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
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1 ( ) 1904 H. Poincaré 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 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
2 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
3 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) 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)
4 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
5 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, )
6 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
7 g(t) Vol(, g(t)) t S 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 ( ) 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]
8 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.
9 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 ( 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 f 2 + 2Ric( f, f) ) e f dv + 2 f 2 e f dv 2 R, f e f dv ( = f 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 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
10 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 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 [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
11 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-
12 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)
13 (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
14 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 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]
15 i ( i, g i ) (X, d) [ ] Perelman [Pe3] 3 Perelman (4 ) 4 S 4 4 S 4 J. ilnor (1956) (exotic sphere) 7 S Freedman (1982) ([ ] ) 38 arxiv.org ( arxiv: arxiv.org [ ] [ ] Perelman [ ] [ ] [ ] [ ] [ ] [ ] [To] [CK], [CLN] Perelman [ ], [ ], [CZ], [KL], [T] [Pe1, Pe2, Pe3] [CP] Perelman Hamilton 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.
16 [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, ; B. Chow and D. Knopf, The Ricci flow: an introduction. athematical Surveys and onographs, 110. American athematical Society, Providence, RI, B. Chow, P. Lu and L. Ni, Hamilton s Ricci flow. Graduate Studies in athematics, 77, AS, Providence, RI; Science Press, New York, R. S. Hamilton, Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), no. 2, ; [CP].,. 7. ( ), 2010,. [KL] B. Kleiner and J. Lott, Notes on Perelman s papers. Geom. Topol. 12 (2008), no. 5, ; arxiv:math/ [ ] [ ] [T] [ ] [ ] [Pe] [Pe1] [Pe2] [Pe3], ( ). 2011,., ,. 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/ , ( ). 2001,.,. 2007,. G. Perelman, Alexandrov s space with curvatures bounded from below II. Preprint, The entropy formula for the Ricci flow and its geometric applications. Preprint, arxiv:math/ , Ricci flow with surgery on three manifolds. Preprint, arxiv:math/ , Finite extinction time for the solutions to the Ricci flow on certain threemanifolds. Preprint, arxiv:math/ [Po] H. Poincaré, Cinquième complément à l analysis situs. Rend. Circ. at. Palermo 18 (1904), [ ] [SY] [ ],. 2009,. T. Shioya and T. Yamaguchi, Volume collapsed three-manifolds with a lower curvature bound. ath. Ann. 333 (2005), no. 1, ; arxiv:math/ :!, 2007,. [Th] W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry. Bull. Amer. ath. Soc. (N.S.) 6 (1982), no. 3, ; [ ], ,. [To] P.. Topping, Lectures on the Ricci flow. LS Lecture Note Series, 325. Cambridge University Press, Cambridge, 2006; maseq/rfnotes.html. [ ], ,.
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