Skip to main content
SpringerLink
Log in

IMCF and the Stability of the PMT and RPI Under \(L^2\) Convergence

  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

We study the stability of the positive mass theorem (PMT) and the Riemannian Penrose inequality (RPI) in the case where a region of an asymptotically flat manifold \(M^3\) can be foliated by a smooth solution of inverse mean curvature flow (IMCF) which is uniformly controlled. We consider a sequence of regions of asymptotically flat manifolds \(U_T^i\subset M_i^3\), foliated by a smooth solution to IMCF which is uniformly controlled, and if \(\partial U_T^i = \Sigma _0^i \cup \Sigma _T^i\) and \(m_H(\Sigma _T^i) \rightarrow 0\) then \(U_T^i\) converges to a flat annulus with respect to \(L^2\) metric convergence. If instead \(m_H(\Sigma _T^i)-m_H(\Sigma _0^i) \rightarrow 0\) and \(m_H(\Sigma _T^i) \rightarrow m >0\) then we show that \(U_T^i\) converges to a topological annulus portion of the Schwarzschild metric with respect to \(L^2\) metric convergence.

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.

Similar content being viewed by others

References

  1. Allen, B.: ODE Maximum Principle at Infinity and Noncompact Solutions of IMCF in Hyperbolic Space. arXiv:1610.01211 [math.DG] (2016)

  2. Bray, H.: Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59(2), 177–267 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bray, H., Finster, F.: Curvature estimates and the positive mass theorem. Commun. Anal. Geom. 2, 291–306 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Corvino, J.: A note on asymptotically flat metrics on \({\mathbb{R}}^3\) which are scalar-flat and admit minimal spheres. Proc. Am. Math. Soc. 12, 3669–3678 (2005). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ding, Q.: The inverse mean curvature flow in rotationally symmetric spaces. Chin. Ann. Math. Ser. B 32, 1–18 (2010)

  6. Dobarro, F., Lami Dozo, E.: Scalar curvature and warped products of Riemannian manifolds. Trans. Am. Math. Soc. 303(1), 160–168 (1987)

    Article  MATH  Google Scholar 

  7. Finster, F.: A level set analysis of the witten spinor with applications to curvature estimates. Math. Res. Lett. 1, 41–55 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Finster, F., Kath, I.: Curvature estimates in asymptotically flat manifolds of positive scalar curvature. Commun. Anal. Geom. 5, 1017–1031 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gerhardt, C.: Flow of nonconvex hypersurfaces into spheres. J. Differ. Geom. 32, 299–314 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gerhardt, C.: Inverse curvature flows in hyperbolic space. J. Differ. Geom. 89, 487–527 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  11. Huang, L.-H., Lee, D., Sormani, C.: Intrinsic flat stability of the positive mass theorem for graphical hypersurfaces of Euclidean space. J. für die reine und ang. Math. (Crelle’s J.) 727, 269–299 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Huisken, G., Ilmanen, T.: The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59, 353–437 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Krylov, N.V.: Nonlinear Elliptic and Parabolic Equations of the Second Order. Mathematics and Its Applications. Reidel, Dordrecht (1987)

    Book  Google Scholar 

  14. Lee, D.: On the near-equality case of the positive mass theorem. Duke Math. J. 1, 63–80 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Lee, D., Sormani, C.: Near-equality in the Penrose inequality for rotationally symmetric Riemannian manifolds. Ann. Henri Poincaré 13, 1537–1556 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Lee, D., Sormani, C.: Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds. J. für die Riene und Ang. Math. 686, 187–220 (2014)

    MathSciNet  MATH  Google Scholar 

  17. LeFloch, P., Sormani, C.: The nonlinear stability of spaces with low regularity. J. Funct. Anal. 268(7), 2005–2065 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, P.: Geometric Analysis. Cambridge Studies in Advanced Mathematics, vol. 134. Cambridge University Press, Cambridge (2012)

    Book  Google Scholar 

  19. Moser, J.: On the volume elements on a manifold. Trans. Am. Math. Soc. 120, 286–294 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  20. Petersen, P., Wei, G.: Relative volume comparison with integral curvature bounds. Geom. Funct. Anal. 7, 1031–1045 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  21. Scheuer, J.: The inverse mean curvature flow in warped cylinders of non-positive radial curvature. Adv. Math. 306, 1130–1163 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  22. Schoen, R., Yau, S.-T.: On the proof of the positive mass conjecture in general relativity. Commun. Math. Phys. 65, 45–46 (1979)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Urbas, J.: On the expansion of starshaped hypersurfaces by symmetric functions of their principal curvatures. Math. A 205, 355–372 (1990)

    MathSciNet  MATH  Google Scholar 

  24. Witten, E.: A new proof of the positive energy theorem. Commun. Math. Phys. 80, 381–402 (1981)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank Christina Sormani for bringing this problem to my attention and for many useful suggestions and discussions. I would also like to thank her for organizing seminars at the CUNY Graduate Center which were important in shaping this paper.

Author information

Authors and Affiliations

  1. Department of Mathematics, United States Military Academy, West Point, NY, 10966, USA

    Brian Allen

Authors
  1. Brian Allen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Brian Allen.

Additional information

Communicated by James A. Isenberg.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allen, B. IMCF and the Stability of the PMT and RPI Under \(L^2\) Convergence. Ann. Henri Poincaré 19, 1283–1306 (2018). https://doi.org/10.1007/s00023-017-0641-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-017-0641-7

Search

Navigation