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
and a pointwise pinching condition
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 \(n=\dim M>4/\phi +1\).
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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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Casteras, JB., Heinonen, E. & Holopainen, I. Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures. J Geom Anal 27, 1106–1130 (2017). https://doi.org/10.1007/s12220-016-9712-0
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DOI: https://doi.org/10.1007/s12220-016-9712-0