c s+1 s+2 b s (4.2) The explicit expressions re b s = (s + 1)(s σ), (4.3) s = s (( + 1)(s 1 σ) m n) λ, (4.4) c s = (s κ)(s + κ σ m n 1). (4.5) We see tht n eigenvector h(s) cn be finitely supported sequence, sy with support [s 1,s 2 ], if nd only if b s1 1 = 0 nd c s2 = 0. If b s1 1 = 0, nd b s 0 for s > s 1, the elements h(s) cn be defined recursively from (4.1), with rbitrry non-zero vlue of h(s 1 ). Ech h(s) is polynomil in λ of degree s s 1, nd the condition of termintion of the sequence t the plce s 2 is c s2 1h(s 2 1) + s2 h(s 2 ) = 0. (4.6) This is polynomil eqution of degree s 2 s 1 +1 in λ which is the condition of hving polynomil solution of degree d = s 2 s 1. Similr condition gives the existence of qusipolynomil solution. Solutions of (4.6) re eigenvlues of (d + 1) (d + 1) Jcobi mtrix obtined by truncting the infinite mtrix (4.2) by leving rows nd columns with indexes from s 1 to s 2. Substituting forml power series H(z) = h(s)(z 1) s s=0 to the eqution (3.7) we obtin nother recurrence reltion of the form (4.1) with pproprite coefficients s,b s nd c s. Since the exponents of (3.7) t the point 1 re 0 nd m + 1, we cn lwys find holomorphic function H whose power series begins with the term (z 1) m+1. But to find power series solution beginning with constnt term, the condition c m 1 h(m 1) + m h(m) = 0 (4.7) 13

14 must be stisfied, nd this is polynomil eqution of degree d = m + 1 in λ. If the condition (4.7) is stisfied, then h(m + 1) cn be chosen rbitrrily. Eqution (4.7) is the trivility condition of the monodromy round 1. Solutions of (4.7) re eigenvlues of the d d Jcobi mtrix obtined by tking rows nd columns of (4.2) with indices from 0 to m. Thus we hve four polynomil conditions which re necessry for Heun s eqution (3.7) to hve two solutions: polynomil nd qusipolynomil. (i) C 1 (λ) = 0 iff there exists polynomil solution, (ii) C 2 (λ) = 0 iff there exists qusipolynomil solution, (iii) C 3 (λ) = 0 iff the monodromy t 1 is trivil, nd (iv) C 4 (λ) = 0 iff the monodromy t is trivil. The degrees of these equtions re: deg C 1 = κ+1, deg C 2 = m+n κ+1, deg C 3 = m+1, deg C 4 = n+1. The first two formuls follow by setting c s2 = 0 in (4.5), nd for the other two one hs to rewrite (3.7) to plce singulr point with integer exponents t 0, nd write the formul for b(s) for this trnsformed eqution (see (4.8) nd (5.3) below). Thus the number of vlues of λ for which ll four polynomils C i vnish is t most min{κ + 1,m + 1,n + 1}, where we used (3.11). Expressing this in terms of the originl exponents α j with the help of (3.9) nd (3.10) we obtin Theorem 4.1 The number of clsses of metrics with prescribed ngles 2πα j t the given points j is t most min{α 1,α 2,κ + 1}, where κ is defined by (3.9), (3.10), (α 1 + α 2 α 0 α 3 )/2 in cse ), κ + 1 = (α 1 + α 2 α 0 α 3 )/2 in cse b). We will lter see tht equlity holds for generic. The crucil fct is 14

15 Proposition 4.2 Of the four polynomils C j, 1 j 4, the polynomil of the smllest degree divides ech of the other three polynomils. Proof. We write Heun s eqution in the form (3.7). 1. Suppose tht C 1 is the polynomil of the smllest degree κ + 1. For every root λ of C 1, we hve polynomil solution p of degree t most κ. So p cnnot hve zero of order α 1 or α 2, becuse by ssumption these numbers re t lest κ + 1. This implies tht the monodromy is trivil t 1 nd. Then the second solution of Heun s eqution lso hs no singulrities t 1 nd. So in this cse C 1 divides C 2, C 3 nd C C 2 cnnot be the polynomil of the smllest degree in view of (3.11). 3. It remins to show tht if C k, k {3, 4} is the polynomil of the smllest degree nd λ is root of C k, then λ is lso root of C 1 nd C 2. Then it will follow tht there is polynomil nd qusipolynomil solutions, so the monodromy will be lso trivil t, tht is, λ will be root of ll three remining polynomils. Proposition 4.3 Let m,n,κ be integers, 0 m n, 0 2κ m + n, nd σ not n integer. Consider the differentil eqution (3.7) which we write s Dy = 0, where Dy = z(z 1)(z ) ( ( σ y z + m z 1 + n ) ) y +κ(σ +1+τ)zy λy, z nd τ = m + n κ κ. Suppose tht m κ, nd tht the monodromy t 1 is trivil, tht is ll solutions re holomorphic t the point 1. Then there exist polynomil solution nd qusipolynomil solution. Proof. Let us trnsform our eqution (3.7) to the form ( ( m z(z 1)(z ) y z + σ z 1 + σ m + n 2κ ) ) y z +κ(κ n 1)zy = λy. (4.8) It hs polynomil solution of degree k simultneously with the originl eqution (3.7). Consider the infinite Jcobi mtrix (4.2). The trivility of the monodromy of (4.8) t 0 mens tht λ is n eigenvlue of the truncted Jcobi mtrix J 0 given by the first m + 1 rows nd columns. The existence 15

16 of polynomil solution of (4.8) of degree k mens tht λ is n eigenvlue of the truncted mtrix J 1 given by the first k + 1 rows nd columns. By explicit formuls for the entries, the mtrix J 1 is upper block-tringulr, the bottom-left (m + 1) k block being equl to zero, nd the top-left (m + 1) (m + 1) block of J 1 equls J 0. Thus every eigenvlue of J 0 is n eigenvlue of J 1. To show the existence of qusipolynomil solution y(z) = z σ+1 q(z), we write the differentil eqution for q (σ will be replced by σ 2) nd then trnsform it to the form (4.8). To summrize the contents of this section, we consider, for ny given α 0,...,α 3 stisfying conditions ) or b) of Theorem 3.1, the polynomil F(,λ) which is the polynomil of the smllest degree of those C j in Proposition 4.2. The condition F(,λ) = 0 (4.9) is equivlent to the sttement tht the monodromy of Heun s eqution is conjugte to subgroup of PSU(2). Thus equivlence clsses of metrics of positive curvture 1 with singulrities t 0, 1,, with prescribed α j re in one-to-one correspondence with solutions of the eqution (4.9). Remrk. The vlue λ in (4.9) depends not only on the qudrilterl (or metric) tht we consider but lso on the choice of the Heun eqution. Different Heun equtions corresponding to the sme qudrilterl cn be obtined by cyclic permuttion of the vertices, nd by the different choices of exponents t the singulrities. The ngle t vertex only fixes the bsolute vlue of the difference of the exponents. The vlues of λ corresponding to the sme qudrilterl but different Heun equtions re relted by frctionlliner trnsformtions. 5 Counting rel solutions In this section we ssume tht is rel nd estimte from below the number of rel Heun s equtions with given conic singulrities t 0, 1,, with prescribed ngles nd unitry monodromy, or, which is the sme, the number of rel solutions λ of eqution (4.9). We will lso show tht for generic we hve equlity in the inequlity of Theorem 4.1 for the number of complex solutions. Our estimtes will be bsed on the following lemm. 16

17 Lemm 5.1 Let J be rel (d + 1) (d + 1) Jcobi mtrix 0 b c 0 1 b c 1 2 b J = c d 2 d 1 b d c d 1 d ) If b j c j > 0, 0 j d 1, then ll eigenvlues of J re rel nd simple. b) If c j 0 for 0 j d 1, then we hve where R = dig(r 0,...,r d ), r 0 = 1 nd J T R = RJ, (5.1) r j = r j 1 b j 1 c j 1, 1 j d. c) Suppose tht the sequence d j = b j c j hs the property d j > 0 for 0 j k nd d j < 0 for k + 1 j d. Then the number of pirs of non-rel eigenvlues, counting multiplicity, is t most [(d k)/2]. Proof. Sttement ) in contined in [20]; we include simple proof for convenience. Consider the mtrix S = dig(s 0,...,s d ), s 0 = 1, nd s j = s j 1 b j 1 /c j 1, 1 j d. Under the ssumption of prt ), the frction under the squre root is positive, nd the mtrix S is the positive squre root of the mtrix R given in prt b), S 2 = R. By explicit clcultion, the mtrix J = SJS 1 is rel nd symmetric, so it is digonlizble nd hs rel eigenvlues. The top right d d submtrix of J λi, where I is the identity mtrix nd λ is n eigenvlue of J, is lower tringulr nd hs the determinnt b0...b d 1 c 0...c d

18 Hence ll eigenvlues of J re simple. Sttement b) is proved by direct clcultion. To prove sttement c), we notice tht our ssumption bout signs of d j implies tht there re [(d k)/2] negtive numbers mong r 1,...,r n, nd the rest re positive. Condition (5.1) mens tht our mtrix J is symmetric with respect to the biliner form (x,y) R = x T Ry, tht is (Jx,y) R = (x,jy) R. Qudrtic form (x,x) R hs [(d k)/2] negtive squres, so our mtrix J hs t most 2[(d k)/2] non-rel eigenvlues, counted with lgebric multiplicities, ccording to the theorem of Pontrjgin [38]. This proves the lemm. We immeditely obtin the following Proposition 5.2 When the prmeter in (3.7) is rel nd is sufficiently close to 0 or 1, then ll metrics in Theorem 4.1 re symmetric with respect to the rel line, nd thus correspond to sphericl qudrilterls. Proof. When is rel, the mtrix J is rel. When 0, it follows from the formuls (4.3) (4.5) tht J tends to n lower tringulr mtrix with distinct elements on the min digonl. All eigenvlues of such mtrix re rel nd distinct. So for smll enough > 0 it lso hs rel distinct eigenvlues. To pply the sme rgument when is rel nd 1, we mke the chnge of the independent vrible z 1 z in (3.7). Our min result on the symmetric cse is the following: Theorem 5.3 Consider the metrics of curvture 1 on the sphere with rel conic singulrities 0, 1, 2, 3 nd the corresponding ngles 2πα 0, 2πα 1, 2πα 2, 2πα 3, where α 1 nd α 2 re integers. Suppose tht conditions of Theorem 3.1 re stisfied. Then (i) If the pirs ( 0, 3 ) nd ( 1, 2 ) do not seprte ech other on the circle R, then ll metrics with these ngles nd singulrities re symmetric. Their number is equl to min{α 1,α 2,κ + 1} where κ is defined in (3.9) nd (3.10). 18

19 (ii) If the pirs ( 0, 3 ) nd ( 1, 2 ) seprte ech other, then the number of of clsses of symmetric metrics is t lest [ ] 1 min{α 1,α 2,κ + 1} 2 2 min {α 1,α 2,δ}, where δ = 1 2 mx{α 1 + α 2 [α 0 ] [α 3 ], 0} (iii) There is n ǫ > 0 depending on the α j such tht if ( 2 0 )( 3 1 ) ( 1 0 )( 3 2 ) < ǫ (5.2) in (2.3) then ll of metrics re symmetric, nd their number is s in (i). The expression under the bsolute vlue in the left hnd side of (5.2) is the cross-rtio which is equl to when the vertices re 0, 1,,. In section 16 we will show tht the estimte in (ii) is chieved sometimes, nd in section 15 we give nother independent proof of this estimte. Corollry 5.4 Suppose tht j re rel, If the pirs ( 0, 3 ) nd ( 1, 2 ) seprte ech other on R, nd [α 0 ] + [α 3 ] + 2 α 1 + α 2 then ll metrics with singulrities t j nd ngles πα j re symmetric with respect to the rel line. If Cse ) of Theorem 3.1 previls, this corollry cn be lso obtined s specil cse of Theorem 5.2 from [32]. Proof of Theorem 5.3. Let us trnsform our eqution (3.7) to the form (4.8). This eqution hs the sme exponents t the singulrities s (3.7), but we plced the point with the smller integer exponent m t 0. The recurrence reltions similr to (4.1) hve in this cse the following coefficients c s = (κ s)(κ n s 1), s = s(s + σ n 2κ + (s 1 m σ)) (5.3) b s = (s + 1)(s m). 19

20 So b s c s = (κ s)(κ n s 1)(s + 1)(s m), (5.4) which is positive for > 0 nd 0 s < min{κ,m}. Thus for > 0 ll eigenvlues re rel nd distinct by Lemm 5.1 ) with d = min{m,κ} + 1. This proves (i). For the cse (ii) tht is < 0, we trnsform eqution (4.8) by the chnge of the vrible z = 1 z into eqution ( ( σ z(z 1)(z ) y z + m z 1 + σ + m + n + 2 2κ ) ) y z +κ(κ n 1)zy = λy. (5.5) Here = 1. The coefficients of the recurrence become So we hve c s = (κ s)(κ n s 1) s = s(s + m + n + 1 2κ) + (s m σ 1)), (5.6) b s = (s + 1)(s σ). c s b s = (κ s)(κ n s 1) (s + 1)(s σ), (5.7) which is positive when > 0, σ + 1 > κ, nd 0 s < κ. Thus under this condition, ll eigenvlues re rel. The rnge < 0 is covered by > 0. If σ + 1 κ, then the Jcobi mtrix with entries (5.6) hs the property described in Lemm 5.1 c), where R hs [σ] + 1 positive squres nd [(κ [σ])/2] negtive squres. So the number of pirs of non-rel eigenvlues, counting lgebric multiplicities, is t most (κ [σ])/2. We recll tht κ + 1 is the degree of the polynomil C 1 in Proposition 4.2. So the polynomil of minimum degree mong the C j hs t most (1/2) min{κ + 1,m + 1,κ [σ]} pirs of non-rel zeros, nd using the vlue of κ from (3.9), (3.10) nd the inequlity (3.8), we obtin (ii). Sttement (iii) is n immedite consequence of Proposition Introduction to nets In this section we begin different tretment of sphericl polygons which is independent of sections

21 Definition 6.1 A sphericl n-gon Q is mrked if one of its corners, lbeled 0, is identified s the first corner, nd the other corners re lbeled so tht 0,..., n 1 re in the counterclockwise order on the boundry of Q. We cll Q sphericl polygon when n is not specified. When n = 2, 3 nd 4, we cll Q sphericl digon, tringle nd qudrilterl, respectively. For n = 1, there is unique mrked 1-gon with the ngle π t its single corner. For convenience, we often drop sphericl nd refer simply to n- gons, polygons, etc. Let Q be mrked sphericl polygon nd f : Q S its developing mp. The imges of the sides ( j, j+1 ) of Q re contined in geodesics (gret circles) on S. These geodesics define prtition P of S into vertices (intersection points of the circles) edges (rcs of circles between the vertices) nd fces (components of the complement to the circles). Some corners of Q my be integer (i.e., with ngles πα where α is n integer). Two sides of Q meeting t its integer corner re mpped by f into the sme circle. The corners of Q with integer (resp., non-integer) ngles re clled its integer (resp., non-integer) corners. The order of corner is the integer prt of its ngle. A removble corner is n integer corner of order 1. A polygon Q with removble corner is isometric to polygon with smller number of corners. A polygon with ll integer corners is clled rtionl. All sides of rtionl polygon mp to the sme circle. Definition 6.2 Preimge of P defines cell decomposition Q of Q, clled the net of Q. The corners of Q re vertices of Q. In ddition, Q my hve side vertices nd interior vertices. If the circles of P re in generl position, interior vertices hve degree 4, nd side vertices hve degree 3. Ech fce F of Q mps one-to-one onto fce of P. An edge e of Q mps either onto n edge of P or onto prt of n edge of P. The ltter possibility my hppen when e hs n end t n integer corner of Q. The djcency reltions of the cells of Q re comptible with the djcency reltions of their imges in S. The net Q is completely defined by its 1-skeleton, connected plnr grph. When it does not led to confusion, we use the sme nottion Q for tht grph. If C is circle of P, then the intersection Q C of Q with the preimge of C is clled the C-net of Q. Note tht the intersection points of Q C with preimges of other circles of P re vertices of Q C. A C-rc of Q (or simply 21

22 n rc when C is not specified) is non-trivil pth γ in the 1-skeleton of Q C tht my hve corner of Q only s its endpoint. If γ is subset of side of Q then it is boundry rc. Otherwise, it is n interior rc. The order of n rc is the number of edges of Q in it. An rc is mximl if it is not contined in lrger rc. Ech side L of Q is mximl boundry rc. The order of L is, ccordingly, the number of edges of Q in L. Definition 6.3 We sy tht Q is reducible if it contins proper polygon with the corners t some (possibly, ll) corners of Q. Otherwise, Q is irreducible. The net of reducible polygon Q contins n interior rc with the ends t two distinct corners of Q. We sy tht Q is primitive if it is irreducible nd its net does not contin n interior rc tht is loop. Definition 6.4 Two polygons Q nd Q re strongly combintorilly equivlent if there is n orienttion preserving homeomorphism h : Q Q mpping the corners of Q to the corners of Q, nd the net Q of Q to the net Q of Q. For mrked polygons Q nd Q, we require lso tht the mrked corner 0 of Q is mpped by h to the mrked corner 0 of Q. Thus strong equivlence clss of sphericl polygons is combintoril object. It is completely determined by the lbeling of the corners nd the djcency reltions. We ll cll such n equivlence clss net when this would not led to confusion. Conversely, given lbeling of the corners nd prtition Q of disk with the djcency reltions comptible with the djcency reltions of P, sphericl polygon with the net Q cn be constructed by gluing together the cells of P ccording to the djcency reltions of Q. Such polygon is unique (up to n isometry preserving the lbels of the corners) if the imge of 0, the direction in which the imge of the edge ( 0, 1 ) is trversed, nd the imges of integer vertices which re different from the vertices of P, re fixed. If polygon Q is not irreducible, Definition 6.4 is not convenient. For exmple, qudrilterls in Fig. 12-c re not strongly equivlent, yet ech of them is obtined by gluing tringle T 0 nd disk long their common side. To reduce the number of cses in clssifiction, we dopt weker equivlence reltion, defined inductively s follows. Definition 6.5 Two irreducible polygons Q nd Q re combintorilly equivlent if they re strongly combintorilly equivlent. 22

23 Two rtionl polygons Q nd Q with ll sides mpped to the sme circle C of P re combintorilly equivlent if there is n orienttion preserving homeomorphism Q Q mpping the net Q C of Q to the net Q C of Q. If Q nd Q re reducible, ech of them represented s the union of two polygons Q 0 nd Q 1 (resp., Q 0 nd Q 1) glued together long their common side, then Q nd Q re combintorilly equivlent when there is n orienttion preserving homeomorphism h : Q Q inducing combintoril equivlence between Q 0 nd Q 0, nd between Q 1 nd Q 1. In wht follows we clssify ll equivlence clsses of nets in the cse when P is defined by two circles. In this cse, the boundry of ech 2-cell of the net Q of Q consists of two segments mpped to the rcs of distinct circles, with the vertices t the common endpoints of the two segments nd, possibly, t some integer corners of Q. By the Uniformiztion Theorem, ech mrked sphericl polygon is conformlly equivlent to closed disk with mrked points on the boundry. In the cse of qudrilterl, we hve four mrked points, so conforml clss of qudrilterl depends on one prmeter, the modulus of the qudrilterl. In section 14 below we will study whether for given permitted ngles of qudrilterl n rbitrry modulus cn be chieved. This will be done by the method of continuity, nd for this we ll need some fcts bout deformtion of sphericl qudrilterls (see section 13). 7 Nets for two-circle prtition Let us consider prtition P of the Riemnn sphere S by two trnsversl circles intersecting t the ngle α (see Fig. 1). Vertices N nd S of P re the intersection points of the two circles. We mesure the ngles in multiples of π, so tht 0 < α < 1, nd the complementry to α ngle is β = 1 α. An ngle tht is n integer multiple of π is clled, ccordingly, n integer ngle. A corner with n integer ngle is clled n integer corner. Let Q be sphericl n-gon over P (see Definition 1.1). We ssume Q to be mrked polygon (see Definition 6.1). Theorem 7.1 An irreducible sphericl polygon Q over the prtition P hs t most two non-integer corners. 23

24 b b b Figure 1: Prtition P of the Riemnn sphere by two circles. Proof. We prove this sttement by induction on the number m of fces of the net Q of Q. If m = 1 then Q mps one-to-one to fce of P, thus it hs exctly two non-integer corners. Let m > 1. Suppose first tht Q hs mximl interior rc γ tht is not loop. Let p nd q be the endpoints of γ. Since γ is mximl, both p nd q re t the boundry of Q. Since Q is irreducible, t lest one of them, sy p, is not corner of Q. Thus γ prtitions Q into two polygons, Q nd Q, ech of them hving less thn m fces of its net. The induction hypothesis pplied to Q nd Q implies tht ech of them hs t most two non-integer corners. But the corners of Q nd Q t p re non-integer, while Q does not hve corner t p. Thus Q hs t most two non-integer corners. Consider now the cse when ll mximl interior rcs of Q re loops. Since the 1-skeleton of Q is connected, there exists mximl interior rc γ of Q with both ends t corner p of Q. The disk D bounded by γ mps oneto-one to the disk in S bounded by the circle C of P to which γ is mpped. Let C be the other circle of P. Then γ intersects with Q C t exctly two points. Let γ be the mximl interior rc of Q C intersecting γ t those two points. By ssumption, γ is loop. If one of those points is p, then p is preimge of vertex of P, nd γ hs single intersection point q with γ inside Q. But this is impossible becuse the complement to the union of the disks bounded by γ nd γ would contin fce of Q whose boundry would not be circle. Thus p is not preimge of vertex of P, nd both intersection points q nd q of γ nd γ re interior vertices of Q. Then γ must contin corner 24

25 p q q p Figure 2: Pseudo-digonl connecting two integer corners of Q. p of Q. Otherwise the complement to the union of the disks bounded by γ nd γ would contin fce of Q whose boundry would not be circle. The sme rguments s bove imply tht p is not preimge of vertex of P, thus the union of γ nd γ is pseudo-digonl of Q shown in Fig. 2. Removing these two loops, we obtin polygon hving m 4 fces in its net, with the sme number of non-integer corners s Q. By the induction hypothesis, Q must hve t most two non-integer corners. This completes the proof of Theorem 7.1. Rtionl sphericl polygons. All corners of rtionl polygon Q re integer, nd ll its sides mp to the sme circle C of P. Thus Q is completely determined, up to combintoril equivlence, by its C-net Q C. Ech mximl rc of Q C connects two corners of Q. If Q is irreducible, Q C does not hve interior rcs. 8 Primitive sphericl polygons with two non-integer corners In this section, Q denotes mrked primitive sphericl n-gon with two noninteger corners over the prtition P, one of these two corners lbeled 0, nd the other one lbeled k where 0 < k < n. The sides of Q re lbeled L j so tht j 1 nd j re the ends of L j, with n identified with 0. We ssume tht the sides L j for 1 j k belong to the preimge of circle C of P, while the sides L j for k < j n belong to the preimge of the circle C C 25

26 of P. Lemm 8.1 The net Q of Q does not hve interior vertices. Proof. Let q be n interior vertex of Q, nd let F be fce of Q djcent to q. Let γ be mximl rc of Q C through q, where C is circle of P. Since γ is n interior mximl rc, it my end either t corner of Q or t side. Since Q is primitive, γ is not loop, nd the ends of γ cnnot be t two distinct corners of Q. If n end p of γ is t side of Q, let L nd γ be the mximl rcs of Q C, where C C, pssing through p nd q, respectively. Note tht tht L is side of Q while γ is n interior rc. We my ssume tht there re two djcent fces, F nd F, of Q hving common segment pq of γ in their boundry. If there re no such fces then we cn replce q by n interior vertex of Q on γ closest to p. Then ech of these two fces must hve n integer corner in its boundry where L nd γ intersect, otherwise the intersection of its boundry with Q C would not be connected. The two corners must be distinct, since they re the ends of side L of Q. This implies tht n interior rc γ hs its ends t two distinct corners of Q, thus Q is not irreducible. Corollry 8.2 Ech interior rc of the net Q of Q is mximl nd hs order one. Lemm 8.3 Ech of the two non-integer corners of Q hs order zero. Proof. Let p be non-integer corner of Q of order greter thn zero. Since p is mpped to vertex of P, there is fce F of the net Q of Q hving p s its vertex, with two interior rcs, γ nd γ, djcent to p in its boundry. The rcs γ nd γ belong to preimges of two different circles of P. The other ends of γ nd γ cnnot be corners of Q, thus they must be side vertices of Q. This implies tht the preimge of ech of the two circles of P in the boundry of F is not connected, contrdiction. Corollry 8.4 Any interior rc of Q hs one of its ends t n integer corner i of Q nd nother end on the side L j, where either 0 < i < k < j n or 0 < j k < i < n. 26

27 Definition 8.5 Let Q be mrked primitive n-gon with two non-integer corners lbeled 0 nd k, nd let Q be the net of Q. For ech pir (i,j), let µ(i,j) be the number of interior rcs of Q with one end t the integer corner i nd the other end on the side L j. Note tht µ(i,j) my be positive only when either 0 < i < k < j n or 0 < j k < i < n, due to Corollry 8.4. We cll the set T of the pirs (i,j) for which µ(i,j) > 0 the (n,k)-type of Q (or simply the type of Q when n nd k re fixed), nd the numbers µ(i,j) the multiplicities. We ll see in Lemm 8.6 below tht the number of pirs in n (n,k)-type is t most n 2. An (n,k)-type with exctly n 2 pirs is clled mximl. Since interior rcs of Q do not intersect inside Q, the (n,k)-type of Q cnnot contin two pirs (i 0,j 0 ) nd (i 1,j 1 ) stisfying ny of the following four conditions: i 0 < i 1 < k < j 0 < j 1, j 0 < j 1 k < i 0 < i 1, i 0 < j 1 k < j 0 < i 1, j 1 i 0 < k < i 1 < j 0. (8.1) (8.1b) (8.1c) (8.1d) Interior rcs of Q cn be cnoniclly ordered, strting from the rc closest to the mrked corner 0, so tht ny two consecutive rcs belong to the boundry of cell of Q. The liner order on the interior rcs of Q induces interior order on the pirs (i,j) in the type T of Q. Lemm 8.6 The number of pirs in the (n,k) type T of Q is t most n 2. Proof. Let i, j T be the first pir, corresponding to the interior rcs of Q closest to 0. We my ssume tht 1 i < k nd n k < j n. Otherwise, we exchnge k nd n k. Let (i,m) T be the pir furthest from 0, with the sme i s the first pir. Then there re t most n m+1 pirs in T with the sme index i. The lst rc of Q with the ends t the vertex i nd the side L m prtitions Q into two polygons, Q nd Q, with Q contining 0. Contrcting Q to point, we obtin (m i)-gon Q with the (m i,k i)- type T obtined from T by deleting ll pirs with the sme i s the first one, nd relbeling vertices nd sides. Since m n nd i > 0, we my ssume inductively tht the type T of Q hs t most m i 2 pirs. This implies tht the type T hs t most n i 1 n 2 pirs. 27

28 Lemm 8.7 Any (n, k)-type cn be obtined from (non-unique) mximl (n,k)-type if some of the multiplicities re permitted to be zero. Proof. Let T be (n,k)-type with less thn n 2 pirs. We wnt to show tht one cn dd pir to T. We prove it by induction on n, the cse n = 2 being trivil. We use nottions of the proof of Lemm 8.6. Note first tht, if i > 1 then pir (1,n) cn be dded to T. Thus we my ssume tht i = 1. Next, T should contin ll n m+1 pirs (1,m),...,(1,n), otherwise missing pir cn be dded to T. Finlly, we cn ssume inductively tht T contins exctly m i 2 = m 3 pirs. Thus the number of pirs in T should be (n m + 1) + (m 3) = n 2. Proposition 8.8 For given n 2 nd k, 0 < k < n, the number M(n,k) of distinct mximl (n,k)-types stisfies the following recurrence: M(n,k) = k n k M(n m,k m + 1) + M(n m,n k m + 1). (8.2) m=1 Since M(2, 1) = 1, this implies tht M(k + l,k)x k y l (1 x)(1 y) = (1 x)(1 y) x(1 y) y(1 x). (8.3) k,l=1 Proof. This recurrence follows from construction in the proof of Lemm 8.6. The following clssifiction is n immedite consequence of Definition 8.5. Theorem 8.9 For ny n 2, ny k, 0 < k < n, ny set T of pirs (i,j) with either 0 < i < k < j n or 0 < j k < i < n, such tht no two pirs (i 0,j 0 ) nd (i 1,j 1 ) in T stisfy ny of the conditions (8.1-d), nd ny positive integers ν(i,j), (i,j) T, there exists mrked primitive sphericl n-gon, unique up to combintoril equivlence, with two non-integer corners, one of them mrked, with the type T nd multiplicities ν(i,j). We re going to pply Theorem 8.9 to clssifiction of sphericl qudrilterls. In this section, we consider primitive qudrilterls (Corollries 8.12 nd 8.13), nd in the next section irreducible (but not necessrily primitive) qudrilterls (Theorem 9.5 nd Corollry 9.6). We lso clssify sphericl digons nd tringles (Corollries 8.10 nd 8.11), used below s building blocks for constructing reducible qudrilterls. m=1 28

29 b N X b S Figure 3: Loction of the point X. Corollry 8.10 Ech primitive digon (n-gon for n = 2) with two noninteger corners mps one-to-one to fce of P, with its two corners mpped to distinct vertices of P. There is single (empty) (2, 1)-type of primitive digon. Let X (see Fig. 3) be point on circle of P shown in solid line, inside disk D bounded by the circle C of P shown in dshed line. For µ 0, let T µ (see Fig. 4) be primitive tringle hving non-integer corner 0 mpping to N, non-integer corner 1 mpped to S (resp., to N) when µ is even (resp., odd), nd n integer corner 2 of order µ+1 mpped to X. The smll blck dots in Fig. 4 indicte the preimges of the vertices of P which re not corners of T µ (though they re vertices of its net). The ngle t the corner 1 of T µ is equl (resp., complementry) to the ngle t its corner 0 when µ is even (resp., odd). Then T µ hs the empty (3, 1)-type when µ = 0, the mximl (3, 1)-type {(2, 1)} when µ > 0, nd the multiplicity µ 2,1 = µ. Corollry 8.11 Every primitive tringle with two non-integer corners over the prtition P is combintorilly equivlent to one of the tringles T µ. A tringle T µ with the (3, 2)-type cn be obtined from the tringle T µ by reflection symmetry, relbeling the corners 1 nd 2. Let X nd Y (see Fig. 5) be two points on the sme rc of circle of P shown in solid line, inside disk D bounded by the circle C of P shown by the dshed line. For µ,ν 0, let R µν (see Fig. 6) be primitive qudrilterl 29

30 T T Figure 4: Primitive tringles T µ. hving non-integer corner 0 mpping to N, non-integer corner 1 mpped to S (resp., to N) when µ + ν is even (resp., odd), nd integer corners 2 of order ν + 1 nd 3 of order µ + 1 mpped to X nd Y (resp., to Y nd X) when µ is even (resp., odd). The side L 1 of R µν is mpped to the circle C trversing it counterclockwise. The ngle t the corner 1 of R µν is equl (resp., complementry) to the ngle t its corner 0 when µ + ν is even (resp., odd). Then R µν hs the empty (4, 1)-type when µ = ν = 0, the (4, 1)-type {(3, 1)} with the multiplicity µ 3,1 = µ when µ > 0, ν = 0, the (4, 1)-type {(2, 1)} with the multiplicity µ 2,1 = ν when µ = 0, ν > 0, nd the (unique) mximl (4, 1)-type {(3, 1), (2, 1)} with the multiplicities µ 3,1 = µ nd µ 2,1 = ν when µ,ν > 0. Note tht when µ = 0 (resp., ν = 0) the corner 3 (resp., 2 ) of R µν is removble, thus R µν is isometric to tringle (to digon when µ = ν = 0). Corollry 8.12 Every primitive qudrilterl over P with two djcent noninteger corners is combintorilly equivlent to one of the qudrilterls R µν. A qudrilterl R µν with the (4, 3)-type cn be obtined from the qudrilterl R µν by reflection symmetry, relbeling the corners 1, 2, 3. 30