Skip to main content
SpringerLink
Log in

On the NNSC fill-ins and asymptotically flat extension

  • Published:
manuscripta mathematica Aims and scope Submit manuscript

Abstract

In this article we will consider the non-existence of fill-ins of \((\Sigma ^n, g, H)\) into (n+1)-dimensional manifolds with nonnegative scalar curvature for \(3\le n\le 6\) where \(\Sigma \) is diffeomorphic to \(S^n\), the scalar curvature of \( (\Sigma , g )\) is positive and \(H>0\). Here we give some explicit estimate for the infimum of H if \((\Sigma ^n, g, H)\) admits a NNSC fill-ins. We also discuss the asymptotically flat extension of the Bartnik data with CMC boundary and provide the upper bound of the Bartnik mass.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bartnik, R.: New definition of quasilocal mass. Phys. Rev. Lett. 62, 2346–2348 (1989)

    Article  MathSciNet  Google Scholar 

  2. Pacheco, A.J.C., Cederbaum, C.: A survey on extensions of Riemannian manifolds and Bartnik mass estimates. arXiv:1904.05830

  3. Pacheco, A.J.C., Cederbaum, C., McCormick, S., Miao, P.: Asymptotically flat extensions of CMC Bartnik data. Class. Quantum Gravity 34, 105001 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  4. Pacheco, A.J.C., Miao, P.: Higher dimensional black hole initial data with prescribed boundary metric. Math. Res. Lett. 25, 937–956 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carr, R.: Construction of manifolds of positive scalar curvature. Trans. Am. Math. Soc. 307, 63–74 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eichmair, M., Huang, L., Lee, D.A., Schoen, R.: The spacetime positive mass theorem in dimensions less than eight. J. Eur. Math. Soc. 18, 83–121 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gromov, M.: Scalar curvature of manifolds with boundaries: natural questions and artificial constructions, arXiv:1811.04311

  8. Gromov, M.: Four lectures on scalar curvature, arXiv:1908.10612

  9. Hawking, S.W.: Black holes in general relativity. Commun. Math. Phys. 25, 152–166 (1972)

    Article  MathSciNet  Google Scholar 

  10. Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jauregui, J.L., Miao, P., Tam, L.-F.: Extensions and fill-ins with non-negative scalar curva- ture. Class. Quantum Gravity 30, 1095007 (2013)

    Article  Google Scholar 

  12. Mantoulidis, C., Schoen, R.: On the Bartnik mass of apparent horizons. Class. Quantum Gravity 32, 205002 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Marques, F.: Deforming three-manifolds with positive scalar curvature. Ann. Math. 176, 815–863 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  14. Miao, P.: Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6, 1163–1182 (2002)

    Article  MathSciNet  Google Scholar 

  15. Miao, P.: On a localized Riemannian Penrose inequality. Commun. Math. Phys. 292, 271–284 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Miao, P.: Nonexistence of NNSC fill-ins with large mean curvature. Proc. Am. Math. Soc. 149, 2705–2709 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Miao, P., Xie, N.: On compact 3-manifolds with nonnegative scalar curvature with a CMC boundary component. Trans. Am. Math. Soc. 370, 5887–5906 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  18. Miao, P., Wang, Y., Xie, N.: On Hawking mass and Bartnik mass of CMC surfaces. Math. Res. Lett. 27, 855–885 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  19. Schoen, R., Yau, S.T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–76 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  20. Schoen, R., Yau, S.T.: Positive scalar curvature and minimal hypersurface singularities. arXiv:1704.05490

  21. Shi, Y., Tam, L.-F.: Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Diff. Geom. 62, 79–125 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Shi, Y., Wang, W., Wei, G.: Total mean curvature of the boundary and nonnegative scalar curvature fill-ins, arXiv:2007.06756

  23. Shi, Y., Wang, W., Wei, G., Zhu, J.: On the fill-in of nonnegative scalar curvature metrics. Math. Ann. 379, 235–270 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wang, Y.: 4-dimensional manifolds with nonnegative scalar curvature and CMC boundary. Commun. Contemp. Math. 23, 1950094 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The work of Y. Wang is supported by the National Natural Science Foundation of China (Grant No.11671089) and the postdoctoral foundation of He’nan University. The authors would like to thank Professor Naqing Xie for showing this problem and giving many valuable suggestions.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, Henan, China

    Mingming Pang & Yaohua Wang

Authors
  1. Mingming Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yaohua Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaohua Wang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pang, M., Wang, Y. On the NNSC fill-ins and asymptotically flat extension. manuscripta math. 171, 85–102 (2023). https://doi.org/10.1007/s00229-022-01378-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00229-022-01378-y

Mathematics Subject Classification

Search

Navigation