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\(L^{2}\) harmonic forms on complete special holonomy manifolds

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Abstract

In this article, we consider \(L^{2}\) harmonic forms on a complete non-compact Riemannian manifold X with a nonzero parallel form \(\omega \). The main result is that if \((X,\omega )\) is a complete \(G_{2}\)- (or \(\textit{Spin}(7)\)-) manifold with a d(linear) \(G_{2}\)- (or \(\textit{Spin}(7)\)-) structure form \(\omega \), then the \(L^{2}\) harmonic 2-forms on X vanish. As an application, we prove that the instanton equation with square-integrable curvature on \((X,\omega )\) only has trivial solution. We would also consider the Hodge theory on the principal G-bundle E over \((X,\omega )\).

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References

  1. Bauer, I., Ivanova, T.A., Lechtenfeld, O., Lubbe, F.: Yang–Mills instantons and dyons on homogeneous \(G_{2}\)-manifolds. JHEP 2010(10), 1–27 (2010)

    Article  MATH  Google Scholar 

  2. Bryant, R.: Metrics with exceptional holonomy. Ann. Math. 126(2), 525–576 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bryant, R.: Some remarks on \(G_{2}\)-structures. In: Proceedings of Gökova Geometry-Topology Conference, pp. 75–109 (2005)

  4. Cao, J.G., Frederico, X.: Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature. Math. Ann. 319, 483–491 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carrión, R.R.: A generalization of the notion of instanton. Differential Geom. Appl. 8(1), 1–20 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, S.Y., Yau, S.T.: Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 333–354 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  7. Corrigan, E., Devchand, C., Fairlie, D.B., Nuyts, J.: First order equations for gauge fields in spaces of dimension great than four. Nucl. Phys. B. 214(3), 452–464 (1983)

    Article  Google Scholar 

  8. Dodziuk, J., Min-Oo, M.: An \(L_{2}\)-isolation theorem for Yang–Mills fields over complete manifolds. Compos. Math. 47, 165–169 (1982)

    MATH  Google Scholar 

  9. Donaldson, S.K., Thomas, R.P.: Gauge Theory in Higher Dimensions, pp. 31–47. The Geometric Universe, Oxford (1998)

    MATH  Google Scholar 

  10. Donaldson S. K., Segal E.: Gauge theory in higher dimensions, II. arXiv:0902.3239 (2009)

  11. Escobar, J.F., Freire, A., Min-Oo, M.: \(L^{2}\) vanishing theorems in positive curvature. Indiana Univ. Math. J. 42(4), 1545–1554 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fubini, S., Nicolai, H.: The octonionic instanton. Phys. Lett. B. 155(5), 369–372 (1985)

    Article  MathSciNet  Google Scholar 

  13. Gemmer, K.P., Lechtenfeld, O., Nölle, C., Popov, A.D.: Yang–Mills instantons on cones and sine-cones over nearly Kähler manifolds. JHEP 9, 103 (2011)

    Article  MATH  Google Scholar 

  14. Gerhardt, G.: An energy gap for Yang–Mills connections. Comm. Math. Phys. 298, 515–522 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Green, M.B., Schwarz, J.H., Witten, E.: Supperstring Theory. Cambridge University Press, Cambridge (1987)

    Google Scholar 

  16. Gromov, M.: Kähler hyperbolicity and \(L_{2}\)-Hodge theory. J. Differential Geom. 33, 263–292 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Harland, D., Ivanova, T.A., Lechtenfeld, O., Popov, A.D.: Yang–Mills flows on nearly Kähler manifolds and \(G_{2}\)-instantons. Comm. Math. Phys. 300(1), 185–204 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hitchin, N.J.: \(L^{2}\) cohomology of hyper-Kähler quotients. Comm. Math. Phys. 211, 153–165 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hitchin, N.J.: The geometry of three-forms in six and seven dimensions. J. Differential Geom. 55(3), 547–576 (2003)

    Article  MATH  Google Scholar 

  20. Huang, T.: Instanton on cylindrical manifolds. Ann. Henri Poincaré 18(2), 623–641 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Huang, T.: Stable Yang–Mills connections on special holonomy manifolds. J. Geom. Phys. 116, 271–280 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Huang T.: Asymptotic behaviour of instantons on cylinder manifolds. arXiv:1801.06959v4

  23. Ivanov, S.: Connections with torsion, parallel spinors and geometry of \(Spin(7)\) manifolds. Math. Res. Lett. 11, 171–186 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ivanova, T.A., Popov, A.D.: Instantons on special holonomy manifolds. Phys. Rev. D 85, 10 (2012)

    Article  Google Scholar 

  25. Ivanova, T.A., Lechtenfeld, O., Popov, A.D., Rahn, T.: Instantons and Yang–Mills flows on coset spaces. Lett. Math. Phys. 89(3), 231–247 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Jost, J., Zuo, K.: Vanishing theorems for \(L^{2}\)-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry. Comm. Anal. Geom. 8, 1–30 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Joyce D.: Compact Riemannian \(7\)-manifolds with holonomy \(G_{2}\), I,II. J. Differ. Geom. 43(2), 291–328. 329–375 (1996)

  28. Joyce, D.: Compact Manifolds with Special Holonomy. Oxford Mathematical Monographs. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  29. Karigiannis, S., Leung, N.C.: Hodge theory for \(G_{2}\)-manifolds: intermediate Jacobians and Abel–Jacobi maps. Proc. Lond. Math. Soc. 99(3), 297–325 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kovalev, A.: Twisted connected sums and special Riemannian holonomy. J. Reine Angew. Math. 565, 125–160 (2003)

    MathSciNet  MATH  Google Scholar 

  31. Lee, J.H., Leung, N.C.: Geometric structures on \(G_{2}\) and \(Spin(7)\)-manifolds. Adv. Theor. Math. Phys. 13(1), 1–31 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Min-Oo, M.: An \(L_{2}\)-isolation theorem for Yang–Mills fields. Compos. Math. 47, 153–163 (1982)

    MathSciNet  MATH  Google Scholar 

  33. Saloff-Coste, L.: Uniformly elliptic operators on Riemannian manifolds. J. Differential Geom. 36, 417–450 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Verbitsky, M.: An intrinsic volume functional on almost complex \(6\)-manifolds and nearly Kähler geometry. Pacific J. Math. 235(2), 323–344 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Verbitsky, M.: Hodge theory on nearly Kähler manifolds. Geom. Topol. 15, 2111–2133 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  36. Verbitsky, M.: Manifolds with parallel differential forms and Kähler identities for \(G_{2}\)-manifolds. J. Geom. Phys. 61(6), 1001–1016 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ward, R.S.: Completely solvable gauge field equations in dimension great than four. Nucl. Phys. B. 236(2), 381–396 (1984)

    Article  Google Scholar 

Download references

Acknowledgements

I would like to thank the anonymous referee for careful reading of my manuscript and helpful comments. I would like to thank Professor Verbitsky for kind comments regarding his article [36]. Also I would like to thank Yuguo Qin for further discussions about this work. This work is supported by Nature Science Foundation of China No. 11801539 and Postdoctoral Science Foundation of China No. 2017M621998, No. 2018T110616.

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  1. School of Mathematical Sciences, Soochow University, Suzhou, 215006, People’s Republic of China

    Teng Huang

  2. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, People’s Republic of China

    Teng Huang

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Correspondence to Teng Huang.

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Huang, T. \(L^{2}\) harmonic forms on complete special holonomy manifolds. Ann Glob Anal Geom 56, 17–36 (2019). https://doi.org/10.1007/s10455-019-09654-z

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