クレイ数学研究所の数学未解決問題７つのうち、唯一解決済みのポアンカレ予想のペレルマン論文を読もうと思った。
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
参考文献(Reference)
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
単語帳
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
英単語帳
３つの論文の英単語帳を作った。
| count | word | 日本語 | 備考 |
|---|---|---|---|
| 1775 | the | その | |
| 1337 | t | t | |
| 803 | a | 一つの | |
| 775 | of | の | |
| 654 | r | r | |
| 638 | x | x | |
| 582 | is | です | |
| 565 | and | そして | |
| 490 | in | に | |
| 459 | to | に | |
| 437 | that | それ | |
| 424 | we | 私達 | |
| 397 | for | にとって | |
| 320 | on | 上に | |
| 264 | with | と | |
| 242 | at | で | |
| 241 | m | m | |
| 212 | can | できる | |
| 212 | y | y | |
| 210 | by | 沿って | |
| 195 | curvature | 曲率 | |
| 192 | f | f | |
| 189 | then | その後 | |
| 184 | h | h | |
| 177 | b | b | |
| 171 | this | この | |
| 169 | solution | 解決 | |
| 168 | be | あります | |
| 167 | flow | 流れ | |
| 164 | c | c | |
| 155 | n | n | |
| 154 | time | 時間 | |
| 153 | ricci | ricci | 人名 |
| 151 | l | l | |
| 148 | gij | gij | |
| 144 | if | もし | |
| 142 | as | なので | |
| 135 | i | 私 | |
| 135 | it | それ | |
| 131 | d | d | |
| 130 | such | そのような | |
| 129 | from | から | |
| 124 | an | 一つの | |
| 122 | k | k | |
| 119 | q | q | |
| 119 | s | s | |
| 118 | g | g | |
| 116 | not | ない | |
| 110 | metric | 測定基準 | |
| 109 | where | どこ | |
| 105 | have | 持ってる | |
| 104 | p | p | |
| 100 | one | 一つ | |
| 100 | w | w | |
| 94 | are | です | |
| 94 | which | これ | |
| 93 | proof | 証明 | |
| 92 | v | v | |
| 87 | let | しましょう | |
| 87 | some | いくつか | |
| 82 | any | どれか | |
| 81 | point | 点 | |
| 80 | manifold | 多様な | |
| 80 | z | z | |
| 79 | limit | 制限 | |
| 77 | now | 今 | |
| 75 | bounded | 跳ねる | |
| 74 | each | 各 | |
| 72 | has | 持っている | |
| 72 | our | 私たちの | |
| 71 | there | そこ | |
| 71 | volume | 量 | |
| 69 | or | または | |
| 68 | case | 場合 | |
| 67 | theorem | 定理 | |
| 66 | ball | 玉 | |
| 65 | all | すべて | |
| 65 | ric | ric | |
| 63 | solutions | 解 | |
| 61 | claim | 請求 | |
| 61 | hamilton | ハミルトン | 人名 |
| 61 | scalar | 変量 | |
| 59 | estimate | 見積もり | |
| 55 | function | 関数 | |
| 52 | get | 取得する | |
| 52 | j | j | |
| 51 | e | e | |
| 51 | satisfies | 満たす | |
| 51 | thus | したがって、 | |
| 51 | zero | ゼロ | |
| 50 | smooth | 滑らかな | |
| 48 | assume | 仮定する | |
| 48 | equation | 方程式 | |
| 48 | rm | rm | |
| 47 | also | また | |
| 47 | defined | 定義済み | |
| 47 | finite | 有限の | |
| 47 | so | そう | |
| 47 | surgery | 手術 | |
| 46 | ct | ct | |
| 46 | small | 小さい | |
| 46 | u | u | |
| 45 | least | 少なくとも | |
| 45 | lemma | 補題 | |
| 45 | radius | 半径 | |
| 44 | dt | dt | |
| 43 | rij | rij | |
| 42 | consider | 検討する | |
| 42 | large | 大 | |
| 42 | positive | 肯定 | |
| 41 | follows | 続く | |
| 41 | neck | 首 | |
| 41 | neighborhood | ご近所 | |
| 41 | other | その他 | |
| 41 | satisfying | 満足 | |
| 41 | suppose | 仮定します | |
| 41 | tk | tk | |
| 40 | nonnegative | 非負 | |
| 39 | interval | 間隔 | |
| 39 | points | 点 | |
| 39 | therefore | したがって、 | |
| 39 | using | を使用して | |
| 38 | assumptions | 仮定 | |
| 38 | every | すべて | |
| 38 | exists | 存在する | |
| 38 | following | 以下 | |
| 38 | three | 三 | |
| 38 | whenever | いつでも | |
| 37 | argument | 引数 | |
| 37 | first | 最初 | |
| 37 | its | その | |
| 37 | manifolds | 多様な | |
| 37 | round | 円形 | |
| 36 | closed | 閉まっている | |
| 36 | may | 五月 | |
| 36 | take | 取る | |
| 35 | ancient | 古代 | |
| 35 | close | 閉じる | |
| 35 | find | 見つける | |
| 35 | only | のみ | |
| 35 | since | 以来 | |
| 34 | bound | 雪の | |
| 34 | curve | 曲線 | |
| 34 | distt | 宛先t | |
| 33 | when | いつ | |
| 32 | assumption | 仮定 | |
| 32 | constant | 絶え間ない | |
| 32 | gradient | 勾配 | |
| 32 | soliton | 孤立波 | |
| 32 | than | より | |
| 31 | above | 上記 | |
| 31 | sequence | 列 | |
| 31 | was | だった | |
| 30 | does | しますか | |
| 30 | hand | 手 | |
| 30 | implies | 意味する | |
| 30 | indeed | 確かに | |
| 30 | inequality | 不平等 | |
| 30 | metrics | 測定基準 | |
| 30 | would | だろう | |
| 29 | curvatures | 曲率 | |
| 29 | same | 同じ | |
| 29 | sectional | 断面 | |
| 28 | clearly | 明らかに | |
| 28 | const | 定数 | |
| 28 | corollary | 当然の結果 | |
| 28 | formula | 式 | |
| 28 | see | 見る | |
| 27 | canonical | 正準 | |
| 27 | either | どちらか | |
| 27 | enough | 足りる | |
| 27 | infinity | 無限 | |
| 27 | scale | 規模 | |
| 26 | apply | 適用する | |
| 26 | given | 与えられた | |
| 26 | math | 数学 | |
| 26 | moreover | さらに | |
| 26 | proposition | 命題 | |
| 26 | section | 節 | |
| 25 | but | だが | |
| 25 | initial | 初期 | |
| 25 | monotonicity | 単調性 | |
| 25 | non | 非 | |
| 25 | particular | 特に | |
| 25 | riemannian | リーマン多様体 | 人名 |
| 25 | times | 時間 | |
| 24 | almost | ほとんど | |
| 24 | complete | 完全な | |
| 24 | distance | 距離 | |
| 24 | property | 特性 | |
| 24 | standard | 標準 | |
| 24 | xk | xk | |
| 23 | along | に沿って | |
| 23 | contradiction | 矛盾 | |
| 23 | dimension | 寸法 | |
| 23 | factor | 因子 | |
| 23 | ij | ij | |
| 23 | no | 否定 | |
| 23 | rk | rk | |
| 22 | more | もっと |
#docker
単語帳はdockerに置いて更新中。
$ docker run -it kaizenjapan/perelman /bin/bash
自己参考資料(self reference)
英語(24)アンの部屋（人名から学ぶ数学：岩波数学辞典）
https://qiita.com/kaizen_nagoya/items/e02cbe23b96d5fb96aa1
岩波数学辞典
https://qiita.com/kaizen_nagoya/items/b37bfd303658cb5ee11e
文書履歴(document history)
ver. 0.01 初稿 20210104
ver. 0.02 誤字訂正、参考資料追記 20210123
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