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The Journal of Geometric Analysis

, Volume 27, Issue 2, pp 1106–1130 | Cite as

Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures

  • Jean-Baptiste Casteras
  • Esko HeinonenEmail author
  • Ilkka Holopainen
Article

Abstract

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan–Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound
$$\begin{aligned} K(P)\le - \frac{\phi (\phi -1)}{r(x)^2} \end{aligned}$$
and a pointwise pinching condition
$$\begin{aligned} |K(P) |\le C_K|K(P') | \end{aligned}$$
for some constants \(\phi >1\) and \(C_K\ge 1\), where P and \(P'\) are any 2-dimensional subspaces of \(T_xM\) containing the (radial) vector \(\nabla r(x)\) and \(r(x)=d(o,x)\) is the distance to a fixed point \(o\in M\). We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions \(n=\dim M>4/\phi +1\).

Keywords

Minimal graph equation Dirichlet problem Hadamard manifold 

Mathematics Subject Classification

Primary 58J32 Secondary 53C21 

Notes

Acknowledgments

J.-B.C. supported by MIS F.4508.14 (FNRS). E.H. supported by the Academy of Finland, Project 252293 and the Wihuri Foundation. I.H. supported by the Academy of Finland, Project 252293. \(\square \)

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Copyright information

© Mathematica Josephina, Inc. 2016

Authors and Affiliations

  • Jean-Baptiste Casteras
    • 1
  • Esko Heinonen
    • 2
    Email author
  • Ilkka Holopainen
    • 2
  1. 1.Departement de MathematiqueUniversite libre de BruxellesBrusselsBelgium
  2. 2.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

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