{
  "abstract": "We prove that if $M$ is a connected complete parabolic Riemannian manifold and $N$ is a connected complete stochastically complete Riemannian manifold, then every bounded harmonic function on $M\\times N$ is independent of the $M$-variable. Equivalently, pullback by the second projection induces an isometric isomorphism from the space of bounded harmonic functions on $N$ onto that on $M\\times N$. In particular, the product of two parabolic manifolds has the bounded Liouville property, thereby answering Problem~16 of Grigor'yan's survey in the affirmative. The key analytic input is a total-variation memory-loss property of the heat kernel on a parabolic manifold. We establish this property by showing that the time-one heat kernel defines an aperiodic Harris recurrent transition kernel and then applying the row-merging theorem of Jamison and Orey.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22942",
  "author_total": 1,
  "authors": [
    {
      "name": "Ruotong Jia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dpcdd2bfdnsx6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22942",
  "fulltext_url": null,
  "id": "dpcdd2bfdnsx6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22942",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7204154139",
  "pdf_url": "https://arxiv.org/pdf/2608.22942",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dpcdd2bfdnsx6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2608.22942",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22942"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Bounded Harmonic Functions on Products with a Parabolic Factor",
  "title_text": "Bounded Harmonic Functions on Products with a Parabolic Factor",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/dpcdd2bfdnsx6",
  "venue": "arXiv",
  "versions": []
}