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( ) 1904 H. Poincaré 2002 03 G. Perelman Perelman 1. Topology Differential Geometry H. Poincaré 1904 [Po] 21 G. Perelman 1 ( ) 3 3 S 3 n n R n := {(x 1,..., x n ) : x 1,..., x n R} R N n R n x = (x 1,..., x n ) R n y = (y 1,..., y n ) R n x, y := n x i y i i=1 x := x, x 3 S 3 4 R 4 0 4 R 4 1 n n S n S n := { x = (x 1,..., x n+1 ) R n+1 : x = 1 } n S n (n + 1) B n+1 B n+1 S n = B n+1 = B n+1 R n+1

X x X X Y X Y R n n n R n R n 0 n 1 B n := {x R n : x < 1} B n := {x R n : x 1} 0 n := (0,..., 0) R n R n 2 S 1 S 1 3 B 2 S 1 (manifold) 2 ( ) n 1 n p p n R n 0 n R n n R n n S n X X n K 1,..., K k k = K i B n R n i=1 K i X (simply connected) X X x 0 X γ(0) = γ(2π) = x 0 γ : [0, 2π] X H : [0, 2π] [0, 1] X s [0, 2π] H(s, 0) = γ(s) H(s, 1) = x 0, u [0, 1] H(0, u) = H(2π, u) = x 0

X γ : S 1 X Γ : B 2 X Γ S 1 = γ s S 1 = B 2 Γ(s) = γ(s) X X π 1 (X) n 2 n S n n 1 n T n := S 1 S 1 X Y π 1 (X) π 1 (Y ) X 2 k π k (X) X ( 1) 2002 03 G. Perelman [Pe1, Pe2] [Pe1, Pe2] [Pe3] Perelman 2. C 4 ( 37 ) 3 oise (1952) n p T p n R n g p T p p g, X := g(x, X) X T p g (, g) 3 n R n R n g R n, R n x R n X, Y T x R n = R n g R n(x, Y ) := X, Y (R n+1, g R n) n S n g S n x S n X, Y T x S n = {v R n+1 : x, v = 0} R n+1 g S n(x, Y ) := X, Y (S n, g S n)

g (distance) (volume) γ : [0, 1] Length g (γ) Length g (γ) := 1 0 g ( dγ ds, dγ ) ds ds p, q d g (p, q) p q d g (p, q) := inf { Length g (γ) : γ γ(0) = p, γ(1) = q } (4) (, d g ) (metric space) p, q γ : [0, 1] g (4) inf min Hopf Rinow (, d g ) sup {d g (p, q) : p, q } (, g) φ : R φ dv (, g) Vol(, g) := dv = 1 dv 1 : R 1 p 1 (p) = 1 dv n (, g) (curvature) Levi-Civita p X, Y, Z, W T p (T p, g) {e i } n i=1 Rm Ric R Rm(X, Y, Z, W ) := g( X Y Z Y X Z [X,Y ] Z, W ), n Ric(X, Y ) := tr Rm(X,, Y, ) = Rm(X, e i, Y, e i ), R := tr Ric(, ) = i=1 n Ric(e i, e i ) tr R : R sectional curvature sec(x, Y ) := i=1 Rm(X, Y, X, Y ) g(x, X)g(Y, Y ) g(x, Y ) 2 (5) (5) 0 X, Y T p 3

2 (, g) K : R p X, Y, Z, W T p Rm(X, Y, Z, W ) = K(p)(g(X, Z)g(Y, W ) g(y, Z)g(X, W )), Ric(X, Y ) = K(p)g(X, Y ), R(p) = 2K(p), sec(x, Y ) = K(p) 6 n (R n, g R n) 0 Rm(X, Y, Z, W ) = 0, Ric(X, Y ) = 0, R = 0 n (S n, g S n) 1 Rm(X, Y, Z, W ) = g S n(x, Z)g S n(y, W ) g S n(y, Z)g S n(x, W ), Ric(X, Y ) = (n 1)g S n(x, Y ), R = n(n 1) 1 (, g) (gradient) (divergence) := div grad φ : R φ := grad φ φ : R (Ricci flow, ) 1 g(t) = 2Ric(g(t)) (7) t 2 g(t), a < t < b 3 (7) g (7) (0, 2)- Ric(g(t)) g(t) (7) u(x, t) = u(x, t), (x, t) (0, ) (8) t R n = n i=1 2 / x 2 i u(x, t) := (4πt) n/2 e x 2 /4t, (x, t) R n (0, )

R n (8) Einstein c R Ric(g) = cg g Einstein 9 (Einsein ) c R g 0 Ric(g 0 ) = cg 0 Einstein g(t) := (1 2ct)g 0, t {t R : 1 2ct > 0} c > 0 T := 1/(2c) g(t) t (, T ) c = 0 g(t) = g 0, t (, ) 10 ( ) N g N (t) g(t) := g N (t) + g R := N R R. Hamilton 1982 [Ha] Hamilton Nash oser D. DeTurck (1983) 11 ( [Ha]) g 0 g(0) = g 0 g(t), t [0, δ] g(t) g(t) t (, g(t)) Vol(, g(t)) d Vol(, g(t)) = dt 1 2 tr g t dv = R dv. 11 g(t) g(t) t g(t) = 2Ric( g(t)) + 2 r(t) g(t). (12) n r(t) := R dv Vol(, g(t)) t R (7) (12) d Vol(, g(t)) = dt 1 tr g 2 t dv = (r(t) R) dv = 0

g(t) Vol(, g(t)) t 2 1 1 S 1 2 13 Gauss Bonnet 2 (, g) χ() R dv = 4πχ() (12) r := t g(t) = (r R) g t R dv Vol(, g(t)) = 4πχ() Vol(, g(t)) 13 (Hamilton (1988), Chow (1991)) g 0 g(0) = g 0 g(t) t [0, ) t g(t) W. Thurston [Th] 3 3 (Geometrization conjecture) 14 ( ) 3 3 3 3 Hamilton [Ha] 15 (Hamilton [Ha]) 3 Ric(g 0 ) > 0 g 0 g(0) = g 0 g(t) t [0, ) t g(t) S 3 15 Hamilton (1986), Böhm Wilking (2008), Brendle Schoen (2009), Brendle (2008) Ric(g 0 ) > 0 4 Hamilton 3 Hamilton (Ricci flow with surgery) Perelman Perelman [Pe1, Pe2]

3. Perelman F- W- 16 (F- ( ) [Pe1, 1.1]) g f : R F(g, f) F(g, f) := (R + f 2 )e f dv. f f f 17 (F- [Pe1, 1.1]) g(t), f(t) g(t) = 2Ric(g(t)), t t f = f + f 2 R (18) d F(g(t), f(t)) = 2 Ric + f 2 e f dv 0. dt f f (0, 2)- :=, g(t) (18) g(t) u := e f u := u u + Ru = 0 (19) t d u dv = u dv = 0, (20) dt u(t) dv t 17 R t R = R + 2 Ric 2. Bochner φ : R 1 2 φ 2 = φ 2 + φ, φ + Ric( φ, φ). φ := tr φ = δdφ φ δ d Bianchi 2δRic = dr.

φ, ψ : R φ, ψ dv = φ( ψ) dv = ( φ)ψ dv. (u dv ) = ( u) dv t ( ) R d F(g(t), f(t)) = dt = t + t f 2 e f dv + ( R + 2 Ric 2 + 2 ( R + f 2 ) u dv. f t, f ( R + f 2 ) ( u) dv ) + 2Ric( f, f) e f dv Bochner 2 f t, f e f dv ( = 2 f + f 2 R ), f e f dv ( = f 2 + 2 f 2 + 2Ric( f, f) ) e f dv + 2 f 2 e f dv 2 R, f e f dv ( = f 2 + 2 f 2 + 2Ric( f, f) 2 R, f ) e f dv. Bianchi 4 Ric, f e f dv = 4 δric( f)e f dv 4 = 2 R, f e f dv + 4 d F(g(t), f(t)) = 2 dt = 2 Ric( f, (e f ))dv Ric( f, f)e f dv. ( Ric 2 + 2 Ric, f + f 2) e f dv Ric + f 2 e f dv 21 g { } λ(g) := inf F(g, f) : e f dv = 1 g(t) λ(g(t)) t

21 t 1 < t 2 e f dv = 1 F(g(t 2 ), f) = λ(t 2 ) f : R f(t 2 ) = f (19) (20) λ(t 1 ) F(g(t 1 ), f(t 1 )) F(g(t 2 ), f(t 2 )) = λ(t 2 ). λ(g(t)) 22 (W- ( ) [Pe1, 3.1]) n g f : R τ W(g, f, τ) [ W(g, f, τ) := τ( f 2 + R) + f n ] (4πτ) n/2 e f dv. 23 (W- [Pe1, 3.1], cf. [Pe1, 9.1]) n g(t), f(t), τ(t) g(t) = 2Ric(g(t)), t d W(g(t), f(t), τ(t)) = dt t f = f + f 2 R + n 2τ, 2τ 1 Ric + f 2τ g 2 dτ dt = 1 (24) (4πτ) n/2 e f dv 0. 25 n g τ { } µ(g, τ) := inf W(g, f, τ) : (4πτ) n/2 e f dv = 1 g(t) (24) τ(t) µ(g(t), τ(t)) t (24) g(t) u := (4πτ) n/2 e f (19) τ t 1 1 23 25 17 21 [Pe1] 5 W- [ ] [ ] 26 ([KL, To] ) n g u : (0, ) τ N (g, u) := u log u dv, Ñ (g, u, τ) := N n 2 (1 + log(4πτ)) u dv g(t), u(t), τ(t) (19) (24) F(g(t), u(t)) = d N (g(t), u(t)), dt W(g(t), u(t), τ(t)) = d (τ(t)ñ (g(t), u(t), τ(t))) dt N (g, u) Shannon

4. Perelman 23 g(t), t [0, T ) T < g(t ) 27 ( ) g(t), t (a, b) (x 0, t 0 ) (a, b) Q > 0 ( ) t g Q (t) := Q g Q + t 0, t (Q(t 0 a), Q(b t 0 )). g Q (t) g(t), t [0, T ) T < sup { Rm (x, t) : (x, t) [0, T )} = i = 1, 2,... (x i, t i ) [0, T ) Q i := Rm (x i, t i ) = max { Rm (x, t) : (x, t) [0, t i ]} i t i T Q i ( ) t g i (t) := Q i g + t i, t ( Q i t i, 0]. Q i {g i (t), t ( Q i t i, 0]} i N (, g i (0)) x i Perelman (, g) x r x r B g (x, r) V g (x, r) (4) d g B g (x, r) := {y : d g (x, y) < r}, V g (x, r) := 1 Bg (x,r) dv 1 B : R B p p B 1 B (p) = 1 p / B 1 B (p) = 0 28 ( [Pe1, 4.2]) ρ, κ n g ρ κ- x r < ρ B g (x, r) B g (x, r) Rm r 2 V g (x, r) κ r n 29 ( I [Pe1, 4.1]) g(t), t [0, T ) n g(0) T κ = κ(g(0), T ) > 0 t [0, T ) g(t) T κ-

29 30 ( I [CZ, To]) g(t), t [0, T ) n g(0) T κ = κ(g(0), T ) > 0 (x 0, t 0 ) [0, T ) 0 < r T B g(t0 )(x 0, r) R(, t 0 ) r 2 V g(t0 )(x 0, r) κ r n n (, g) x r > 0 ( V g (x, r) = ω n r n R(x) ) 6(n + 2) rn+2 + o(r n+2 ) (31) ω n n B n 30 g(t), t [0, T ) T R (, g) x 0 B g (x 0, r) Rm r 2 V g (x 0, r) κ r n Cheeger x 0 30 µ 0 := inf {µ(g(0), τ) : 0 < τ 2T }, { κ := min exp (µ 0 + n 1 + n ) 2 log(4π) 36 2n, ω } n 2 0 < r T B g(t0 )(x 0, r) R(, t 0 ) r 2 (x 0, t 0 ) [0, T ) µ 0 > B g(t0 )(x 0, r) V g(t0 )(x 0, r) B t0 (x 0, r) V t0 (x 0, r) ε C φ : [ε, ) x B t0 (x 0, r/2) φ(x) = C, x / B t0 (x 0, r) φ(x) = ε, x φ (x) 3C/r. C (4πr2 ) n/2 φ 2 dv = 1 1 = (4πr 2 ) n/2 φ 2 dv (4πr 2 ) n/2 C 2 V t0 (x 0, r/2). (32) f := 2 log φ φ 2 = e f 25 µ 0 µ(g(0), r 2 + t 0 ) µ(g(t 0 ), r 2 ) W(g(t 0 ), f, r 2 ) W(g, f, r 2 ) = (4πr 2 ) n/2 [ r 2 (4 φ 2 + Rφ 2 ) φ 2 log φ 2 nφ 2] dv. (33)

(33) ε = 0 (32) 36(4πr 2 ) n/2 C 2 V t0 (x 0, r) 36 V t 0 (x 0, r) V t0 (x 0, r/2), + 1 n. 34 (Jensen ) G : R R φ : R dµ ( G(φ) dµ dµ G φ dµ ) dµ G(a) := a log a dµ := 1 Bt0 (x 0,r)dV Jensen log V t 0 (x 0, r) (4πr 2 ) n/2 µ 0 36 V t 0 (x 0, r) V t0 (x 0, r/2) + log V t 0 (x 0, r) + 1 n. (4πr 2 ) n/2 V t0 (x 0, r) < κ r n κ V t0 (x 0, r/2) < κ(r/2) n i N (31) i V t0 (x 0, r/2 i ) < κ(r/2 i ) n ω n 2 κ > V t 0 (x 0, r/2 i ) (r/2 i ) n ω n V t0 (x 0, r) κ r n Hamilton (1995) 29 {(, g i (t), x i ), t ( Q i t i, 0]} i N n g (t), t (, 0] 35 ( ) (, 0) g(t), t (, 0] ancient solution

3 Hamilton Ivey 3 κ (κ-solution) Hamilton (1993) Harnack [Pe2, 1.5] Perelman [Pe1] 6, 7 (thermostat) L n g(t), t [0, T ] (reduced volume) τ := T t Ṽ (p,0) (τ) := (4πτ) n/2 e l (p,0)(,τ) dv I II F- W- Perelman 5. Hamilton Perelman [Pe2] 3 3 g(t), t [0, T ) T < [Pe2, 5] 14 (Thick-thin decomposition) [Pe2, 7] 36 ( [SY], cf. Perelman [Pe2, 7.4]) v 0 > 0 3 (, g) 1 Vol(, g) v 0 π 1 () Perelman 36 36 v 0 1 Vol( i, g i ) i Vol( i, g i ) 0 3 {( i, g i )} i N {( i, d gi )} i N (X, d) Gromov Hausdorff (X, d) 1 Alexandrov (X, d) 3 {( i, g i )} i N (1996) Perelman [Pe]

i ( i, g i ) (X, d) [ ] Perelman [Pe3] 3 Perelman 6. 37 (4 ) 4 S 4 4 S 4 J. ilnor (1956) (exotic sphere) 7 S 7 7 4 4 4 4. Freedman (1982) 37 4 4 4 ([ ] ) 38 arxiv.org (http://arxiv.org) arxiv: arxiv.org [ ] [ ] Perelman [ ] [ ] [ ] [ ] [ ] [ ] [To] [CK], [CLN] Perelman [ ], [ ], [CZ], [KL], [T] [Pe1, Pe2, Pe3] [CP] Perelman Hamilton https://math.berkeley.edu/ lott/ricciflow/perelman.html Perelman [CP] Collected papers on Ricci flow. Edited by H. D. Cao, B. Chow, S. C. Chu and S. T. Yau. Series in Geometry and Topology, 37. International Press, Somerville, A, 2003.

[CZ] [CK] [CLN] [Ha] [ ] H.-D. Cao and X.-P. Zhu, A complete proof of the Poincaré and geometrization conjectures application of the Hamilton-Perelman theory of the Ricci flow. Asian J. ath. 10 (2006), no. 2, 165 492; http://projecteuclid.org/euclid.ajm/1154098947. B. Chow and D. Knopf, The Ricci flow: an introduction. athematical Surveys and onographs, 110. American athematical Society, Providence, RI, 2004. B. Chow, P. Lu and L. Ni, Hamilton s Ricci flow. Graduate Studies in athematics, 77, AS, Providence, RI; Science Press, New York, 2006. R. S. Hamilton, Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), no. 2, 255 306; [CP].,. 7. ( ), 2010,. [KL] B. Kleiner and J. Lott, Notes on Perelman s papers. Geom. Topol. 12 (2008), no. 5, 2587 2855; arxiv:math/0605667. [ ] [ ] [T] [ ] [ ] [Pe] [Pe1] [Pe2] [Pe3], ( ). 2011,., 22 3. 2002,. J. organ and G. Tian, Ricci flow and the Poincaré conjecture. Clay athematics onographs, 3. American athematical Society, Providence, RI; Clay athematics Institute, Cambridge, A, 2007; arxiv:math/0607607., ( ). 2001,.,. 2007,. G. Perelman, Alexandrov s space with curvatures bounded from below II. Preprint, http://www.math.psu.edu/petrunin/papers/. The entropy formula for the Ricci flow and its geometric applications. Preprint, arxiv:math/0211159., Ricci flow with surgery on three manifolds. Preprint, arxiv:math/0303109., Finite extinction time for the solutions to the Ricci flow on certain threemanifolds. Preprint, arxiv:math/0307245. [Po] H. Poincaré, Cinquième complément à l analysis situs. Rend. Circ. at. Palermo 18 (1904), 45 110. [ ] [SY] [ ],. 2009,. T. Shioya and T. Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound. ath. Ann. 333 (2005), no. 1, 131 155; arxiv:math/0304472. :!, 2007,. [Th] W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. ath. Soc. (N.S.) 6 (1982), no. 3, 357 381; http://projecteuclid.org/euclid.bams/1183548782. [ ], 3. 2007,. [To] P.. Topping, Lectures on the Ricci flow. LS Lecture Note Series, 325. Cambridge University Press, Cambridge, 2006; http://homepages.warwick.ac.uk/ maseq/rfnotes.html. [ ], 1. 2005,.

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