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

  • Matthew McGonagle
  • Ling XiaoEmail author
Article
  • 9 Downloads

Abstract

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.

Keywords

Minimal graph Graphical mean curvature flow Manifolds 

Mathematics Subject Classification

35K20 53C44 53A10 

Notes

Acknowledgements

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.

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Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

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

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