Skip to main content
SpringerLink
Log in

Spacelike mean curvature flow solitons, polynomial volume growth and stochastic completeness of spacelike hypersurfaces immersed into pp-vave spacetimes

  • Published:
Collectanea Mathematica Aims and scope Submit manuscript

Abstract

Our purpose in this paper is to study some geometric properties of spacelike hypersurfaces immersed into a pp-wave spacetime, namely, a connected Lorentzian manifold admitting a parallel lightlike vector field. Initially, by applying a new form of maximum principle for smooth functions on a complete noncompact Riemannian manifold, we obtain sufficient conditions which guarantee that a complete noncompact spacelike hypersurface with polynomial volume growth is either totally geodesic, maximal or 1-maximal. As a consequence, we establish nonexistence results concerning such spacelike hypersurfaces. Next, using a weak form of the Omori–Yau maximum principle, we get uniqueness and nonexistence results for stochastically complete spacelike hypersurface with constant mean curvature. Finally, we establish the notion of spacelike mean curvature flow soliton in pp-wave spacetimes and we provide some geometric conditions that allow us to guarantee how close a complete spacelike mean curvature flow soliton is to a totally geodesic immersion.

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. Aarons, M.: Mean curvature flow with a forcing term in Minkowski space. Calc. Var. PDE 25, 205–246 (2005)

    MathSciNet  Google Scholar 

  2. Albujer, A.L., Camargo, F., de Lima, H.F.: Complete spacelike hypersurfaces in a Robertson–Walker spacetime. Math. Proc. Camb. Philos. Soc. 151, 271–282 (2011)

    MathSciNet  Google Scholar 

  3. Alías, L.J., Caminha, A., do Nascimento, F.Y.: A maximum principle related to volume growth and applications. Ann. Mat. Pura Appl. 200, 1637–1650 (2021)

  4. Alías, L.J., de Lira, J.H., Rigoli, M.: Mean curvature flow solitons in the presence of conformal vector fields. J. Geom. Anal. 30, 1466–1529 (2020)

    MathSciNet  Google Scholar 

  5. Alías, L.J., Romero, A., Sánchez, M.: Uniqueness of complete spacelike hypersurfaces of constant mean curvature in generalized Robertson-Walker spacetimes. Gen. Relat. Grav. 27, 71–84 (1995)

    MathSciNet  Google Scholar 

  6. Angenent, S.: On the formation of singularities in the shortening flow. J. Differ. Geom. 33, 601–633 (1991)

    MathSciNet  Google Scholar 

  7. Bernstein, S.: Sur un théorème de géométrie et ses applications aux équations aux dérivées partielles du type elliptique. Commun. Soc. Math. Kharkov 15, 38–45 (1915)

    Google Scholar 

  8. Brinkmann, H.: Einstein spaces which are mapped conformally on each other. Math. Ann. 18, 119–145 (1925)

    MathSciNet  Google Scholar 

  9. Calabi, E.: Examples of Bernstein problems for some nonlinear equations. In: Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, CA, 1968), Amer. Math. Soc., Providence, pp. 223–230 (1970)

  10. Caballero, M., Romero, A., Rubio, R.M.: Uniqueness of maximal surfaces in generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems. J. Geom. Phys. 60, 394–402 (2010)

    MathSciNet  Google Scholar 

  11. Caballero, M., Romero, A., Rubio, R.M.: Complete cmc spacelike surfaces with bounded hyperbolic angle in generalized Robertson–Walker spacetimes. Int. J. Geom. Methods Mod. Phys. 7, 961–978 (2010)

    MathSciNet  Google Scholar 

  12. Chen, Q., Qiu, H.: Rigidity of self-shrinkers and translating solitons of mean curvature flows. Adv. Math. 294, 517–531 (2016)

    MathSciNet  Google Scholar 

  13. Cheng, S.Y., Yau, S.T.: Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces. Ann. Math. 104, 407–419 (1976)

    MathSciNet  Google Scholar 

  14. Colombo, G., Mari, L., Rigoli, M.: Remarks on mean curvature flow solitons in warped products. Disc. Cont. Dyn. Syst. 13, 1957–1991 (2020)

    MathSciNet  Google Scholar 

  15. de Lima, H.F., Parente, U.: On the geometry of maximal spacelike hypersurfaces immersed in a generalized Robertson–Walker spacetime. Ann. Mat. Pure Appl. 192, 649–663 (2013)

    MathSciNet  Google Scholar 

  16. de Lima, H.F., Velásquez, M.A.L.: On the totally geodesic spacelike hypersurfaces in conformally stationary spacetimes. Osaka J. Math. 51, 1027–1052 (2014)

    MathSciNet  Google Scholar 

  17. de Lima, H.F., Velásquez, M.A.L.: Uniqueness of complete spacelike hypersurfaces via their higher order mean curvatures in a conformally stationary spacetime. Math. Nachr. 287, 1223–1240 (2014)

    MathSciNet  Google Scholar 

  18. de Lira, J.H., Martín, F.: Translating solitons in Riemannian products. J. Differ. Equ. 266, 7780–7812 (2019)

    MathSciNet  Google Scholar 

  19. Ecker, K.: On mean curvature flow of spacelike hypersurfaces in asymptotically flat spacetime. J. Austral. Math. Soc. Ser. A 55, 41–59 (1993)

    MathSciNet  Google Scholar 

  20. Ecker, K.: Interior estimates and longtime solutions for mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space. J. Differ. Geom. 46, 481–498 (1997)

    MathSciNet  Google Scholar 

  21. Ecker, K.: Mean curvature flow of spacelike hypersurfaces near null initial data. Commun. Anal. Geom. 11, 181–205 (2003)

    MathSciNet  Google Scholar 

  22. Ecker, K., Huisken, G.: Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes. Commun. Math. Phys. 135, 595–613 (1991)

    MathSciNet  Google Scholar 

  23. Einstein, A., Rosen, N.: On gravitational waves. J. Franklin Inst. 223, 43–54 (1937)

    MathSciNet  Google Scholar 

  24. Émery, M.: Stochastic Calculus on Manifolds. Springer, Berlin (1989)

    Google Scholar 

  25. Gao, S., Li, G., Wu, C.: Translating spacelike graphs by mean curvature flow with prescribed contact angle. Arch. Math. 103, 499–508 (2014)

    MathSciNet  Google Scholar 

  26. Gerhardt, C.: Hypersurfaces of prescribed mean curvature in Lorentzian manifolds. Math. Z. 235, 83–97 (2000)

    MathSciNet  Google Scholar 

  27. Grigor’yan, A.: Stochastically complete manifolds and summable harmonic functions, Izv. Akad. Nauk SSSR Ser. Mat. 52,: 1102–1108; translation in Math. USSR-Izv. 33(1989), 425–532 (1988)

  28. Grigor’yan, A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135–249 (1999)

    MathSciNet  Google Scholar 

  29. Hamilton, R.S.: Harnack estimate for the mean curvature flow. J. Differ. Geom. 41, 215–226 (1995)

    MathSciNet  Google Scholar 

  30. Huisken, G.: Flow by mean curvature of convex surfaces into spheres. J. Differ. Geom. 20, 237–266 (1984)

    MathSciNet  Google Scholar 

  31. Huisken, G., Yau, S.T.: Definition of center of mass for isolated physical system and unique foliations by stable spheres with constant curvature. Invent. Math. 124, 281–311 (1996)

    MathSciNet  Google Scholar 

  32. Hungerbühler, N., Mettler, T.: Soliton solutions of the mean curvature flow and minimal hypersurfaces. Proc. Am. Math. Soc. 140, 2117–2126 (2012)

    MathSciNet  Google Scholar 

  33. Hungerbühler, N., Roost, B.: Mean curvature flow solitons, Analytic aspects of problems in Riemannian geometry: elliptic PDEs, solitons and computer imaging, pp. 129–158, Sémin. Congr., 22, Soc. Math. France, Paris (2011)

  34. Hungerbühler, N., Smoczyk, K.: Soliton solutions for the mean curvature flow. Differ. Integral Equ. 13, 1321–1345 (2000)

    MathSciNet  Google Scholar 

  35. Jian, H.: Translating solitons of mean curvature flow of noncompact spacelike hypersurfaces in Minkowski space. J. Differ. Equ. 220, 147–162 (2006)

    MathSciNet  Google Scholar 

  36. Lambert, B.: A note on the oblique derivative problem for graphical mean curvature flow in Minkowski space. Abh. Math. Semin. Univ. Hambg. 82, 115–120 (2012)

    MathSciNet  Google Scholar 

  37. Lambert, B.: The perpendicular Neumann problem for mean curvature flow with a timelike cone boundary condition. Trans. Am. Math. Soc. 366, 3373–3388 (2014)

    MathSciNet  Google Scholar 

  38. Li, G., Salavessa, I.: Mean curvature flow of spacelike graphs. Math. Z. 269, 697–719 (2011)

    MathSciNet  Google Scholar 

  39. Marsden, J., Tipler, F.: Maximal hypersurfaces and foliations of constant mean curvature in general relativity. Phys. Rep. 66, 109–139 (1980)

    MathSciNet  Google Scholar 

  40. O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (1983)

    Google Scholar 

  41. Omori, H.: Isometric immersions of Riemannian manifolds. J. Math. Soc. Jpn. 19, 205–214 (1967)

    MathSciNet  Google Scholar 

  42. Pelegrín, J., Romero, A., Rubio, R.: On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field. Class. Q. Grav. 33, 055003 (2016)

    MathSciNet  Google Scholar 

  43. Pigola, S., Rigoli, M., Setti, A.G.: A remark on the maximum principle and stochastic completeness. Proc. Am. Math. Soc. 131, 1283–1288 (2003)

    MathSciNet  Google Scholar 

  44. Pigola, S., Rigoli, M., Setti, A.G.: Maximum principles on Riemannian manifolds and applications, Mem. Am. Math. Soc. 174(822), x+99pp (2005)

  45. Romero, A.: Constant mean curvature spacelike hypersurfaces in spacetimes with certain causal symmetries, 20th IWHSS, Daegu Korea,: Springer Proceedings in Mathematics and Statistics 203(2017), 1–15 (2016)

  46. Romero, A., Rubio, R.M.: On the mean curvature of spacelike surfaces in certain three-dimensional Robertson–Walker spacetimes and Calabi–Bernstein’s type problems. Ann. Glob. Anal. Geom. 37, 21–31 (2010)

    MathSciNet  Google Scholar 

  47. Romero, A., Rubio, R.M., Salamanca, J.J.: Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes. Class. Quantum Grav. 30, 1–13 (2013)

    MathSciNet  Google Scholar 

  48. Sachs, R.K., Wu, H.: General Relativity for Mathematicians, Graduate Texts in Mathematics 48. Springer, New York (1977)

    Google Scholar 

  49. Smoczyk, K.: A Relation between mean curvature flow solitons and minimal submanifolds. Math. Nachr. 229, 175–186 (2001)

    MathSciNet  Google Scholar 

  50. Spruck, J., Xiao, L.: Entire downward translating solitons of the mean curvature flow in Minkowski space. Proc. Am. Math. Soc. 144, 3517–3526 (2016)

    MathSciNet  Google Scholar 

  51. Stephani, H., Kramer, D., MacCallum, M., Hoenselaersand, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  52. Stroock, D.: An Introduction to the Analysis of Paths on a Riemannian Manifold, Math. Surveys and Monographs, volume 4, American Mathematical Society (2000)

  53. Stumbles, S.: Hypersurfaces of constant mean curvature. Ann. Phys. 133, 28–56 (1981)

    MathSciNet  Google Scholar 

  54. Velásquez, M.A.L., de Lima, H.F.: Complete spacelike hypersurfaces immersed in pp-wave spacetimes. Gen. Relativ. Gravit. 52, 41 (2020)

    MathSciNet  Google Scholar 

  55. Xu, R., Liu, T.: Rigidity of complete spacelike translating solitons in pseudo-Euclidean space. J. Math. Anal. Appl. 477, 692–707 (2019)

    MathSciNet  Google Scholar 

  56. Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)

    MathSciNet  Google Scholar 

  57. Yau, S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for reading the manuscript in great detail and for his/her valuable suggestions and useful comments which improved the paper.

Funding

The first author is partially supported by CNPq, Brazil, grant 304891/2021-5. The second author is partially supported by CNPq, Brazil, grant 301970/2019-0. The third author is partially supported by CAPES, Brazil.

Author information

Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal de Campina Grande, Campina Grande, Paraíba, 58.429-970, Brazil

    Marco A. L. Velásquez, Henrique F. de Lima & José H. H. de Lacerda

Authors
  1. Marco A. L. Velásquez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henrique F. de Lima
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José H. H. de Lacerda
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco A. L. Velásquez.

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

Velásquez, M.A.L., de Lima, H.F. & de Lacerda, J.H.H. Spacelike mean curvature flow solitons, polynomial volume growth and stochastic completeness of spacelike hypersurfaces immersed into pp-vave spacetimes. Collect. Math. 75, 189–211 (2024). https://doi.org/10.1007/s13348-022-00384-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13348-022-00384-3

Keywords

Mathematics Subject Classification

Search

Navigation