E.., (2) g t = e 2' g E. g t = g ij (t u k )du i du j, i j k =1 2. (u 1 0 0) u2 2 U, - v, w, g 0 (v w) = g ij (0 u k 0)v i w j = 0, (t) = g ij (t u k

2007 10 (545) 517.929..,.. 1. g t M, d dt g t = ;2 Ric(g t ) (1) Ric(g) g.,, -, - (., [1], [2]).,,.,, f t - (M G), g t = ft G,, (1)., -, -, (, ), - (,, [3]). t - E 3, g t, t E 3, (1). t -., -. (M g) Ric = Kg, K, d dt g t = ;2K(g t )g t : (2),. 1. U R 2 (u 1 u 2 ) f t : U! E 3 -. G E 3 g t = ft G - U. g t - (2), g t (u v) =e 2'(t u1 u 2 ) g E, g E = du 12 + du 22. ' = '(u 1 u 2 t) ' t = e;2' E ' (3) 29

E.., (2) g t = e 2' g E. g t = g ij (t u k )du i du j, i j k =1 2. (u 1 0 0) u2 2 U, - v, w, g 0 (v w) = g ij (0 u k 0)v i w j = 0, (t) = g ij (t u k 0)v i w j. K(t u 1 u 2 ) g t (u 1 u 2 ). (2) d = ;2K(t dt uk 0)(t). (0) = 0, (t) =0., v w g 0, - g t., g t g 0., g 0 g E (., [4],. 111), g ij (t u k ) = e 2'(t uk ) (du 12 +du 22 ). g ij (t u k ) K(t u k )=;e ;2'(t uk ) E '(t u k ) (., [4],. 113)., (2) (3). (3) '(0 u 1 u 2 )=' 0 (u 1 u 2 ) (., [5],. 327).,, [6]., t. [7] [8]., g ij, -., g = g ij (u k )du i du j E 3 ( ), g - (., [9],. 191 [10],. 216217)., z = z(u 1 u 2 ) " pq " rs r p z r r q z s 2(1 ; g pq z p z q ) = K z i = i z, " ij g K g, g, z, E 3 (. [10],. 216217).,,. :,..,?,,. t u, t ~r(t u v)=a(t u)~e(v)+b(t u) ~ k: (4) u a(t u) 2 + u b(t u) 2 = a(t u) 2 : (5) g t = a(t u) 2 (du 2 + dv 2 ) 1 a(t u) a t = 1 a E log a: (6) - a(0 u)=a 0 (u) ([5],. 327), b(u t) (5): Z q b(t u)= a(t u) 2 ; u a(t u) 2 du: (7) 30

,,, (4), a(t u) (6), b(t u) (7). [11], -,,. 2. 2.1.., (.,.,.. [12], [13]). (.,. [14] ) -.,. t 0 ~r = ~r(t u i ). ~r i (u i t)= i ~r(t u i ), ~n(t u i ) t. t ~r(t ui )= e i (t u i )~r i (t u i )+e(t u i )~n(t u i ): (8) r - g ij (t u i ) t, h ij (t u i ). (t u i ). 2 (, ). t ~r i = s i ~r s + i ~n (9) ~n t = ;s ~r s (10) j i = r i e j ; eh j i (11) i = i e + h is e s : (12). i ~r j =; k ij ~r k + h ij ~n i ~n = ;h s i ~r s (13) ; k ij h ij. (8) u j (13), (9), j i i (11) (12)., t ~n ~n =1,~n ~r i = 0 (9), (10). - - j i i, (11) (12). g ij, ij = i sg js, ij = g is s. j 3. t ~r = ~r(t u i ). g ij (t u i ), h ij (t u i ) ij (t u i ), - t, " ij (t u a ) t, K(t u i ) H(t u i ) 31

t. t g ij = ij + ji t " ij = s i " sj ; j" s si t gij = ; ij ; ji (14) t "ij = " js i s ; " is s j (15) t h ij = r i j + h is s j = r i ( j e)+r i h e js s + h is r e j s + h js r e i s ; eg sp h is h jp (16) t ij = h js r i s + h is r j s (17) t K =(Hgij ; h ij )r i j ; Kg ij ij (18) t H = gij r i j ; h ij ij : (19). g ij = ~r i ~r j (9), (14). (14) g is g js = j i t g t ij. (15) " ij =(~r i ~r j ~n) t, ( ), (9) (10). (16), h ij = ; i ~n~r j t t ~n ~r j ; i ~n t ~r j = ; i t ~n ~r j + h s i~r s t ~r j: (20) t h ij = ; i (10) u i, (13) ; i t ~n = r i s ~r s + h is s ~n: (21) (20), (16), ( (11) (12)). ij = i ~n j ~n (21) (17). H = g ij h ij t (14) (16), (19)., ij ; Hh ij + Kg ij = 0 (., [10],. 194, (8)) (18). 2.2.. t, t 2 (" "), 0 ~r = ~r(t u i ). ~r _ = ~r(t t ui )j t=0-0, t. (8) _~r = e i (0 u i )~r i (0 u i )+e(0 u i )~n(0 u i ): (22) (u i ) = e (0 u i ), (u i ) = e(0 u i ). i (u i )~r i (0 u i ) _ ~r 0,. (u i )~n(0 u i ). 4. t, t 2 (;" "), ~r = ~r(u i t) _ ~r = s ~r s + ~n (. (22)) 0. 32

~ = s ~r s 0 r i j + r j i = ;2Kg ij +2h ij (23) g ij, h ij 0, r - g ij, K 0.. (14) ij + ji = ;2Kg ij. (11) (23). ~v = s ~r s +~n, 0,, s ~r s (23). -, (4),,,.,. (23) 1. 0. -,, 0.. i = 0, (23) t. 2.3. -., ~ = i ~r i - 0. S 2 (E 2 ) (2 0) (u v)=g ik g jm u ij v km = u ij v ij = u ij v ij : (24) E 2, " ij, - E 2 (., [10],. 26). j : " i = g js " is " ij = g ik g jm " km. (u v w) =" kl " mn " pq u kq v lm w np u v w 2 S 2 (E 2 ) (3 0) S 2 (E 2 ). ~e 1 ~e 2 S 2 (E 2 )., (u v w)=(e 12 ) 3 det u 11 u 12 u 22 v 11 v 12 v 22 w 11 w 12 w 22 " ij, u ij, v ij, w ij ", u, v, w ~e i. -, 3- S 2 (E 2 ). S 2 (E 2 ), S 2 (E 2 ) -. ~e i E 2, 1 p (e 2 1 e 2 + e 2 e 1 ) e 2 e 2 e 1 e 1 S 2 (E 2 )., S 2 (E 2 ) (u v w) = p 2(u v w)= p 2 " kl " mn " pq u kq v lm w np : (25) 33

, [ ] S 2 (E 2 ), ([u v] w)=(u v w). S 2 (E 2 ),, t = [u v] u v 2 S 2 (E 2 ) t ij = ; 1 p 2 (" i k " j m + " j k " i m )" pq u kp v mq : 5. 0. i, i (" mi h j m + " mj h i m)r i j =0 (26) (H 2 ; 2K)(div ) ; Hh ij r i j + K(H 2 ; 4K)=0: (27). 0, - g ij h ij 0. u ij, g ij h ij, (g h u) =0. (25) " kl " p k = ;g lp, u ij, u = ag + bh 2 S 2 (E 2 ),,, a = ( ) (24). " mn h p mu np =0: (28) (u g)(h h) ; (h g)(u h) (g g)(h h) ; (g h) 2 (29) (u g)=g ij u ij (u h)=h ij u ij (h g)=g ij h ij = H (h h) =h ij h ij = k 2 1 + k 2 2 = H 2 ; 2K K, H, k 1, k 2 0. uij = r i j + r j i. (28) (26)., (u g)=2g ij r i j = 2 div( ~ ), (u h)=2h ij r i j. (23), a = ;2K, (29) (27). 1. (5), (26) -, r i j +r j i g ij h ij,. (26) ~v = s ~r s + ~n. 3. 0 ~v = ~ + ~, ~ = s ~r s ~ = ~n, 0., 0. 3.1. -. 0, p 2 (u 1 u 2 ), -, g 11 = g 22 = 1 g 12 =0 h 11 =1 h 12 =0 h 22 = ;1 (30) 34

(. [15],. 44). - ; 1 11 = ; 1 2 ;1 12 = ; 2 2 ;1 22 = 1 2 ; 2 11 = 2 ; 2 12 = ; 1 ; 2 22 = ; 2 2, (23), -, : 1 1 + 1 2 ; 2 2 2 2 ; 1 1 + 2 2 1 2 + 2 1 + 2 2 : (31) 2 2 = + (32) 2 2 = ; (33) 1 + 1 2 =0: (34) 3.2. (32)(34). (32) (33), 2 = 1 1 + 2 2 (35) 2 = 1 1 ; 2 2 + 1 1 ; 2 2 : (36) K - log(;4k)=4k, g (., [15], 3.1.3, [10],. 238, (2)). K = ; 2,, E log = ;2 (37) E. (35) 1 1 + 2 2 + 1 ( 1 log )+ 2 ( 2 log )=0 1 ( 1 + 1 log )+ 2 ( 2 + 2 log )=0:, ', 1 + 1 log =2 2 ' 2 + 2 log = ;2 1 ': (38) (34) 1 ( 2 )+ 2 ( 1 )=0: i (38), 2 ( 2 ') ; 1 ( 1 ') ; 12 =0: (39) 6. (32)(34) 1 =2 2 ' ; 1 =2 12 ' + 1 2 = ;2 1 ' ; 2 2 ' + 2 1 ' + 22 ; 11 ' (40) 2 (41) 11 ' ; 22 ' + 1 1 ' ; 2 2 ' + 12 =0: (42) 35

., (32)(34) (40), (41), ' (42) ( (39))., i, (40),, (41), ' (42), (32)(34). i (32), ; 1 2 E log ;=0, (37). (33). - i (34), (39). 2. i, (40) (41), (32)(33) '. (34) (42) '. 3.3. '. - (. [10],. X). W E 3, ~x, ~y, W. r i x j = i y j, r -, (.. - i ~x). ~t, ~ = i i =2 -. ~t W div(~t )=g ij r i t j = K, K., ~w div( ~w) = K, ~w. 0. H =0, (27) div( ~ )=;2K, ~t = ; 1 2~ - W 0. ~x, W. ~ 1, 0,. 7. '(p) ~x(p) ~ 1 (p). 1: ' (40), ~ = ;2~t, ~t W. 2: ' (42), 1- i = h is s.., 3.2. ~ 1 = p ~r 1 ~x = p (cos '~r 1 + sin '~r 2 ) ~y = p (; sin '~r 1 + cos '~r 2 ): i r i x j = i y j. x 1 = p 1 cos ', x 2 = p 1 sin ', y 1 = ; p 1 sin ', y 2 = p 1 cos '. (31), - r 1 x 1 =( 1 ' + 2 2 )y 1 r 2 x 1 =( 2 ' ; 1 2 )y 1: 1 = 1 ' + 2 2 2 = 2 ' ; 1 : 36 2

~t t 1 = ; 2, t 2 = 1 ( - ). ~t ~ = ;2~t, 1 =2 2 2 = ;2 1, (40)... h 1 1 =, h 2 2 = ;, h 2 1 = h 1 2 = 0, 1 = h 1 1 1 =2 2 ' ; 1 2 = h 2 2 2 =2 1 ' + 2 : 1 2 ; 2 1 =0, ' (42). 3. (40) ' -. 4. ~t = ; 1 2~ 1- i = h s i s, 1- i = h s i t s. 7, - 0 ~t, 1- ( ~ X)=h(~t ~ X). 3.4.,. - t ~r = ~r(u i t), ~v = ~ + ~, ~ = s ~r s ~ = ~n, 0. 8. = 2K, K - 0.. t H(t) t, H, j t t=0 = 0. (19), 0 g ij r i j ; h ij ij = 0. (u 1 u 2 ), (30), r 1 1 + r 2 2 ; ( 11 ; 22 )=0: (43) (12) 1 = 1 + 1 2 = 2 ; 2., (31), r 1 1 + r 2 2 = 1 1 + 2 2 = 11 + 22 + 1 ( 1 ) ; 2 ( 2 ): (11), 0 ij = r i j ; h ij, 11 ; 22 = r 1 1 ;r 2 2 ; 2. (31), r 1 1 = 1 1 + 1 2 1 ; 2 2 r 2 2 = 2 2 ; 1 1 + 2 2 : 2 2 2 11 ; 22 = 1 1 ; 2 2 + 1 1 ; 2 2 ; 2 ( 11 ; 22 )= 1 ( 1 ) ; 2 ( 2 ) ; 2:, (43) 11 + 22 +2 =0 (44). 37

t, (41), ' (42).,, (44),.., -. 3.5., -. ~v = + ~n 0 0, = 0.,, ', (41) (42), 12 ' + 1 2 ' + 2 1 ' + 22 ; 11 2 2 4 11 ' ; 22 ' + 1 1 ' ; 2 2 ' + 12 =0 (45) =0: (46), u = 10 u 1 + u 2, u = 20 u 1 ; u 2 (45) (46) (, ). '. ' = p 1, (45) (46) 22 ; 11 ; 22 + 11 2 4 2 1 12 + 2 ; 12 + 11 ; 22 4 2 2 4 p ; ( 2) 2 ; ( 1 ) 2,,, + 12 p =0 (47) =0: (48) 9. -.. ~r(u 1 u 2 ) = cosh(u 1 )~e(u 2 ) ; u 1 ~ k, (u 1 u 2 1 )=. (47), (48) cosh 2 (u 1 ) 11 (u 1 u 2 ) ; 22 (u 1 u 2 )+a(u 1 ) (u 1 u 2 )=0 (49) 12 (u 1 u 2 )+b(u 1 )=0 (50) 2 a(u 1 )= cosh 2 (u 1 ) ; 1 )= 2 cosh2 (u 1 ) ; 3 b(u1 : (51) 2 cosh 3 (u 1 ) a b. (50) 122 =0, 112 = ;b 0, 1122 = 0. (49) ; 2222 + a 22 = 0, u 1, a 0 22 = 0. a 0 6= 0, 22 = 0,, (u 1 u 2 ) = f(u 1 )u 2 + g(u 2 ). (49) (50), f 00 + af =0 g 00 + ag =0 f 0 = ;b:, ;b 0 + af =0, f = b 0 =a. (b 0 =a) 0 = ;b,, a b, (51),.. (b 0 =a) 0 = 2cosh6 (u 1 ) ; 27 cosh 4 (u 1 ) + 54 cosh 2 (u 1 ) ; 36 : 2cosh 7 (u 1 ) ; 8 cosh 5 (u 1 ) + 8 cosh 3 (u 1 ),,.., 38

\ " t =(,, - ).....,..,. 1. Chow B., Knopf D. The Ricci ow: an introduction. Mathematical Surveys and Monographs, V. 110. American Mathematical Society, Providence, RI, 2004. 2. Bennett Chow, Sun-Chin Chu, Chia-Yi, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci ow: techniques and applications. Part I: Geometric aspects. Mathematical Surveys and Monographs, V. 135. American Mathematical Society, Providence, RI, 2007. 3. Zhu Xi-Ping. Lectures on mean curvature ows. AMS/IP Studies in Advanced Mathematics-32. Providence, RI: American Mathematical Society Somerville: International Press. ix, 2002. 150 p. 4...,..,... -..:, 1979. 759. 5. Taylor M.E. Partial dierential equations. III Nonlinear equations. Springer-Verlag, 1996. 608 p. 6. Carstea S.A., Visinescu M. Special solutions for Ricci ow equation in 2D using the linearization approach // Mod. Phys. Lett. A 20 (2005), 39, 2993-3002 (. http://arxiv.org/abs/hepth/0506113). 7. Hamilton R.S. Three-manifolds with positive Ricci curvature // J. Dierential Geom. 1982. V. 17. 2.P. 255306. 8. DeTurck D.M. Deforming metrics in the direction of their Ricci tensor // J. Dierential Geom. 1983. V. 18. 1. P. 157162. 9..... 1. -,.. I..., -, 1947. 512. 10.....:, 1956. 260. 11. Rubinstein J., Sinclair R. Visualizing Ricci ow of manifolds of revolution // Experimental Mathematics. 2005. V. 14. P. 285298. 12... //.. 1996..59.. 2.. 284290. 13... //.. 2003.. 74.. 34.. 438444. 14....:, 1957. 223. 15. Palais R.S., Terng Chuu-lian, Critical point theory and submanifold geometry. Lecture Notes in Math. 1988. V. 1353. Los Andes (, ) 13.04.2007 39