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The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 951–1000 | Cite as

Asymptotics of the Self-Dual Deformation Complex

  • Antonio G. Ache
  • Jeff A. ViaclovskyEmail author
Article

Abstract

We analyze the indicial roots of the self-dual deformation complex on a cylinder \((\mathbb{R}\times Y^{3}, dt^{2} + g_{Y})\), where Y 3 is a space of constant curvature. An application is the optimal decay rate of solutions on a self-dual manifold with cylindrical ends having cross-section Y 3, which is crucial in gluing results for orbifolds in the case of cross-section Y 3=S 3/Γ. We also resolve a conjecture of Kovalev–Singer in the case where Y 3 is a hyperbolic rational homology 3-sphere, and show that there are infinitely many examples for which the conjecture is true, and infinitely many examples for which the conjecture is false.

Keywords

Anti-self-dual metrics Gluing theory Indicial roots 

Mathematics Subject Classification

53C25 

Notes

Acknowledgements

The authors would like to thank Claude LeBrun for several discussions about the paper [17], and the relation with the gluing theorems given in [14]. We would also like to thank Richard Kent for crucial help with the hyperbolic examples in Theorem 1.17. Finally, the authors would like to thank the anonymous referee for valuable comments which improved the exposition.

References

  1. 1.
    Ache, A.G., Viaclovsky, J.A.: Obstruction-flat asymptotically locally Euclidean metrics. Geom. Funct. Anal. 22(4), 832–877 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Chen, S.-Y.S.: Optimal curvature decays on asymptotically locally Euclidean manifolds (2009). arXiv:0911.5538 [math.DG]
  3. 3.
    DeBlois, J.: Totally geodesic surfaces and homology. Algebr. Geom. Topol. 6, 1413–1428 (2006). (electronic) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Donaldson, S., Friedman, R.: Connected sums of self-dual manifolds and deformations of singular spaces. Nonlinearity 2(2), 197–239 (1989) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-Manifolds. Oxford Mathematical Monographs. Clarendon, Oxford (1990) zbMATHGoogle Scholar
  6. 6.
    Douglis, A., Nirenberg, L.: Interior estimates for elliptic systems of partial differential equations. Commun. Pure Appl. Math. 8, 503–538 (1955) CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Floer, A.: Self-dual conformal structures on \(l{\bf C}{\rm P}^{2}\). J. Differ. Geom. 33(2), 551–573 (1991) zbMATHMathSciNetGoogle Scholar
  8. 8.
    Folland, G.B.: Harmonic analysis of the de Rham complex on the sphere. J. Reine Angew. Math. 398, 130–143 (1989) zbMATHMathSciNetGoogle Scholar
  9. 9.
    Hodgson, C.D., Kerckhoff, S.P.: Universal bounds for hyperbolic Dehn surgery. Ann. Math. 162(1), 367–421 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Itoh, M.: The Weitzenböck formula for the Bach operator. Nagoya Math. J. 137, 149–181 (1995) zbMATHMathSciNetGoogle Scholar
  11. 11.
    Kapovich, M.: Deformations of representations of discrete subgroups of SO(3,1). Math. Ann. 299(2), 341–354 (1994) CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    King, A.D., Kotschick, D.: The deformation theory of anti-self-dual conformal structures. Math. Ann. 294(4), 591–609 (1992) CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Koiso, N.: Nondeformability of Einstein metrics. Osaka J. Math. 15(2), 419–433 (1978) zbMATHMathSciNetGoogle Scholar
  14. 14.
    Kovalev, A., Singer, M.: Gluing theorems for complete anti-self-dual spaces. Geom. Funct. Anal. 11(6), 1229–1281 (2001) CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Lafontaine, J.: Modules de structures conformes plates et cohomologie de groupes discrets. C. R. Math. Acad. Sci. 297(13), 655–658 (1983) zbMATHMathSciNetGoogle Scholar
  16. 16.
    Lockhart, R.B., McOwen, R.C.: Elliptic differential operators on noncompact manifolds. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 12(3), 409–447 (1985) zbMATHMathSciNetGoogle Scholar
  17. 17.
    LeBrun, C., Maskit, B.: On optimal 4-dimensional metrics. J. Geom. Anal. 18(2), 537–564 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    LeBrun, C., Singer, M.: A Kummer-type construction of self-dual 4-manifolds. Math. Ann. 300(1), 165–180 (1994) CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Sun Poon, Y.: Compact self-dual manifolds with positive scalar curvature. J. Differ. Geom. 24(1), 97–132 (1986) zbMATHGoogle Scholar
  20. 20.
    Petronio, C., Porti, J.: Negatively oriented ideal triangulations and a proof of Thurston’s hyperbolic Dehn filling theorem. Expo. Math. 18(1), 1–35 (2000) zbMATHMathSciNetGoogle Scholar
  21. 21.
    Yann, R., Singer, M.: Non-minimal scalar-flat Kähler surfaces and parabolic stability. Invent. Math. 162(2), 235–270 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Streets, J.: Asymptotic curvature decay and removal of singularities of Bach-flat metrics. Trans. Am. Math. Soc. 362(3), 1301–1324 (2010) CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Tian, G., Viaclovsky, J.: Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160(2), 357–415 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Viaclovsky, J.A.: An index theorem on anti-self-dual orbifolds. Int. Math. Res. Not. rns160, 1–20 (2012) Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Department of MathematicsPrinceton UniversityPrincetonUSA
  2. 2.Department of MathematicsUniversity of WisconsinMadisonUSA

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