{
  "id": "2108.02844",
  "version": "v1",
  "published": "2021-08-05T20:52:48.000Z",
  "updated": "2021-08-05T20:52:48.000Z",
  "title": "A Moser/Bernstein type theorem in a Lie group with a left invariant metric under a gradient decay condition",
  "authors": [
    "Ari Aiolfi",
    "Leonardo Bonorino",
    "Jaime Ripoll",
    "Marc Soret",
    "Marina Ville"
  ],
  "categories": [
    "math.DG"
  ],
  "abstract": "We say that a PDE in a Riemannian manifold $M$ is geometric if,$\\ $whenever $u$ is a solution of the PDE on a domain $\\Omega$ of $M$, the composition $u_{\\phi}:=u\\circ\\phi$ is also solution on $\\phi^{-1}\\left( \\Omega\\right) $, for any isometry $\\phi$ of $M.$ We prove that if $u\\in C^{1}\\left( \\mathbb{H}^{n}\\right) $ is a solution of a geometric PDE satisfying the comparison principle, where $\\mathbb{H}^{n}$ is the hyperbolic space of constant sectional curvature $-1,$ $n\\geq2,$ and if \\[ \\limsup_{R\\rightarrow\\infty}\\left( e^{R}\\sup_{S_{R}}\\left\\Vert \\nabla u\\right\\Vert \\right) =0, \\] where $S_{R}$ is a geodesic sphere of $\\mathbb{H}^{n}$ centered at fixed point $o\\in\\mathbb{H}^{n}$ with radius $R,$ then $u$ is constant. Moreover, given $C>0,$ there is a bounded non-constant harmonic function $v\\in C^{\\infty }\\left( \\mathbb{H}^{n}\\right) $ such that \\[ \\lim_{R\\rightarrow\\infty}\\left( e^{R}\\sup_{S_{R}}\\left\\Vert \\nabla v\\right\\Vert \\right) =C. \\] The first part of the above result is a consequence of a more general theorem proved in the paper which asserts that if $G$ is a non compact Lie group with a left invariant metric, $u\\in C^{1}\\left( G\\right) $ a solution of a left invariant PDE (that is, if $v$ is a solution of the PDE on a domain $\\Omega$ of $G$, the composition $v_{g}:=v\\circ L_{g}$ of $v$ with a left translation $L_{g}:G\\rightarrow G,$ $L_{g}\\left( h\\right) =gh,$ is also solution on $L_{g}^{-1}\\left( \\Omega\\right) $ for any $g\\in G),$ the PDE satisfies the comparison principle and% \\[ \\limsup_{R\\rightarrow\\infty}\\left( \\sup_{g\\in B_{R}}\\left\\Vert \\operatorname*{Ad}\\nolimits_{g}\\right\\Vert \\sup_{S_{R}}\\left\\Vert \\nabla u\\right\\Vert \\right) =0, \\] where $\\operatorname*{Ad}\\nolimits_{g}:\\mathfrak{g}\\rightarrow\\mathfrak{g}$ is the adjoint map of $G$ and $\\mathfrak{g}$ the Lie algebra of $G,$ then $u$ is constant.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-05T20:52:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C42",
      "35B08"
    ],
    "keywords": [
      "left invariant metric",
      "moser/bernstein type theorem",
      "gradient decay condition",
      "comparison principle",
      "non compact lie group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}