Minimal Graphs and Graphical Mean Curvature Flow in \(M^n\times \mathbb {R}\)

  • Matthew McGonagle
  • Ling XiaoEmail author


In this paper, we investigate the problem of finding minimal graphs in \(M^n\times \mathbb {R}\) with general boundary conditions using a variational approach. Following Giusti (Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel, 1984), we study the so-called generalized solutions that minimize the adapted area functional (AAF) (1.2). We also show that when the boundary data \(\varphi \) satisfy certain conditions, the generalized solution is actually the classical solution. This generalizes the results in Schulz–Williams (Analysis 7(3–4):359–374, 1987) to \(M^n\times \mathbb {R}.\) Finally, following the idea of Oliker–Ural’tseva (Commun Pure Appl Math 46(1):97–135, 1993 and Topol Methods Nonlinear Anal 9(1):17–28, 1997), we consider the long time existence and convergence of the graphical mean curvature flow (1.3). We show that as \(t\rightarrow \infty ,\)\(u(\cdot , t)\rightarrow \bar{u},\) where \(\bar{u}\) is a generalized solution to the associated Dirichlet problem.


Minimal graph Graphical mean curvature flow Manifolds 

Mathematics Subject Classification

35K20 53C44 53A10 



We would like to thank Professor Joel Spruck for his guidance and suggestions. This paper is based on the work supported by the National Science Foundation under Grant No. 0932078000, while the second author was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall semester of 2013.


  1. 1.
    De Silva, D., Spruck, J.: Rearrangements and radial graphs of constant mean curvature in hyperbolic space. Calc. Var. Partial Differ. Equ. 34(1), 73–95 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ding, Q., Jost, J., Xin, Y.L.: Existence and non-existence of area-minimizing hypersurfaces in manifolds of non-negative Ricci curvature. Am. J. Math. 138(2), 287–327 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    do Carmo, M.P.: Riemannian Geometry. Mathematics: Theory & Applications. Birkhäuser, Boston (1992). Translated from the second Portuguese edition by Francis FlahertyGoogle Scholar
  4. 4.
    Ecker, K., Huisken, G.: Mean curvature evolution of entire graphs. Ann. Math. (2) 130(3), 453–471 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn, vol. 224. Springer, Berlin (1983)Google Scholar
  6. 6.
    Giusti, E.: Minimal Surfaces and Functions of Bounded Variation. Monographs in Mathematics, vol. 80. Birkhäuser Verlag, Basel (1984)CrossRefzbMATHGoogle Scholar
  7. 7.
    Guan, B.: The Dirichlet problem for Monge-Ampère equations in non-convex domains and spacelike hypersurfaces of constant Gauss curvature. Trans. Am. Math. Soc. 350(12), 4955–4971 (1998)CrossRefzbMATHGoogle Scholar
  8. 8.
    Gudmundsson, S., Kappos, E.: On the geometry of tangent bundles. Expo. Math. 20(1), 1–41 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Commun. Pure Appl. Math. 27, 715–727 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kowalski, O.: Curvature of the induced Riemannian metric on the tangent bundle of a Riemannian manifold. J. Reine Angew. Math. 250, 124–129 (1971)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Ladyzhenskaya, O.A., Ural’tseva, N.N.: Local estimates for gradients of solutions of non-uniformly elliptic and parabolic equations. Commun. Pure Appl. Math. 23, 677–703 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Miranda Jr., M.: Functions of bounded variation on “good” metric spaces. J. Math. Pures Appl. (9) 82(8), 975–1004 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Oliker, V.I., Uraltseva, N.N.: Evolution of nonparametric surfaces with speed depending on curvature. II. The mean curvature case. Commun. Pure Appl. Math. 46(1), 97–135 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Oliker, V.I., Ural’tseva, N.N.: Long time behavior of flows moving by mean curvature II. Topol. Methods Nonlinear Anal. 9(1), 17–28 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Rosenberg, H., Schulze, F., Spruck, J.: The half-space property and entire positive minimal graphs in \(M\times \mathbb{R}\). J. Differ. Geom. 95(2), 321–336 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Schulz, F., Williams, G.: Barriers and existence results for a class of equations of mean curvature type. Analysis 7(3–4), 359–374 (1987)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Serrin, J.: The Dirichlet problem for surfaces of constant mean curvature. Proc. Lond. Math. Soc. (3) 21, 361–384 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University Centre for Mathematical Analysis, Canberra (1983)Google Scholar
  19. 19.
    Spruck, J.: Interior gradient estimates and existence theorems for constant mean curvature graphs in \(M^n\times { R}\). Pure Appl. Math. Q. 3(3, Special Issue: In honor of Leon Simon. Part 2), 785–800 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wang, M.-T.: Long-time existence and convergence of graphic mean curvature flow in arbitrary codimension. Invent. Math. 148(3), 525–543 (2002)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA
  2. 2.Department of MathematicsUniversity of ConnecticutStorrsUSA

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