3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]). 3 H 3 CMC 1 Bryant ([B, UY1]).

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1 3 de Sitter CMC 1 (Shoichi Fujimori) Department of Mathematics, Kobe University 3 de Sitter S (CMC 1), 1 ( [AA]) 3 H 3 CMC 1 Bryant ([B, UY1]) H 3 CMC 1, Bryant ([CHR, RUY1, RUY2, UY1, UY2, UY3, Yu]),, S 3 1 CMC 1 ([Ak, R]), 3 Euclid R 3 (Weierstrass ) 3 Lorentz R 3 1 ( [Ko, Mc], ),,, R 3 1, maxface maxface ([UY4]), S 3 1 CMC 1 ( CMC 1 face ), R 3, Osserman ([O1, Theorem 32]) [JM, Theorem 4], Riemann Cohn-Vossen Osserman, Gauss 1

2 , H 3 CMC 1 R 3 1 maxface ( ) ([UY1, UY2] [UY4] ) S 3 1 CMC 1 face ([F, Theorem 39]), CMC 1 face 1 CMC 1 faces R Lorentz,, Lorentz (x 0, x 1, x 2, x 3 ), (y 0, y 1, y 2, y 3 ) = x 0 y 0 + x 1 y 1 + x 2 y 2 + x 3 y 3, S 3 1 = S 3 1(1) = {(x 0, x 1, x 2, x 3 ) R 4 1 x x x x 2 3 = 1}, 1 Lorentz S de Sitter, R Hermite ( ) 3 R 4 x 0 + x 3 x 1 + ix 2 1 X = (x 0, x 1, x 2, x 3 ) X = x k e k = x 1 ix 2 x 0 x 3 k=0 ( ) ( ) ( ) i e 0 =, e 1 =, e 2 =, e 3 = i 0 S 3 1 ( S 3 1 = {X X = X, det X = 1} = {F e 3 F F SL(2, C)} ( X = t X), ) X, Y = 1 2 trace ( Xe 2 ( t Y )e 2 ), X, X = det X S 3 1,, S 3 1 CMC 1, 2

3 11 ( [AA]) D C, z 0 D g D (C { }) \ {z C z 1}, ω D, 1 (11) dŝ 2 = (1 + g 2 ) 2 ω ω D Riemann, F = (F jk ) D SL(2, C) ( ) F 1 g g 2 (12) df = ω, F (z 0 ) = e 0 1 g, (13) f = F e 3 F D S 3 1 CMC 1 D ds 2 = f (ds 2 S f 1), 3 2 h, f Gauss G (14) ds 2 = (1 g 2 ) 2 ω ω, h = Q + Q + ds 2, G = df 11 df 21 = df 12 df 22, Q = ωdg f Hopf CMC (1) g 2 Gauss, (g, ω) Weierstrass data (2) f N D H 3 ( ) 1 N = g 2 1 F g g F, 2ḡ g H 3 = {X X = X, det X = 1, trx > 0} = {F F F SL(2, C)} ([KY, Lemma 31] ) (3) ( Gauss ) S 3 1 ( x 0 > 0) S 2, Riemann C { } f D S 3 1 f(z) 3

4 N(z), S 2 Gauss G(z) (4) (12) F, ˆf = F F D H 3 CMC 1, ˆf (ds 2 H ) 3 (11) dŝ 2, 2 Q Q + dŝ 2 ˆf Gauss f Gauss G (5) [UY1] (26), G g Q 2Q = S(g) S(G) S(g) = S z (g)dz 2, ( ) g S z (g) = 1 ( ) g 2 ( = d/dz) g 2 g g Schwarz,, ([UY4, Definition 22] ) 13 M 2, f M S 3 1 ds 2 = f (ds 2 S (1) (3) f CMC 1) 3 1 face (1) W M, f W W S 3 1 CMC 1, (2) p M \W p U C 1 β U W R +, βds 2 U C 1 Riemann, (3) M p df(p) 0 CMC 1 face f M S 3 1,, f M, 14 M 2 f M S 3 1 CMC 1 face, W M f W CMC 1 4

5 , (1) (2) M J (1) f W J, (2) J F M SL(2, C) det(df ) = 0 f ϱ = F e 3 F, ϱ M M M ( F f ), CMC 1 face f M S 3 1 M,, M Riemann 15 M Riemann, F M SL(2, C), (0, 2) (15) det[d(f e 3 F )], f = F e 3 F M S 3 1 CMC 1 face, p M f det[d(f e 3 F )] p = 0, f p M (16) det[d(f F )] p (0, 2) 14, 15, ( ) CMC 1 face 16 M Riemann, z 0 M g, ω M 1, (11) dŝ 2 M Riemann M M, g 1 F = (F jk ) M SL(2, C) (12), (13) f M S 3 1 CMC 1 face M ds2, f 2 h, Gauss G (14) CMC 1 face {p M g(p) = 1}, M Riemann, f M S 3 1 CMC 1 face, M g 1 ω, g 1 5

6 , dŝ 2 M Riemann, (13) F M SL(2, C) (12) 17 F M, 12 (3), G M, (14), Hopf Q M 2 CMC 1 faces S 3 1 CMC 1, ([Ak, R], 41 ) CMC 1 face ([UY4, Definition 41] ) 21 M Riemann, f M S 3 1 CMC 1 face ds 2 = f (ds 2 S f (resp ), 1) 3 C M, M (0, 2) T, T M \ C, ds 2 + T (resp ) Riemann 22 CMC 1 face f M S 3 1 M ( ), f M, f ϱ M S 3 1 ( ) ϱ M M M f M S 3 1 CMC 1 face (M, ds 2 + T ) Riemann, [H], M Riemann, CMC 1 face ϱ M M M F M SL(2, C) f γ [0, 1] M M τ M γ, F Φ γ F τ = F Φ γ, f = F e 3 F M well-defined, γ Φ γ SU(1, 1) Φ γ 6

7 1 ( ) ( ) ( ) e iθ 0 e s (21) E =, H = ±, P = ± 0 e iθ 0 e s 0 1 θ [0, 2π) s R \ {0} 23 f M S 3 1 CMC 1 face, F f, E, H, P 24 SU(1, 1) ( ) p q X = SU(1, 1) q p, H 2 w (pw + q)/( qw + p) H 2 Poincaré H 2 = ({w C w < 1}, ds 2 H 2 1 = 4dwd w/(1 w 2 ) 2 ) X, H 2 1 X, H 2 H 2 2, H 2 H SU(2) E, H 3 CMC 1, S 3 1 CMC 1 face CMC 1 face 25 f M S 3 1 CMC 1 face f, Riemann M p 1,, p n M M M \ {p 1,, p n }, f Hopf Q M 2 26 f M = M \ {p 1,, p n } S 3 1 Gauss G p j (j = 1,, n), p j, 1,, 7

8 f Hopf Q, 12 (4), (5) 27 [B, Proposition 6] p j (j = 1,, n), Q p j 2 3 Osserman f M = M \ {p 1,, p n } S 3 1 CMC 1 face G, Q f Gauss, Hopf 31 M Riemann dŝ 2 (31), dŝ 2 = (1 + G 2 ) 2 Q dg dˆσ 2 = ( K dŝ 2)dŝ 2 = ( ) Q dg 4dGdG (1 + G 2 ) 2 32 G, Q M, dŝ 2, dˆσ 2 M 33 ([UY3, Definition 21], [Yam] ) M dς 2 p j m j, dς 2 u j z p j 2m j dzd z (u j 0) m j Ord pj (dς 2 ), dς 2 p j Riemann, Ord pj (dς 2 ) = 0 = {z C z < 1}, = \ {0} 34 f S 3 1 z = 0 CMC 1 face, (32) Ord 0 (dˆσ 2 ) Ord 0 (Q) 2, 8

9 Proof, [UY1, Lemma 53] [UY2, Lemma 3], F SL(2, C) f (, F e 3 F = f), F (12) [UY1] (15), f [UY1, Lemma 53], Λ SL(2, C) ( ) z λ 1 a(z) z λ 2 b(z) (33) ΛF = z λ 1 m 1 c(z) z λ 2 m 2 d(z) a, b, c, d, λ 1, λ 2, m 1, m 2 (1) Ord 0 Q = 2, m 1 = m 2, λ 1 = ( µ + m j )/2 < λ 2 = (µ + m j )/2, (2) Ord 0 Q 1, m 1 = (ν + 1) < m 2 = 2µ + ν + 1, λ 1 = 0 < λ 2 = m 2, ν = Ord 0 Q µ + 1 ˆf = (ΛF )(ΛF ), ( ) e 2πλ 1i 0 (ΛF ) τ = ΛF P, P =, 0 e 2πλ 2i, f ˆf, f ˆf Hopf Q, [UY2, Lemma 3] (32), m 1 < m 2 (ΛF )(ΛF ) (ΛF )e 3 (ΛF ) leading term, f ˆf 35 ([KTUY, Lemma 41], [Yu] ) f M S 3 1 CMC 1 face f f, dŝ 2 M, (34) Ord pj (dŝ 2 ) 2, j = 1,, n 36 (Osserman ) f M S 3 1 CMC 1 face f n, 9

10 G f Gauss, (35) 2 deg(g) χ(m) + n, deg(g) G (G, deg(g) = ), χ(m) M Euler,, Proof 25, M = M \{p 1,, p n }, M Riemann, p 1,, p n M f, G deg(g) =, (35) f Riemann-Hurwicz Gauss (36) dŝ 2 dˆσ 2 = 4QQ, 2 deg(g) = χ(m) + p M Ord p dˆσ 2 = χ(m) + p M ( Ordp Q Ord p dŝ 2) = χ(m) + Ord p Q Ord p dŝ 2 p M p M n = χ(m) Ord pj dŝ 2 j=1 χ(m) + 2n = χ(m) + n ((34) ) n Ord pj dŝ 2 j=1, 34 (36) 10

4 CMC 1 face, [KY, LY, Yan] S 3 1 hollow ball model, S3 1 ( ) x 0 + x 3 x 1 + ix 2 (x 0, x 1, x 2, x 3 ) S 3 1 x 1 ix 2 x 0 x 3, (y 1, y 2, y 3 ) y k = earctan x 0 1 + x 2 0 x k, k = 1, 2, 3 e π < y1

11 4 CMC 1 face, [KY, LY, Yan] S 3 1 hollow ball model, S3 1 ( ) x 0 + x 3 x 1 + ix 2 (x 0, x 1, x 2, x 3 ) S 3 1 x 1 ix 2 x 0 x 3, (y 1, y 2, y 3 ) y k = earctan x x 2 0 x k, k = 1, 2, 3 e π < y1 2 + y2 2 + y3 2 < e π (x 0, x 1, x 2, x 3 ) (y 1, y 2, y 3 ) S 3 1 hollow ball H = {(y 1, y 2, y 3 ) R 3 e π < y y y 2 3 < e π } hollow ball H S 3 1, H 3 CMC M = C, (g, ω) = (c 1, c 2 dz), c 1 C, c 2 C \ {0}, H 3 horosphere CMC 1 face CMC 1 face CMC 1 {z C z <5, 0 arg z π } {z C z <10, 0 arg z π } 1 41 c 1 = 12 c 2 = 1 c 1 = 0 c 2 = 1 11

12 42 M = C, (g, ω) = (z, cdz), c R \ {0}, H 3 Enneper cousin CMC 1 face M, CMC 1 face Ord Q = 4 < 2,, (35) {z C z < 13} {z C 2 42 (c = 1) 08< z <13 π 1<arg z<π+1 } 43 M = C, (g, ω) = (e z, ice z dz), c R\{0}, H 3 helicoid cousin CMC 1 face CMC 1 face {z C Re(z) = 0} CMC 1 face 44 M = C \ {0}, (g, ω) = (z µ, (1 µ 2 )dz/4µz µ+1 ), µ R + \ {1}, H 3 catenoid cousin CMC 1 face M e µπi, e µπi S 1, CMC 1 face Ord 0 Q = Ord Q = 2,, CMC 1 face (35) CMC 1 face, H 3 CMC 1 S 3 1 CMC 1 face 12

13 {z C 09<Rez<09 4π<Imz<4π } 08<Rez<08 {z C 03<Imz<03 } 3 43 (c = 1) {z C e 5 < z <e 5 0<arg z<π } {z C e 5 < z <e 5 π<arg z<(3/2)π } {z C e 2 < z <e 2 0<arg z<π } 4 44 (µ = 08) {z C e 5 < z <e 5 0<arg z<π } {z C e 5 < z <e 5 π<arg z<(3/2)π } {z C e 2 < z <e 2 0<arg z<π } 5 44 (µ = 12) 13

14 ˆf M = M \ {p 1,, p n } H 3 CMC 1, dŝ 2 ˆf, f F, U(1) = {( e iθ 0 0 e iθ ) θ R (g, ω) F Weierstrass data,, (g, ω) ( ) F 1 g g 2 df = ω 1 g, F g, ω M, p 1,, p n 2 Gauss g 1 f = F e 3 F M, 45 CMC 1 face f M S 3 1, f, Weierstrass data (g, ω), λ R \ {0}, (λg, λ 1 ω) CMC 1 M ([UY1, Theorem 33]) 46 ˆf M H 3 CMC 1 n f F, U(1) f, (g, ω) F Weierstrass data, m (0 m n) λ 1,, λ m R +, λ R \ {0, ±λ 1,, ±λ m } (λg, λ 1 ω) f λ M S 3 1 CMC 1 face,, H 3 CMC 1, [RUY1, RUY2] 46, 14 }

15 47 CMC 1 face f M S 3 1, O(0), O( 5), O( 2, 3), O( 1, 1, 2), O( 4), O( 6), O( 2, 4), O( 1, 2, 2), O( 2, 2), O( 1, 4), O( 3, 3), O( 2, 2, 2), f O(d 1,, d n ) M = (C { }) \ {p 1,, p n } f Hopf Q p j d j [AA] [Ak] R Aiyama and K Akutagawa, Kenmotsu-Bryant type representation formulas for constant mean curvature surfaces in H 3 ( c 2 ) and S 3 1(c 2 ), Ann Global Anal Geom (1) 17 (1998), K Akutagawa, On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math Z 196 (1987), [B] R Bryant, Surfaces of Mean Curvature One in Hyperbolic Space, [CHR] [ER] Astérisque (1987), P Collin, L Hauswirth and H Rosenberg, The geometry of finite topology Bryant surfaces, Ann of Math (2) 153 (2001), F J M Estudillo and A Romero, Generalized maximal surfaces in Lorentz-Minkowski space L 3, Math Proc Cambridge Philos Soc (3) 111 (1992), [F] S Fujimori, Spacelike CMC 1 surfaces with elliptic ends in de Sitter 3- [H] [JM] [KY] [Ko] Space, preprint, arxivmathdg/ A Huber, On subharmonic functions and differential geometry in the large, Comment Math Helv 32 (1957), L Jorge and W H Meeks III, The topology of complete minimal surfaces of finite total Gaussian curvature, Topology, (2) 22 (1983), Y-W Kim and S-D Yang, The goedesic reflection principle for spacelike constant mean curvature surfaces in de Sitter three-space, preprint O Kobayashi, Maximal surfaces in the 3-dimensional Minkowski 3-space L 3, Tokyo J Math 6 (1983), [KTUY] M Kokubu, M Takahashi, M Umeharara and K Yamada, An analogue [L] of minimal surface theory in SL(n, C)/SU(n), Trans Amer Math Soc 354 (2002), H B Lawson Jr, Lectures on minimal submanifolds, Vol I, Second edition, Mathematics Lecture Series, 9 Publish or Perish, Inc, (1980) 15

16 [LY] [Mc] [O1] S Lee and S-D Yang, A spinor representation for spacelike surfaces of constant mean curvature 1 in de Sitter three-space, preprint L McNertney, One-parameter families of surfaces with constant curvature in Lorentz 3-space, PhD Thesis, Brown Univ, (1980) R Osserman, Global properties of minimal surfaces in E 3 and E n, Ann of Math (2) 80 (1964), [O2], A survey of minimal surfaces, Dover Publications (1986) [P] B Palmer, Spacelike constant mean curvature surfaces in pseudo- [R] [RUY1] [RUY2] [UY1] [UY2] [UY3] [UY4] [Yam] [Yan] [Yu] Riemannian space forms, Ann Global Anal Geom (3) 8 (1990), J Ramanathan, Complete spacelike hypersurfaces of constant mean curvature in de Sitter space, Indiana Univ Math J (2) 36 (1987), W Rossman, M Umehara and K Yamada, Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature I, Hiroshima Math J 34 (2004), 21 56, Mean curvature 1 surfaces in hyperbolic 3-space with low total curvature II, Tohoku Math J (2) 55 (2003), M Umehara and K Yamada, Complete surfaces of constant mean curvature 1 in the hyperbolic 3-space, Ann of Math (2) 137 (1993), , A duality on CMC-1 surfaces in hyperbolic space, and a hyperbolic analogue of the Osserman inequality, Tsukuba J Math (1) 21 (1997), , Surfaces of constant mean curvature c in H 3 ( c 2 ) with prescribed hyperbolic Gauss map, Math Ann 304 (1996), , Maximal surfaces with singularities in Minkowski space, preprint, arxivmathdg/ K Yamada, 1, 1206 (2001), S-D Yang, Björling formula for spacelike surfaces of constant mean curvature in the de Sitter space, preprint Z-H Yu, The value distribution of the hyperbolic Gauss map, Proc Amer Math Soc 125 (1997), address fujimori@mathkobe-uacjp 16

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