| Summary: | We say that a PDE on the hyperbolic space Hn of constant sectional curvature - 1 , n≥ 2 , is geometric if, whenever u is a solution of the PDE on a domain Ω of Hn, the composition uϕ: = u∘ ϕ is also a solution on ϕ- 1(Ω) for any isometry ϕ of Hn. We prove that if u∈ C1(Hn) is a solution of a geometric PDE satisfying the comparison principle and if lim supr→∞(e2rsupSr∥∇u∥)=0,where Sr is a geodesic sphere of Hn centered at a fixed point o∈ Hn with radius r, then u is constant. However, given C> 0 , there exists a bounded non-constant harmonic function v∈ C∞(Hn) such that limr→∞(ersupSr∥∇v∥)=C.We prove (1) by showing a similar result for left invariant PDEs on a Lie group and by endowing Hn with a Lie group structure. © 2022, Springer Nature Switzerland AG.
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