{
  "id": "2309.06820",
  "version": "v1",
  "published": "2023-09-13T09:08:16.000Z",
  "updated": "2023-09-13T09:08:16.000Z",
  "title": "Liouville theorem for $V$-harmonic maps under non-negative $(m, V)$-Ricci curvature for non-positive $m$",
  "authors": [
    "Kazuhiro Kuwae",
    "Songzi Li",
    "Xiangdong Li",
    "Yohei Sakurai"
  ],
  "categories": [
    "math.DG",
    "math.PR"
  ],
  "abstract": "Let $V$ be a $C^1$-vector field on an $n$-dimensional complete Riemannian manifold $(M, g)$. We prove a Liouville theorem for $V$-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative $(m, V)$-Ricci curvature for $m\\in\\,[\\,-\\infty,\\,0\\,]\\,\\cup\\,[\\,n,\\,+\\infty\\,]$ into Cartan-Hadam\\-ard manifolds, which extends Cheng's Liouville theorem proved S.~Y.~Cheng for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan-Hadamard manifolds. We also prove a Liouville theorem for $V$-harmonic maps from complete Riemannian manifolds with non-negative $(m, V)$-Ricci curvature for $m\\in\\,[\\,-\\infty,\\,0\\,]\\,\\cup\\,[\\,n,\\,+\\infty\\,]$ into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt-Jost-Wideman and Choi. Our probabilistic proof of Liouville theorem for several growth $V$-harmonic maps into Hadamard manifolds enhances an incomplete argument by Stafford. Our results extend the results due to Chen-Jost-Qiu\\cite{ChenJostQiu} and Qiu\\cite{Qiu} in the case of $m=+\\infty$ on the Liouville theorem for bounded $V$-harmonic maps from complete Riemannian manifolds with non-negative $(\\infty, V)$-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of $V$-harmonic maps and the recurrence property of $\\Delta_V$-diffusion processes on manifolds. Our results are new even in the case $V=\\nabla f$ for $f\\in C^2(M)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-13T09:08:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "53C21",
      "53C43",
      "53C20",
      "58J65",
      "60J60"
    ],
    "keywords": [
      "harmonic maps",
      "liouville theorem",
      "ricci curvature",
      "complete riemannian manifold",
      "sectional curvature upper bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}