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mean curvature equation

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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Existence and non-existence of minimal graphic and p-harmonic functions
Part of
Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 1 / February 2020

Published online by Cambridge University Press:

25 January 2019, pp. 341-366

Print publication:

February 2020
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We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and p-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.