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Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds with Asymptotically Hyperbolic Ends

Published 15 Feb 2024 in math.DG and math.GT | (2402.10366v1)

Abstract: We construct infinitely many examples of finite volume 4-manifolds with $T3$ ends that do not admit any cusped asymptotically hyperbolic Einstein metrics yet satisfy a strict logarithmic version of the Hitchin-Thorpe inequality due to Dai-Wei. This is done by using estimates from Seiberg-Witten theory due to LeBrun as well as a method for constructing solutions to the Seiberg-Witten equations on noncompact manifolds due to Biquard. We also use constructions coming from the $Pin-(2)$ monopole equations to obtain a larger class of manifolds where these techniques apply.

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References (33)
  1. Michael T. Anderson. Dehn filling and Einstein metrics in higher dimensions. J. Differential Geom., 73(2):219–261, 2006. URL: http://projecteuclid.org/euclid.jdg/1146169911.
  2. Richard H. Bamler. Construction of einstein metrics by generalized dehn filling. J. Eur. Math. Soc., 14:887–909, 2012.
  3. Arthur L. Besse. Einstein manifolds, volume 10 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1987. doi:10.1007/978-3-540-74311-8.
  4. Exotic 4-manifolds with signature zero, 2023. arXiv:2305.10908.
  5. Olivier Biquard. Métriques d’einstein à cusps et équations de seiberg-witten. J. Reine Angew. Math., 490:129–154, 1997.
  6. V. Braungardt and D. Kotschick. Einstein metrics and the number of smooth structures on a four-manifold. Topology, 44(3):641–659, 2005. doi:10.1016/j.top.2004.11.001.
  7. Luca Fabrizio Di Cerbo. Aspects of the Seiberg-Witten equations on manifolds with cusps. ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–State University of New York at Stony Brook. URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3458214.
  8. Luca F. Di Cerbo. Seiberg–witten equations on certain manifolds with cusps. New York J. Math., 17:491–512, 2011.
  9. Luca F. Di Cerbo. Seiberg-witten equations on surfaces of logarithmic general type. Internat. J. Math., 24(9), 2013.
  10. Heberto Del Rio Guerra. Seiberg-Witten invariants of nonsimple type and Einstein metrics. Ann. Global Anal. Geom., 21(4):319–339, 2002. doi:10.1023/A:1015621002823.
  11. Hitchin–thorpe inequality for noncompact einstein 4-manifolds. Adv. Math., 214:551–570, 2007.
  12. Matthew P. Gaffney. A special stokes’s theorem for complete riemannian manifolds. Ann. of Math., 60(2):140–145, 1954.
  13. Nigel Hitchin. Compact four-dimensional Einstein manifolds. J. Differential Geometry, 9:435–441, 1974. URL: http://projecteuclid.org/euclid.jdg/1214432419.
  14. Spin manifolds, Einstein metrics, and differential topology. Math. Res. Lett., 9(2-3):229–240, 2002. doi:10.4310/MRL.2002.v9.n2.a9.
  15. Complements of tori and Klein bottles in the 4-sphere that have hyperbolic structure. Algebr. Geom. Topol., 5:999–1026, 2005.
  16. Masashi Ishida. Einstein metrics and exotic smooth structures. Pacific J. Math., 258(2):327–348, 2012. doi:10.2140/pjm.2012.258.327.
  17. Dubravko Ivanšić. Hyperbolic structure on a complement of tori in the 4-sphere. Adv. Geom., 4(1):119–139, 2004. doi:10.1515/advg.2004.002.
  18. Dieter Kotschick. Einstein metrics and smooth structures. Geom. Topol., 2:1–10, 1998.
  19. P. B. Kronheimer. Minimal genus in S1×M3superscript𝑆1superscript𝑀3S^{1}\times M^{3}italic_S start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT × italic_M start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Invent. Math., 135(1):45–61, 1999. doi:10.1007/s002220050279.
  20. Claude LeBrun. Einstein metrics and Mostow rigidity. Math. Res. Lett., 2(1):1–8, 1995. doi:10.4310/MRL.1995.v2.n1.a1.
  21. Claude LeBrun. Four-manifolds without Einstein metrics. Math. Res. Lett., 3(2):133–147, 1996. doi:10.4310/MRL.1996.v3.n2.a1.
  22. Claude LeBrun. Ricci curvature, minimal volumes, and Seiberg-Witten theory. Invent. Math., 145(2):279–316, 2001. doi:10.1007/s002220100148.
  23. On the geometric boundaries of hyperbolic 4444-manifolds. Geom. Topol., 4:171–178, 2000. doi:10.2140/gt.2000.4.171.
  24. John Morgan. The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. Princeton University Press, 1996.
  25. Hodge theory on hyperbolic manifolds. Duke Math. J., 60(2):509–559, 1990.
  26. Nobuhiro Nakamura. Pin−⁢(2)superscriptPin2\rm{Pin}^{-}(2)roman_Pin start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( 2 )-monopole equations and intersection forms with local coefficients of four-manifolds. Math. Ann., 357(3):915–939, 2013.
  27. Nobuhiro Nakamura. Pin−⁢(2)superscriptPin2\rm{Pin}^{-}(2)roman_Pin start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ( 2 )-monopole invariants. J. Differential Geom., 101(3):507–549, 2015.
  28. Jongil Park. The geography of symplectic 4-manifolds with an arbitrary fundamental group. Proc. Amer. Math. Soc., 135(7):2301–2307, 2007. doi:10.1090/S0002-9939-07-08818-1.
  29. Ulf Persson. Chern invariants of surfaces of general type. Compositio Math., 43(1):3–58, 1981. URL: http://www.numdam.org/item?id=CM_1981__43_1_3_0.
  30. Chern slopes of simply connected complex surfaces of general type are dense in [2,3]23[2,3][ 2 , 3 ]. Ann. of Math. (2), 182(1):287–306, 2015. doi:10.4007/annals.2015.182.1.6.
  31. Hemanth Saratchandran. Finite volume hyperbolic complements of 2-tori and klein bottles in closed smooth simply connected 4-manifolds. New York J. Math., 24:443–450, 2018.
  32. Savage surfaces. J. Eur. Math. Soc. (JEMS), 25(10):4205–4219, 2023. doi:10.4171/jems/1284.
  33. Steven Zucker. L2subscript𝐿2L_{2}italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT cohomology of warped products and arithmetic groups. Invent. Math., 70(2):169–218, 1982/83.

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Authors (1)

  1. Alex Xu 

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