Read the Perelman treatise of Poincare conjecture

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

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

Vocabulary Book

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

English word length

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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