{
  "abstract": "In a Hilbert space, it is known that a monotone operator is maximal if and only if the \\emph{surjectivity condition}, \\ie its resolvent has a full domain, holds. This equivalence is an open question in a $\\mathrm{CAT}(0)$ space, knowing only that the surjectivity condition implies the monotone maximality. In fact, there have been three approaches to generalize the monotonicity from a Hilbert space into a $\\mathrm{CAT}(0)$ space $X$ --- \\textbf{A:} the ones of dual space $X^{\\ast}$, \\textbf{B:} of linear dual space $X^{\\diamond}$, and \\textbf{C:} of tangent bundle $TX$. In this paper, we answer \\emph{negatively} on the first two approaches by constructing counterexamples on an $\\mathbb{R}$-tree. We then go on proving that the equivalence is \\emph{affirmative} on the same $\\mathbb{R}$-tree, without any assumption on the domain. Moreover, we show that the answer is still \\emph{positive} in some other cases, through a flat slice principle that transfers Minty's principle from Hilbert spaces to flat convex subsets, and through Minty's principle on proper Hilbert-Hadamard spaces satisfying a convex half-space condition. The question of equivalence \\emph{remains open} under the general setting with the tangential approach.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25080",
  "author_total": 1,
  "authors": [
    {
      "name": "Parin Chaipunya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0yy95s2d1ntxy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25080",
  "fulltext_url": null,
  "id": "0yy95s2d1ntxy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25080",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7223061719",
  "pdf_url": "https://arxiv.org/pdf/2610.25080",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0yy95s2d1ntxy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25080",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25080"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On an open question about the equivalence of monotone maximality and surjectivity condition in $\\mathrm{CAT}(0)$ spaces: Negative and partially positive answers",
  "title_text": "On an open question about the equivalence of monotone maximality and surjectivity condition in CAT(0) spaces: Negative and partially positive answers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/0yy95s2d1ntxy",
  "venue": "arXiv",
  "versions": []
}