Classification of Spherical Quadrilaterals

Transcription

1 Clssifiction of Sphericl Qudrilterls Alexndre Eremenko, Andrei Gbrielov, Vitly Trsov November 28, 2014

2 R 01 S 11 U 11 V 11 W 11 1

3 R 11 S 11 U 11 V 11 W 11 2

4 A sphericl polygon is surfce homeomorphic to the closed disk, with severl mrked points on the boundry clled corners, equipped with Riemnnin metric of constnt curvture K = 1, such tht the sides (rcs between the corners) re geodesic, nd the metric hs conicl singulrities t the corners. A conicl singulrity is point ner which the length element of the metric is ds = 2α z α 1 dz 1 + z 2, where z is locl conforml coordinte. The number 2πα > 0 is the ngle t the conicl singulrity. The interior ngle of our polygon is πα. These ngles cn be rbitrrily lrge. 3

5 Every polygon cn be mpped conformlly onto the unit disk. We consider the problem of clssifiction up to isometry of polygons with prescribed ngles nd prescribed corners. By prescribed corners we men tht the imges of the corners on the unit circle re prescribed. The necessry condition on the ngles, αj > n 2, follows from the Guss Bonnet formul. If 0 < α j < 1, then we hve existence nd uniqueness (M. Troynov, 1991, F. Luo nd G. Tin, 1992). 4

6 Sphericl tringles were clssified by F. Klein, 1890, A. Eremenko, 2004, S. Fujimori et l, If ll α j re not integers, the necessry nd sufficient condition for the existence of sphericl tringle is cos 2 α 0 + cos 2 α 1 + cos 2 α cos α 0 cos α 1 cos α 2 < 1, nd the tringle is unique. If α 0 is n integer, then the necessry nd sufficient condition is tht either α 1 + α 2 or α 1 α 2 is n integer m < α 0, with m nd α 0 of opposite prity. The tringle with n integer corner is not unique: there is 1-prmetric fmily when only one ngle is integer, nd 2-prmetric fmily when ll ngles re integer. 5

7 Developing mp. A surfce D of constnt curvture 1 is loclly isometric to region on the stndrd sphere S. This isometry is conforml, hs n nlytic continution to the whole polygon, nd is clled the developing mp f : D S. We sy tht sphericl polygons re equivlent if their developing mps differ by post-composition with frctionl-liner trnsformtion. Let us choose the upper hlf-plne H s the conforml model of our polygon, with n corners 0,..., n 1, nd choose n 1 =. Accordingly, we sometimes denote α n 1 s α. The other corners re rel numbers. 6

8 Then f : H S is meromorphic function mpping the sides into gret circles. By the Symmetry Principle, f hs n nlytic continution to multi-vlued function in C \ { 0,..., n 1 } whose monodromy is subgroup of P SU(2). Such function must be rtio of two linerly independent solutions of the Fuchsin differentil eqution w + n 2 k=0 1 α k z k w + P (z) (z k ) w = 0, where P is rel polynomil of degree n 3 whose top coefficient cn be expressed in terms of the α j. The remining n 3 coefficients of P re clled the ccessory prmeters. The monodromy group of this eqution must be conjugte to subgroup of P SU(2). 7

9 In the opposite direction, if Fuchsin differentil eqution with rel singulrities nd rel coefficients hs the monodromy group conjugte to subgroup of P SU(2), then the rtio of two linerly independent solutions restricted to H is developing mp of sphericl polygon. Thus clssifiction of sphericl polygons with given ngles nd corners is equivlent to the following problem: For Fuchsin eqution with given rel prmeters j, α j, to find the rel vlues of ccessory prmeters for which the monodromy group of tht eqution is conjugte to subgroup of P SU(2). These vlues of ccessory prmeters re in bijective correspondence with the equivlence clsses of sphericl polygons. 8

10 Sphericl polygons with ll integer ngles. In this cse, the developing mp is rel rtionl function with rel criticl points. The multiplicities of the criticl points re α j 1. Such functions hve been studied in gret detil (A. Eremenko nd A. Gbrielov, 2002, 2011, I. Scherbk, 2002, A. Eremenko, A. Gbrielov, M. Shpiro, F. Vinshtein, 2006). The necessry nd sufficient condition on the ngles is (αj 1) = 2d 2, where d = deg f is n integer, nd α j d for ll j. For given ngles, there exist exctly K(α 0 1,..., α n 1 1) of the equivlence clsses of polygons, where K is the Kostk number: it is the number of wys to fill in tble with two rows of length d 1 with α 0 1 zeros, α 1 1 ones, etc., so tht the entries re non-decresing in the rows nd incresing in the columns. 9

11 Polygons with two non-integer ngles. Let α 0 nd α n 1 be non-integer, while the rest of the ngles α j re integer. Assuming 0 = 0 nd n 1 = we conclude tht the developing mp hs the form f(z) = z α P (z) Q(z), where α (0, 1) nd P, Q re rel polynomils. 10

12 For this cse, necessry nd sufficient condition on the ngles is the following Theorem 1. Let σ := α α n 2 n + 2. ) If σ + [α 0 ] + [α n 1 ] is even, then α 0 α n 1 is n integer of the sme prity s σ, nd α 0 α n 1 σ. b) If σ + [α 0 ] + [α n 1 ] is odd, then α 0 + α n 1 is n integer of the sme prity s σ, nd α 0 + α n 1 σ. 11

13 Finding ll polygons with prescribed ngles is equivlent in this cse to solving the eqution z(p Q P Q ) + αp Q = R with respect to rel polynomils P nd Q of degrees p nd q, respectively, where R is given rel polynomil of degree p + q. The degree of the mp equls W α : (P, Q) z(p Q P Q ) + αp Q ( p + q p (it is liner projection of Veronese vriety), nd one cn show tht when ll roots of R re non-negtive, ll solutions (P, Q) Wα 1 (R) re rel. ) 12

14 Enumertion of polygons with two djcent noninteger ngles. An importnt specil cse is when 0 nd n 1 re djcent corners of the polygon, 2α 0 nd 2α n 1 re odd integers, while ll other α j re integers. Equivlence clsses of such polygons re in bijective correspondence with odd rel rtionl functions with ll criticl points rel, given by f(z) = g( z), where f is the developing mp of our polygon nd g is rtionl function s bove. 13

15 By deformtion rgument, this gives the following Theorem 2. If the ngles stisfy the necessry nd sufficient condition given bove, nd the corners 0 = 0 nd n 1 = re djcent, then there re exctly E(2α 0 1, α 1 1,..., α n 2 1, 2α n 1 1) equivlence clsses of polygons, where E(m 0,..., m n 1 ) is the number of chord digrms in H, symmetric with respect to z z, with the vertices 0 = 0 < 1 <... < n 2 < n 1 = nd 1,..., n 2, nd m j chords ending t ech vertex j. If 0 nd n 1 re not djcent, E gives n upper bound on the number of equivlence clsses of polygons. 14

16 One cn express E in terms of the Kostk numbers. Proposition. Let m 0 nd m n 1 be even. Then E(m 0, m 1,..., m n 2, m n 1 ) = K(r, m 1,..., m n 2, s), where positive integers r nd s stisfy r + s > m m n 2, (1) nd cn be defined s follows: If µ := (m 0 + m n 1 )/2 + m m n 2 is even, then r = m 0 /2 + k, s = m n 1 /2 + k, where k is lrge enough, so tht (1) is stisfied. If µ is odd, then r = (m 0 + m n 1 )/2 + k + 1, s = k, nd k is lrge enough, so tht (1) is stisfied. 15

17 Sphericl qudrilterls. Heun s eqution. In the cse n = 4 the Fuchsin eqution for the developing mp is the Heun s eqution ( w 1 α0 + z + 1 α 1 z α 2 z ) w Az λ + z(z 1)(z ) w = 0, where A cn be expressed in terms of α j, nd λ is the ccessory prmeter. We cn plce three singulrities t rbitrry points, so we choose 0 = 0, 1 = 1, 2 =, 3 =. The condition tht the monodromy belongs to P SU(2) is equivlent to n eqution of the form F (, λ) = 0. This eqution is lgebric if t lest one or the ngles is integer. 16

18 Theorem 2 in the cse of qudrilterls with two integer nd two non-integer ngles specilizes to the following Theorem 3. The number of clsses of qudrilterls with two integer nd two non-integer ngles is t most where k + 1 = min{α 1, α 2, k + 1}, { (α1 + α 2 α 0 α 3 )/2 in cse ) (α 1 + α 2 α 0 α 3 )/2 in cse b). If > 0 we hve equlity. Here cses ) (when α 0 α 3 is integer) nd b) (when α 0 +α 3 is integer) re s in Theorem 1. Condition > 0 mens tht the corners 1 nd 2 with integer ngles re djcent. 17

19 Qudrilterls with non-djcent integer ngles. Let δ = mx(0, α 1 + α 2 [α 0 ] [α 3 ])/2. Theorem 4. The number of equivlence clsses of qudrilterls with non-djcent corners 1 nd 2, with integer ngles α 1 nd α 2, is t lest min{α 1, α 2, k + 1} 2 2 min {α 1, α 2, δ} where k is the sme s in Theorem 3. [ 1 ], (2) Notice tht in cse b) of Theorems 1 nd 3, the lower bound (2) becomes 0 when min{α 1, α 2, k + 1} is even nd 1 if min{α 1, α 2, k + 1} is odd. 18

20 Nets. The developing mp is locl homeomorphism, except t the corners, of closed disk D to the stndrd sphere S. The sides re mpped to gret circles. These gret circles define prtition (cell decomposition) of the sphere. Tking the f-preimge of this prtition, nd dding vertices corresponding to the integer corners, we obtin cell decomposition of D which is clled net. Two nets re considered equivlent if they cn be mpped to ech other by n orienttion-preserving homeomorphism of the disk, respecting lbeling of the corners. It is esy to see tht net, together with the prtition of the sphere by the gret circles, define the polygon completely. 19

21 b N b S Fig. 1. Prtition of the Riemnn sphere by two gret circles. 20

22 R 21 R R R Fig. 2. Primitive nets, two djcent integer corners. 21

23 U _ U 12 _ U 12 U X 12 0 b b 3 3 b b 2 0 b b 1 1 b b _ 2 X 21 Fig. 3. Primitive nets, two opposite integer corners. 22

24 Fig. 4. Pseudo-digonl, two opposite integer corners. 23

25 Fig. 5. Non-uniqueness, two opposite integer corners. 24

26 ) 3 3 c) b) Fig. 6. A chin of nets, two opposite integer corners. 25

27 Qudrilterls with three non-integer ngles. Suppose tht α 3 is integer while the rest of the ngles re not. The necessry nd sufficient condition for the existence of qudrilterl with the given ngles is the sme s in the cse of tringles, nd the number of qudrilterls with the given ngles is t lest α 3 2 [ min ( α3 2, [α 1] + 1, δ where δ = mx(0, [α 1 ] + α 3 [α 0 ] [α 2 ]). )] 26

28 Fig. 7. Prtition of the Riemnn sphere by three gret circles. 27

29 R 11 3 U 11 3 V _ R _ U 12 _ V Fig. 8. Primitive nets, three non-integer corners. 28

30 3 _ X 21 X 22 3 Z Fig. 9. Primitive nets, three non-integer corners. ) 3 b) 3 c) Fig. 10. A chin of nets, three non-integer corners. 29

31 Fig. 11. Pseudo-digonl, three non-integer corners. 30

32 ) b) Fig. 12. Prtition of the Riemnn sphere by four gret circles (two views). 31

33 R 11 S 11 U 11 W 22 X 13 X 22 V 11 Z 11 Fig. 13. Primitive nets, four non-integer corners. 32

34 Fig. 14. Pseudo-digonl, four non-integer corners. 33

35 Fig. 15. Prtition of the Riemnn sphere by non-generic four gret circles. 34

36 RS 11 UV 11 XX 12 ZZ 22 VW 22 Fig. 16. Some nets for non-generic four gret circles. 35

37 Fig. 17. Pseudo-digonl for non-generic four gret circles. 36

38 A B 4 C D E F 4 G H Fig. 18. Prtitions of the Riemnn sphere by four non-geodesic circles. 37