Geometry & Topology Volume 25, issue 1 (2021)

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Complete noncompact Spin(7) manifolds from self-dual Einstein $4$–orbifolds

Lorenzo Foscolo

Geometry & Topology 25 (2021) 339–408

DOI: 10.2140/gt.2021.25.339

Bibliography

1	A Adem, J Leida, Y Ruan, Orbifolds and stringy topology, 171, Cambridge Univ. Press (2007) MR2359514
2	V Apostolov, S Salamon, Kähler reduction of metrics with holonomy G₂, Comm. Math. Phys. 246 (2004) 43 MR2044890
3	C Bär, Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993) 509 MR1224089
4	R Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986) 661 MR849427
5	Y V Bazaikin, On new examples of complete noncompact metrics with Spin(7) holonomy group, Sibirsk. Mat. Zh. 48 (2007) 11 MR2304875
6	Y V Bazaikin, Noncompact Riemann spaces with the holonomy group Spin(7) and 3–Sasakian manifolds, Tr. Mat. Inst. Steklova 263 (2008) 6 MR2599368
7	O Biquard, V Minerbe, A Kummer construction for gravitational instantons, Comm. Math. Phys. 308 (2011) 773 MR2855540
8	J M Bismut, The Atiyah–Singer index theorem for families of Dirac operators : two heat equation proofs, Invent. Math. 83 (1986) 91 MR813584
9	J E Borzellino, Orbifolds of maximal diameter, Indiana Univ. Math. J. 42 (1993) 37 MR1218706
10	J E Borzellino, S H Zhu, The splitting theorem for orbifolds, Illinois J. Math. 38 (1994) 679 MR1283015
11	C P Boyer, K Galicki, Sasakian geometry, Oxford Univ. Press (2008) MR2382957
12	C P Boyer, K Galicki, B M Mann, The geometry and topology of 3–Sasakian manifolds, J. Reine Angew. Math. 455 (1994) 183 MR1293878
13	C P Boyer, K Galicki, B M Mann, Hypercomplex structures from 3–Sasakian structures, J. Reine Angew. Math. 501 (1998) 115 MR1637849
14	C P Boyer, K Galicki, B M Mann, E G Rees, Compact 3–Sasakian 7–manifolds with arbitrary second Betti number, Invent. Math. 131 (1998) 321 MR1608567
15	R L Bryant, Metrics with exceptional holonomy, Ann. of Math. 126 (1987) 525 MR916718
16	R L Bryant, Some remarks on G₂–structures, from: "Proceedings of Gökova Geometry-Topology Conference 2005" (editors S Akbulut, T Önder, R J Stern), GGT (2006) 75 MR2282011
17	R L Bryant, S M Salamon, On the construction of some complete metrics with exceptional holonomy, Duke Math. J. 58 (1989) 829 MR1016448
18	E Calabi, Métriques kählériennes et fibrés holomorphes, Ann. Sci. École Norm. Sup. 12 (1979) 269 MR543218
19	D M J Calderbank, M A Singer, Toric self-dual Einstein metrics on compact orbifolds, Duke Math. J. 133 (2006) 237 MR2225692
20	J Cheeger, T H Colding, Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. of Math. 144 (1996) 189 MR1405949
21	J Cheeger, G Tian, On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math. 118 (1994) 493 MR1296356
22	C van Coevering, Ricci-flat Kähler metrics on crepant resolutions of Kähler cones, Math. Ann. 347 (2010) 581 MR2640044
23	T C Collins, G Székelyhidi, Sasaki–Einstein metrics and K–stability, Geom. Topol. 23 (2019) 1339 MR3956894
24	R J Conlon, H J Hein, Asymptotically conical Calabi–Yau manifolds, I, Duke Math. J. 162 (2013) 2855 MR3161306
25	M Cvetič, G W Gibbons, H Lü, C N Pope, Almost special holonomy in type IIA and M–theory, Nuclear Phys. B 638 (2002) 186 MR1924090
26	M Cvetič, G W Gibbons, H Lü, C N Pope, Cohomogeneity one manifolds of Spin(7) and G₂ holonomy, Phys. Rev. D 65 (2002) MR1919035
27	M Cvetič, G W Gibbons, H Lü, C N Pope, New complete noncompact Spin(7) manifolds, Nuclear Phys. B 620 (2002) 29 MR1873145
28	M Cvetič, G W Gibbons, H Lü, C N Pope, New cohomogeneity one metrics with Spin(7) holonomy, J. Geom. Phys. 49 (2004) 350 MR2076750
29	M Cvetič, H Lü, D N Page, C N Pope, New Einstein–Sasaki spaces in five and higher dimensions, Phys. Rev. Lett. 95 (2005) MR2167018
30	J Eells, S Salamon, Twistorial construction of harmonic maps of surfaces into four-manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985) 589 MR848842
31	L Foscolo, Deformation theory of nearly Kähler manifolds, J. Lond. Math. Soc. 95 (2017) 586 MR3656283
32	L Foscolo, M Haskins, J Nordström, Complete non-compact G₂–manifolds from asymptotically conical Calabi–Yau 3–folds, preprint (2017) arXiv:1709.04904
33	L Foscolo, M Haskins, J Nordström, Infinitely many new families of complete cohomogeneity one G₂–manifolds : G₂ analogues of the Taub–NUT and Eguchi–Hanson spaces, preprint (2018) arXiv:1805.02612
34	T Friedrich, I Kath, A Moroianu, U Semmelmann, On nearly parallel G₂–structures, J. Geom. Phys. 23 (1997) 259 MR1484591
35	K Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, from: "Recent topics in differential and analytic geometry" (editor T Ochiai), Adv. Stud. Pure Math. 18, Academic (1990) 143 MR1145256
36	A Futaki, H Ono, G Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki–Einstein manifolds, J. Differential Geom. 83 (2009) 585 MR2581358
37	K Galicki, H B Lawson Jr., Quaternionic reduction and quaternionic orbifolds, Math. Ann. 282 (1988) 1 MR960830
38	K Galicki, T Nitta, Nonzero scalar curvature generalizations of the ALE hyper-Kähler metrics, J. Math. Phys. 33 (1992) 1765 MR1158996
39	K Galicki, S Salamon, Betti numbers of 3–Sasakian manifolds, Geom. Dedicata 63 (1996) 45 MR1413621
40	J P Gauntlett, D Martelli, J Sparks, D Waldram, Sasaki–Einstein metrics on S² × S³, Adv. Theor. Math. Phys. 8 (2004) 711 MR2141499
41	G Gibbons, S Hawking, Gravitational multi-instantons, Phys. Lett. B 78 (1978) 430
42	S Goette, Adiabatic limits of Seifert fibrations, Dedekind sums, and the diffeomorphism type of certain 7–manifolds, J. Eur. Math. Soc. 16 (2014) 2499 MR3293803
43	R Goto, Calabi–Yau structures and Einstein–Sasakian structures on crepant resolutions of isolated singularities, J. Math. Soc. Japan 64 (2012) 1005 MR2965437
44	V W Guillemin, S Sternberg, Supersymmetry and equivariant de Rham theory, Springer (1999) MR1689252
45	S Gukov, J Sparks, M–theory on Spin(7) manifolds, Nuclear Phys. B 625 (2002) 3 MR1890885
46	S Gukov, J Sparks, D Tong, Conifold transitions and five-brane condensation in M–theory on Spin(7) manifolds, Classical Quantum Gravity 20 (2003) 665 MR1959397
47	M Haskins, H J Hein, J Nordström, Asymptotically cylindrical Calabi–Yau manifolds, J. Differential Geom. 101 (2015) 213 MR3399097
48	T Hausel, E Hunsicker, R Mazzeo, Hodge cohomology of gravitational instantons, Duke Math. J. 122 (2004) 485 MR2057017
49	A Haydys, HyperKähler and quaternionic Kähler manifolds with S¹–symmetries, J. Geom. Phys. 58 (2008) 293 MR2394039
50	H J Hein, S Sun, Calabi–Yau manifolds with isolated conical singularities, Publ. Math. Inst. Hautes Études Sci. 126 (2017) 73 MR3735865
51	N J Hitchin, Kählerian twistor spaces, Proc. Lond. Math. Soc. 43 (1981) 133 MR623721
52	N J Hitchin, A new family of Einstein metrics, from: "Manifolds and geometry" (editors P de Bartolomeis, F Tricerri, E Vesentini), Sympos. Math. 36, Cambridge Univ. Press (1996) 190 MR1410073
53	N Hitchin, On the hyperKähler/quaternion Kähler correspondence, Comm. Math. Phys. 324 (2013) 77 MR3116317
54	N Hitchin, The hyperholomorphic line bundle, from: "Algebraic and complex geometry" (editors A Frühbis-Krüger, R N Kloosterman, M Schütt), Springer Proc. Math. Stat. 71, Springer (2014) 209 MR3278575
55	N J Hitchin, A Karlhede, U Lindström, M Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987) 535 MR877637
56	G R Jensen, Einstein metrics on principal fibre bundles, J. Differential Geom. 8 (1973) 599 MR353209
57	D D Joyce, Compact 8–manifolds with holonomy Spin(7), Invent. Math. 123 (1996) 507 MR1383960
58	D Joyce, A new construction of compact 8–manifolds with holonomy Spin(7), J. Differential Geom. 53 (1999) 89 MR1776092
59	D D Joyce, Compact manifolds with special holonomy, Oxford Univ. Press (2000) MR1787733
60	H Kanno, Y Yasui, On Spin(7) holonomy metric based on SU(3)∕U(1), I, J. Geom. Phys. 43 (2002) 293 MR1929908
61	H Kanno, Y Yasui, On Spin(7) holonomy metric based on SU(3)∕U(1), II, J. Geom. Phys. 43 (2002) 310 MR1929909
62	S Karigiannis, J D Lotay, Deformation theory of G₂ conifolds, Comm. Anal. Geom. 28 (2020) 1057 MR4165315
63	P Kaste, R Minasian, M Petrini, A Tomasiello, Kaluza–Klein bundles and manifolds of exceptional holonomy, J. High Energy Phys. 2002 (2002) MR1949076
64	M Konishi, On manifolds with Sasakian 3–structure over quaternion Kähler manifolds, Kōdai Math. Sem. Rep. 26 (1975) 194 MR377782
65	A Kovalev, Asymptotically cylindrical manifolds with holonomy Spin(7), I, preprint (2013) arXiv:1309.5027
66	P B Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989) 665 MR992334
67	P B Kronheimer, A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989) 685 MR992335
68	P B Kronheimer, H Nakajima, Yang–Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990) 263 MR1075769
69	F Lehmann, Geometric transitions with Spin(7) holonomy via a dynamical system, preprint (2020) arXiv:2012.11758
70	R Lockhart, Fredholm, Hodge and Liouville theorems on noncompact manifolds, Trans. Amer. Math. Soc. 301 (1987) 1 MR879560
71	R B Lockhart, R C McOwen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12 (1985) 409 MR837256
72	T B Madsen, A Swann, Toric geometry of G₂–manifolds, Geom. Topol. 23 (2019) 3459 MR4047648
73	D Martelli, J Sparks, Toric Sasaki–Einstein metrics on S² × S³, Phys. Lett. B 621 (2005) 208 MR2152673
74	D Martelli, J Sparks, Resolutions of non-regular Ricci-flat Kähler cones, J. Geom. Phys. 59 (2009) 1175 MR2547337
75	P Molino, Riemannian foliations, 73, Birkhäuser (1988) MR932463
76	L Nirenberg, H F Walker, The null spaces of elliptic partial differential operators in ℝⁿ, J. Math. Anal. Appl. 42 (1973) 271 MR320821
77	J Nordström, Deformations of asymptotically cylindrical G₂–manifolds, Math. Proc. Cambridge Philos. Soc. 145 (2008) 311 MR2442130
78	F Pacard, Connected sum constructions in geometry and nonlinear analysis, lecture notes (2008)
79	T H Parker, A Morse theory for equivariant Yang–Mills, Duke Math. J. 66 (1992) 337 MR1162193
80	F Reidegeld, Exceptional holonomy and Einstein metrics constructed from Aloff–Wallach spaces, Proc. Lond. Math. Soc. 102 (2011) 1127 MR2806102
81	B L Reinhart, Differential geometry of foliations, 99, Springer (1983) MR705126
82	S Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982) 143 MR664330
83	T Shioya, Eigenvalues and suspension structure of compact Riemannian orbifolds with positive Ricci curvature, Manuscripta Math. 99 (1999) 509 MR1713808
84	S Smale, On the structure of 5–manifolds, Ann. of Math. 75 (1962) 38 MR141133
85	A Swann, Hyper-Kähler and quaternionic Kähler geometry, Math. Ann. 289 (1991) 421 MR1096180
86	P Tondeur, Geometry of foliations, 90, Birkhäuser (1997) MR1456994