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

最後までおよみいただきありがとうございました。

いいね　💚、フォローをお願いします。

Thank you very much for reading to the last sentence.

Please press the like icon 💚　and follow me for your happy life.