Chapter 69: Complex structure degenerations and collapsing of Calabi-Yau metrics 001X

Sun, Song; Zhang, Ruobing

Original paper · arXiv:1906.03368v1

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Source edition: 1906.03368v1

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Statements and objects

1. Theorem 1.1 . [04YZ]
2. Remark 1.1.1 . [04Z0]
3. Remark 1.1.2 . [04Z1]
4. Remark 1.1.3 . [04Z2]
5. Remark 1.1.4 . [04Z3]
6. Definition 3.1 (Normal coordinates) . [04ZE]
7. Lemma 3.2 (Generalized Gauss Lemma) . [04ZF]
8. Definition 3.3 (Normal regularity order) . [04ZG]
9. Lemma 3.4 . [04ZH]
10. Proof. [04ZI]
11. Definition 3.5 ( k -current) . [04ZK]
12. Definition 3.6 (Green’s current) . [04ZL]
13. Example 3.7 . [04ZM]
14. Proposition 3.8 . [04ZN]
15. Proof. [04ZP]
16. Notation 3.9 . [04ZQ]
17. Remark 3.9.1 . [04ZR]
18. Theorem 3.10 . [04ZS]
19. Remark 3.10.1 . [04ZT]
20. Remark 3.10.2 . [04ZU]
21. Remark 3.10.3 . [04ZV]
22. Remark 3.10.4 . [04ZW]
23. Lemma 3.11 . [04ZX]
24. Proof. [04ZY]
25. Lemma 3.12 . [04ZZ]
26. Proof. [0500]
27. Lemma 3.13 . [0501]
28. Proof. [0502]
29. Lemma 3.14 (Rearrangement Lemma) . [0503]
30. Proof. [0504]
31. Proposition 3.15 . [0505]
32. Proof. [0506]
33. Lemma 3.16 (Cancellation Lemma) . [0507]
34. Proof. [0508]
35. Lemma 3.17 . [0509]
36. Proof. [050A]
37. Lemma 3.18 . [050B]
38. Proof. [050C]
39. Lemma 3.19 . [050D]
40. Proof. [050E]
41. Proposition 3.20 . [050F]
42. Proof. [050G]
43. Lemma 3.21 . [050H]
44. Proof. [050I]
45. Lemma 3.22 . [050J]
46. Proof. [050K]
47. Proof of Theorem 3.10 . [050L]
48. Lemma 3.23 . [050M]
49. Proof. [050N]
50. Proposition 3.24 . [050Q]
51. Proof. [050R]
52. Corollary 3.24.1 . [050S]
53. Proof. [050T]
54. Lemma 3.25 . [050U]
55. Proof. [050V]
56. Proposition 3.26 . [050W]
57. Lemma 3.27 . [050X]
58. Proof. [050Y]
59. Proof of Proposition 3.26 . [050Z]
60. Proposition 3.28 . [0510]
61. Lemma 3.29 . [0511]
62. Remark 3.29.1 . [0512]
63. Proof of Lemma 3.29 . [0513]
64. Proof of Proposition 3.28 . [0514]
65. Proposition 3.30 . [0515]
66. Proof. [0516]
67. Proposition 3.31 . [0518]
68. Remark 3.31.1 . [0519]
69. Lemma 3.32 . [051A]
70. Proof of Proposition 3.31 . [051B]
71. Lemma 3.33 . [051C]
72. Proof. [051D]
73. Lemma 4.1 . [051G]
74. Proof. [051H]
75. Lemma 4.2 . [051I]
76. Proof. [051J]
77. Definition 4.3 . [051K]
78. Remark 4.3.1 . [051L]
79. Remark 4.3.2 . [051M]
80. Lemma 4.4 . [051N]
81. Proof. [051P]
82. Proposition 4.5 . [051Q]
83. Proof. [051R]
84. Remark 4.5.1 . [051S]
85. Lemma 4.6 . [051T]
86. Proof. [051U]
87. Proposition 4.7 . [051V]
88. Proof. [051W]
89. Proposition 4.8 . [051Z]
90. Remark 4.8.1 . [0520]
91. Proof. [0521]
92. Remark 4.8.2 . [0522]
93. Lemma 4.9 . [0523]
94. Proof. [0524]
95. Lemma 4.10 . [0525]
96. Proof. [0526]
97. Proposition 4.11 . [0527]
98. Proof. [0528]
99. Corollary 4.11.1 . [0529]
100. Proof. [052A]
101. Proposition 4.12 . [052C]
102. Remark 4.12.1 . [052D]
103. Proof. [052E]
104. Remark 4.12.2 . [052F]
105. Remark 4.12.3 . [052G]
106. Lemma 4.13 . [052H]
107. Proof. [052I]
108. Definition 4.14 (Local regularity) . [052K]
109. Definition 4.15 ( C k , α -regularity scale) . [052L]
110. Example 4.16 . [052M]
111. Example 4.17 . [052N]
112. Remark 4.17.1 . [052P]
113. Proposition 4.18 (Regularity scale on ℳ T ) . [052Q]
114. Remark 4.18.1 . [052R]
115. Remark 4.18.2 . [052S]
116. Remark 4.18.3 . [052T]
117. Corollary 4.18.1 (Harnack inequality for the regularity scale) . [052U]
118. Definition 4.19 (Weight function) . [052W]
119. Remark 4.19.1 . [052X]
120. Remark 4.19.2 . [052Y]
121. Remark 4.19.3 . [052Z]
122. Lemma 4.20 (Lower bound estimate for the weight function) . [0530]
123. Proof. [0531]
124. Definition 4.21 (Weighted Hölder space) . [0532]
125. Remark 4.21.1 . [0533]
126. Proposition 4.22 (Weighted Schauder estimate, the local version) . [0534]
127. Remark 4.22.1 . [0535]
128. Proof. [0536]
129. Proposition 4.23 (Weighted error estimate) . [0537]
130. Proof. [0538]
131. Proposition 4.24 . [053A]
132. Lemma 4.25 . [053B]
133. Proof. [053C]
134. Remark 4.25.1 . [053D]
135. Definition 5.1 ( δ -aysmptotically Calabi space) . [053F]
136. Theorem 5.2 (Liouville Theorem) . [053G]
137. Remark 5.2.1 . [053H]
138. Remark 5.2.2 . [053I]
139. Remark 5.2.3 . [053K]
140. Remark 5.2.4 . [053L]
141. Proposition 5.3 . [053N]
142. Proof. [053P]
143. Corollary 5.3.1 . [053Q]
144. Proof. [053R]
145. Lemma 5.4 . [053S]
146. Proposition 5.5 . [053T]
147. Proof. [053U]
148. Corollary 5.5.1 . [053V]
149. Proposition 5.6 . [053X]
150. Proof. [053Y]
151. Lemma 5.7 . [053Z]
152. Lemma 5.8 . [0540]
153. Proof. [0541]
154. Proposition 5.9 . [0542]
155. Proof. [0543]
156. Corollary 5.9.1 . [0544]
157. Proposition 5.10 . [0545]
158. Proof. [0546]
159. Lemma 5.11 . [0547]
160. Proof. [0548]
161. Lemma 5.12 . [0549]
162. Remark 5.12.1 . [054A]
163. Proof. [054B]
164. Corollary 5.12.1 . [054C]
165. Lemma 5.13 . [054E]
166. Proof. [054F]
167. Proposition 5.14 (Asymptotics of harmonic functions) . [054G]
168. Proof. [054H]
169. Lemma 5.15 . [054J]
170. Proof. [054K]
171. Proposition 5.16 . [054L]
172. Proof. [054M]
173. Lemma 5.17 . [054P]
174. Proof of Theorem 5.2 . [054Q]
175. Lemma 6.1 (Implicit function theorem) . [054T]
176. Remark 6.1.1 . [054U]
177. Lemma 6.2 . [054V]
178. Proof. [054W]
179. Theorem 6.3 (Existence of S 1 -invariant Calabi-Yau metrics) . [054X]
180. Lemma 6.4 (Nonlinear error estimate) . [054Y]
181. Proof. [054Z]
182. Lemma 6.5 (Removable singularity) . [0551]
183. Proof. [0552]
184. Lemma 6.6 (Liouville theorem on ℝ m + n ) . [0553]
185. Proof. [0554]
186. Lemma 6.7 (Liouville theorem on a cylinder) . [0555]
187. Proposition 6.8 (Uniform injectivity estimate on the neck) . [0557]
188. Proposition 6.9 (Weighted Schauder estimate on the neck, the global version) . [0558]
189. Proof. [0559]
190. Proposition 6.10 (Uniform injectivity estimate on the neck) . [055A]
191. Proof. [055B]
192. Proof of Theorem 6.3 . [055C]
193. Definition 6.11 (Measured Gromov-Hausdorff convergence) . [055E]
194. Remark 6.11.1 . [055F]
195. Remark 6.11.2 . [055G]
196. Remark 6.11.3 . [055H]
197. Lemma 7.1 . [055M]
198. Proof. [055N]
199. Lemma 7.2 . [055Q]
200. Proof. [055R]
201. Lemma 7.3 . [055S]
202. Proof. [055T]
203. Proposition 7.4 ( [ TY90 ] , see also [ HSVZ18 ] ) . [055V]
204. Lemma 7.5 . [055W]
205. Remark 7.5.1 . [055Y]
206. Lemma 7.6 . [0562]
207. Proof. [0563]
208. Proposition 7.7 . [0564]
209. Proof. [0565]
210. Proposition 7.8 . [0566]
211. Proof. [0567]
212. Proposition 7.9 . [0568]
213. Proof. [0569]
214. Proposition 7.10 . [056A]
215. Proposition 7.11 . [056B]
216. Proposition 7.12 . [056C]
217. Remark 7.12.1 . [056D]
218. Proposition 7.13 (Nonlinear error estimate) . [056F]
219. Proposition 7.14 (Weighted Schauder estimate, the global version) . [056G]
220. Proposition 7.15 (Global injectivity estimates) . [056H]
221. Lemma 8.1 . [056K]
222. Proof. [056L]
223. Remark 8.1.1 . [056M]
224. Lemma A.1 . [056R]
225. Proof. [056S]
226. Lemma A.2 . [056U]
227. Proof. [056V]
228. Lemma A.3 . [056W]
229. Proof. [056X]
230. Lemma A.4 . [056Y]
231. Proof. [056Z]
232. Lemma A.5 . [0570]
233. Proof. [0571]
234. Lemma A.6 (Kummer’s transformation law) . [0572]
235. Proof. [0573]
236. Lemma A.7 . [0574]
237. Proof. [0575]
238. Lemma A.8 . [0576]
239. Proof. [0577]
240. Corollary A.8.1 . [0578]
241. Proof. [0579]

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