Of the seven unsolved mathematics problems at the Clay Mathematics Institute, I decided to read the only solved Poincare conjecture Perelman treatise.
Perelman, Grisha (11 November 2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159。https://arxiv.org/abs/math.DG/0211159
Perelman, Grisha (10 March 2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109。https://arxiv.org/abs/math.DG/0303109
Perelman, Grisha (17 July 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245。https://arxiv.org/abs/math.DG/0307245
The entropy formula for the Ricci flow and its geometric applications [A] M.T.Anderson Scalar curvature and geometrization conjecture for three-manifolds. Comparison Geometry (Berkeley, 1993-94), MSRI Publ. 30 (1997), 49-82. [B-Em] D.Bakry, M.Emery Diffusions hypercontractives. Seminaire de Probabilites XIX, 1983-84, Lecture Notes in Math. 1123 (1985), 177-206. [Cao-C] H.-D. Cao, B.Chow Recent developments on the Ricci flow. Bull. AMS 36 (1999), 59-74. [Ch-Co] J.Cheeger, T.H.Colding On the structure of spaces with Ricci curvature bounded below I. Jour. Diff. Geom. 46 (1997), 406-480. [C] B.Chow Entropy estimate for Ricci flow on compact two-orbifolds. Jour. Diff. Geom. 33 (1991), 597-600. [C-Chu 1] B.Chow, S.-C. Chu A geometric interpretation of Hamilton’s Harnack inequality for the Ricci flow. Math. Res. Let. 2 (1995), 701-718. [C-Chu 2] B.Chow, S.-C. Chu A geometric approach to the linear trace Harnack inequality for the Ricci flow. Math. Res. Let. 3 (1996), 549-568. [D] E.D’Hoker String theory. Quantum fields and strings: a course for mathematicians (Princeton, 1996-97), 807-1011. [E 1] K.Ecker Logarithmic Sobolev inequalities on submanifolds of euclidean space. Jour. Reine Angew. Mat. 522 (2000), 105-118. [E 2] K.Ecker A local monotonicity formula for mean curvature flow. Ann. Math. 154 (2001), 503-525. [E-Hu] K.Ecker, G.Huisken In terior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105 (1991), 547-569. [Gaw] K.Gawedzki Lectures on conformal field theory. Quantum fields and strings: a course for mathematicians (Princeton, 1996-97), 727-805. [G] L.Gross Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms (Varenna, 1992) Lecture Notes in Math. 1563 (1993), 54-88. [H 1] R.S.Hamilton Three manifolds with positive Ricci curvature. Jour. Diff. Geom. 17 (1982), 255-306. [H 2] R.S.Hamilton Four manifolds with positive curvature operator. Jour. Diff. Geom. 24 (1986), 153-179. [H 3] R.S.Hamilton The Harnack estimate for the Ricci flow. Jour. Diff. Geom. 37 (1993), 225-243. [H 4] R.S.Hamilton Formation of singularities in the Ricci flow. Surveys in Diff. Geom. 2 (1995), 7-136. 38 [H 5] R.S.Hamilton Four-manifolds with positive isotropic curvature. Commun. Anal. Geom. 5 (1997), 1-92. [H 6] R.S.Hamilton Non-singular solutions of the Ricci flow on threemanifolds. Commun. Anal. Geom. 7 (1999), 695-729. [H 7] R.S.Hamilton A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1 (1993), 113-126. [H 8] R.S.Hamilton Monotonicity formulas for parabolic flows on manifolds. Commun. Anal. Geom. 1 (1993), 127-137. [H 9] R.S.Hamilton A compactness property for solutions of the Ricci flow. Amer. Jour. Math. 117 (1995), 545-572. [H 10] R.S.Hamilton The Ricci flow on surfaces. Contemp. Math. 71 (1988), 237-261. [Hu] G.Huisken Asymptotic behavior for singularities of the mean curvature flow. Jour. Diff. Geom. 31 (1990), 285-299. [I] T.Ivey Ricci solitons on compact three-manifolds. Diff. Geo. Appl. 3 (1993), 301-307. [L-Y] P.Li, S.-T. Yau On the parabolic kernel of the Schrodinger operator. Acta Math. 156 (1986), 153-201. [Lott] J.Lott Some geometric properties of the Bakry-Emery-Ricci tensor. arXiv:math.DG/0211065. https://arxiv.org/abs/math/0211065
Ricci flow with surgery on three-manifolds
[I] G.Perelman The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 v1 [A] M.T.Anderson Scalar curvature and geometrization conjecture for threemanifolds. Comparison Geometry (Berkeley, 1993-94), MSRI Publ. 30 (1997), 49-82. [C-G] J.Cheeger, M.Gromov Collapsing Riemannian manifolds while keeping their curvature bounded I. Jour. Diff. Geom. 23 (1986), 309-346. [G-L] M.Gromov, H.B.Lawson Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. IHES 58 (1983), 83-196. [H 1] R.S.Hamilton Three-manifolds with positive Ricci curvature. Jour. Diff. Geom. 17 (1982), 255-306. [H 2] R.S.Hamilton Formation of singularities in the Ricci flow. Surveys in Diff. Geom. 2 (1995), 7-136. [H 3] R.S.Hamilton The Harnack estimate for the Ricci flow. Jour. Diff. Geom. 37 (1993), 225-243. [H 4] R.S.Hamilton Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7 (1999), 695-729. [H 5] R.S.Hamilton Four-manifolds with positive isotropic curvature. Commun. Anal. Geom. 5 (1997), 1-92. G.Perelman Spaces with curvature bounded below. Proceedings of ICM- 1994, 517-525. F.Waldhausen Eine Klasse von 3-dimensionalen Mannigfaltigkeiten I,II. Invent. Math. 3 (1967), 308-333 and 4 (1967), 87-117.
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
[A-G] S.Altschuler, M.Grayson Shortening space curves and flow through singularities. Jour. Diff. Geom. 35 (1992), 283-298. [B] S.Bando Real analyticity of solutions of Hamilton’s equation. Math. Zeit. 195 (1987), 93-97. [E-Hu] K.Ecker, G.Huisken Interior estimates for hypersurfaces moving by mean curvature. Invent. Math. 105 (1991), 547-569. [G-H] M.Gage, R.S.Hamilton The heat equation shrinking convex plane curves. Jour. Diff. Geom. 23 (1986), 69-96. [H] R.S.Hamilton Non-singular solutions of the Ricci flow on three-manifolds. Commun. Anal. Geom. 7 (1999), 695-729. [Hi] S.Hildebrandt Boundary behavior of minimal surfaces. Arch. Rat. Mech. Anal. 35 (1969), 47-82. [M] C.B.Morrey The problem of Plateau on a riemannian manifold. Ann. Math. 49 (1948), 807-851. [P] G.Perelman Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109 v1 https://arxiv.org/abs/math.DG/0303109
The entropy formula for the Ricci flow and its geometric applications Ricci flow equation positive Ricci curvature Richard Hamilton Riemannian metric arbitrary (smooth) metric curvature tensor closed manifold. evolution equation metric tensor implies quadratic expression of the curvatures. scalar curvature maximum principle
Ricci flow with surgery on three-manifolds
Finite extinction time for the solutions to the Ricci flow on certain three-manifolds
I made an English word book for three treatises.
| count | word | Japanese | Remarks |
|---|---|---|---|
| 1775 | the | That | |
| 1337 | t | t | |
| 803 | a | A | |
| 775 | of | of | |
| 654 | r | r | |
| 638 | x | x | |
| 582 | is | is | |
| 565 | and | And | |
| 490 | in | To | |
| 459 | to | To | |
| 437 | that | It | |
| 424 | we | we | |
| 397 | for | for | |
| 320 | on | above | |
| 264 | with | When | |
| 242 | at | so | |
| 241 | m | m | |
| 212 | can | it can | |
| 212 | y | y | |
| 210 | by | Along | |
| 195 | curvature | curvature | |
| 192 | f | f | |
| 189 | then | afterwards | |
| 184 | h | h | |
| 177 | b | b | |
| 171 | this | this | |
| 169 | solution | Solution | |
| 168 | be | There is | |
| 167 | flow | flow | |
| 164 | c | c | |
| 155 | n | n | |
| 154 | time | time | |
| 153 | ricci | ricci | Personal name |
| 151 | l | l | |
| 148 | gij | gij | |
| 144 | if | if | |
| 142 | as | So | |
| 135 | i | I | |
| 135 | it | It | |
| 131 | d | d | |
| 130 | such | like that | |
| 129 | from | From | |
| 124 | an | AN | |
| 122 | k | k | |
| 119 | q | q | |
| 119 | s | s | |
| 118 | g | g | |
| 116 | not | Absent | |
| 110 | metric | Measurement standard | |
| 109 | where | Where | |
| 105 | have | Have | |
| 104 | p | p | |
| 100 | one | 1 | |
| 100 | w | w | |
| 94 | are | is | |
| 94 | which | this | |
| 93 | proof | Proof | |
| 92 | v | v | |
| 87 | let | let's do it | |
| 87 | some | A few | |
| 82 | any | Any | |
| 81 | point | point | |
| 80 | manifold | Various | |
| 80 | z | z | |
| 79 | limit | Limits | |
| 77 | now | now | |
| 75 | bounded | Bounce | |
| 74 | each | each | |
| 72 | has | have | |
| 72 | our | our | |
| 71 | there | There | |
| 71 | volume | amount | |
| 69 | or | Or | |
| 68 | case | If | |
| 67 | theorem | theorem | |
| 66 | ball | ball | |
| 65 | all | all | |
| 65 | ric | ric | |
| 63 | solutions | Solution | |
| 61 | claim | Claim | |
| 61 | hamilton | Hamilton | Personal name |
| 61 | scalar | variable | |
| 59 | estimate | Estimate | |
| 55 | function | function | |
| 52 | get | get | |
| 52 | j | j | |
| 51 | e | e | |
| 51 | satisfies | Fulfill | |
| 51 | thus | Therefore, | |
| 51 | zero | zero | |
| 50 | smooth | Smooth | |
| 48 | assume | Assuming | |
| 48 | equation | equation | |
| 48 | rm | rm | |
| 47 | also | Also | |
| 47 | defined | Predefined | |
| 47 | finite | Finite | |
| 47 | so | so | |
| 47 | surgery | Surgery | |
| 46 | ct | ct | |
| 46 | small | small | |
| 46 | u | u | |
| 45 | least | at least | |
| 45 | lemma | Lemma | |
| 45 | radius | radius | |
| 44 | dt | dt | |
| 43 | rij | rij | |
| 42 | consider | consider | |
| 42 | large | Big | |
| 42 | positive | positive | |
| 41 | follows | Continue | |
| 41 | neck | neck | |
| 41 | neighborhood | neighborhood | |
| 41 | other | Other | |
| 41 | satisfying | Satisfaction | |
| 41 | suppose | Suppose | |
| 41 | tk | tk | |
| 40 | nonnegative | Non-negative | |
| 39 | interval | interval | |
| 39 | points | point | |
| 39 | therefore | Therefore, | |
| 39 | using | using | |
| 38 | assumptions | Assumption | |
| 38 | every | all | |
| 38 | exists | Exists | |
| 38 | following | Less than | |
| 38 | three | three | |
| 38 | whenever | anytime | |
| 37 | argument | argument | |
| 37 | first | the first | |
| 37 | its | That | |
| 37 | manifolds | Various | |
| 37 | round | Round | |
| 36 | closed | Closed | |
| 36 | may | May | |
| 36 | take | take | |
| 35 | ancient | Ancient | |
| 35 | close | close | |
| 35 | find | locate | |
| 35 | only | only | |
| 35 | since | Since then | |
| 34 | bound | the snow's | |
| 34 | curve | curve | |
| 34 | distt | Destination t | |
| 33 | when | When | |
| 32 | assumption | Assumption | |
| 32 | constant | Continuous | |
| 32 | gradient | Slope | |
| 32 | soliton | Solitary wave | |
| 32 | than | Than | |
| 31 | above | the above | |
| 31 | sequence | Column | |
| 31 | was | was | |
| 30 | does | will you do | |
| 30 | hand | hand | |
| 30 | implies | means | |
| 30 | indeed | surely | |
| 30 | inequality | inequality | |
| 30 | metrics | Measurement standard | |
| 30 | would | right | |
| 29 | curvatures | curvature | |
| 29 | same | the same | |
| 29 | sectional | cross section | |
| 28 | clearly | clearly | |
| 28 | const | constant | |
| 28 | corollary | Natural result | |
| 28 | formula | formula | |
| 28 | see | to see | |
| 27 | canonical | Canonical | |
| 27 | either | which one | |
| 27 | enough | Sufficient | |
| 27 | infinity | infinite | |
| 27 | scale | scale | |
| 26 | apply | Apply | |
| 26 | given | Given the | |
| 26 | math | Math | |
| 26 | moreover | further | |
| 26 | proposition | proposition | |
| 26 | section | section | |
| 25 | but | But | |
| 25 | initial | initial | |
| 25 | monotonicity | Monotonic | |
| 25 | non | Non | |
| 25 | particular | In particular | |
| 25 | riemannian | Riemannian manifold | |
| 25 | times | time | |
| 24 | almost | Almost | |
| 24 | complete | Complete | |
| 24 | distance | distance | |
| 24 | property | Characteristic | |
| 24 | standard | standard | |
| 24 | xk | xk | |
| 23 | along | along | |
| 23 | contradiction | Contradiction | |
| 23 | dimension | Size | |
| 23 | factor | factor | |
| 23 | ij | ij | |
| 23 | no | denial | |
| 23 | rk | rk | |
| 22 | more | More |
docker The word book is being updated in docker.
$ docker run -it kaizenjapan/perelman /bin/bash
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