A Moser/Bernstein type theorem in a Lie group with a left invariant metric under a gradient decay condition

• Main Authors: Aiolfi, A. (Author), Bonorino, L. (Author), Ripoll, J. (Author), Soret, M. (Author), Ville, M. (Author)
• Format: Article
• Language: English
• Published: Birkhauser 2022
• Subjects: Entire minimal graphs; Entire PDE solutions; Harmonic functions; Lie groups
• Online Access: View Fulltext in Publisher

 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.