Advertisement

Science China Mathematics

, Volume 62, Issue 3, pp 585–596 | Cite as

The boundary behavior of domains with complete translating, minimal and CMC graphs in N2×ℝ

  • Hengyu ZhouEmail author
Articles
  • 16 Downloads

Abstract

In this paper, we discuss graphs over a domain Ω ⊂ N2 in the product manifold N2×ℝ. Here N2 is a complete Riemannian surface and Ω has piecewise smooth boundary. Let γ ⊂ ∂Ω be a smooth connected arc and Σ be a complete graph in N2×ℝ over Ω. We show that if Σ is a minimal or translating graph, then γ is a geodesic in N2. Moreover if Σ is a CMC graph, then γ has constant principal curvature in N2. This explains the infinity value boundary condition upon domains having Jenkins-Serrin theorems on minimal and constant mean curvature (CMC) graphs in N2×ℝ.

Keywords

CMC graphs minimal graphs translating surface Jenkins-Serrin theorem product manifold 

MSC(2010)

53A35 53A10 35J93 49Q05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11261378 and 11521101). The author is very grateful to the encouragement from Professor Lixin Liu. The author also thanks the referees for careful readings and helpful suggestions.

References

  1. 1.
    Altschuler S J, Wu L F. Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle. Calc Var Partial Differential Equations, 1994, 2: 101–111MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angenent S B, Velazquez J J L. Asymptotic shape of cusp singularities in curve shortening. Duke Math J, 1995, 77: 71–110MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Angenent S B, Velazquez J J L. Degenerate neckpinches in mean curvature flow. J Reine Angew Math, 1997, 482: 15–66MathSciNetzbMATHGoogle Scholar
  4. 4.
    Clutterbuck J, Schnürer O C, Schulze F. Stability of translating solutions to mean curvature flow. Calc Var Partial Differential Equations, 2007, 29: 281–293MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Colding T H, Minicozzi W P. Estimates for parametric elliptic integrands. Int Math Res Not IMRN, 2002, 6: 291–297MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Collin P, Rosenberg H. Construction of harmonic diffeomorphisms and minimal graphs. Ann of Math (2), 2010, 172: 1879–1906MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    do Carmo M P A. Riemannian Geometry. Boston: Birkhäuser, 1992CrossRefzbMATHGoogle Scholar
  8. 8.
    Eichmair M. The plateau problem for marginally outer trapped surfaces. J Differential Geom, 2010, 83: 551–584MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Eichmair M, Metzger J. Jenkins-Serrin-type results for the Jang equation. J Differential Geom, 2016, 102: 207–242MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gui C, Jian H, Ju H. Properties of translating solutions to mean curvature flow. Discrete Contin Dyn Syst, 2010, 28: 441–453MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hauswirth L, Rosenberg H, Spruck J. On complete mean curvature \(\frac{1}{2}\) surfaces in ℍ2 × ℝ. Comm Anal Geom, 2008, 16: 989–1005MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Huisken G, Polden A. Geometric evolution equations for hypersurfaces. In: Calculus of Variations and Geometric Evolution Problems. Lecture Notes in Mathematics, vol. 1713. Berlin: Springer, 1999, 45–84Google Scholar
  13. 13.
    Huisken G, Sinestrari C. Mean curvature flow with surgeries of two-convex hypersurfaces. Invent Math, 2008, 175: 137–221MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Jenkins H, Serrin J. The Dirichlet problem for the minimal surface equation in higher dimensions. J Reine Angew Math, 1968, 229: 170–187MathSciNetzbMATHGoogle Scholar
  15. 15.
    Pinheiro A L. A Jenkins-Serrin theorem in M 2 × ℝ. Bull Braz Math Soc (NS), 2009, 40: 117–148CrossRefzbMATHGoogle Scholar
  16. 16.
    Schoen R. Estimates for stable minimal surfaces in three dimensional manifolds. In: Seminar on Minimal Submanifolds, vol. 103. Princeton: Princeton University Press, 1983, 111–126Google Scholar
  17. 17.
    Shahriyari L. Translating graphs by mean curvature flow. Geom Dedicata, 2015, 175: 57–64MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Spruck J. Infinite boundary value problems for surfaces of constant mean curvature. Arch Ration Mech Anal, 1972, 49: 1–31MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sun J. Lagrangian L-stability of Lagrangian translating solitons. ArXiv:1612.06815, 2016Google Scholar
  20. 20.
    Wang X-J. Convex solutions to the mean curvature flow. Ann of Math (2), 2011, 173: 1185–1239MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zhang S. Curvature estimates for CMC surfaces in three dimensional manifolds. Math Z, 2005, 249: 613–624MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhou H. Inverse mean curvature flows in warped product manifolds. J Geom Anal, 2018, 28: 1749–1772MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-Sen UniversityGuangzhouChina

Personalised recommendations