{
  "abstract": "Let $Ω\\subset\\mathbb R^3$ be an unbounded domain with smooth boundary and let $Σ$ be a complete, noncompact, orientable, immersed free boundary minimal surface in $Ω$, with compact boundary $\\partial Σ\\subset\\partial Ω$ and finite Morse index. We prove that if the mean curvature of $\\partial Ω$ satisfies $H_{\\partial Ω}\\ge 0$ along $\\partial Σ$ and $H_{\\partialΩ}>0$ at some point of $\\partialΣ$, then \\[ \\textrm{Ind}(Σ)\\ \\ge\\ \\frac13\\Bigl(2g+k+2\\sum_{j=1}^r(d_j+1)-2\\Bigr), \\] where $g$ is the genus of $Σ$, $k$ is the number of connected components of $\\partialΣ$, $r$ is the number of ends of $Σ$ and $d_1,\\dots,d_r$ are their respective multiplicities. When $H_{\\partialΩ}$ is only assumed to be nonnegative we obtain $\\textrm{Ind}(Σ)\\ge(2g+k-2)/3$, with the sharp bound $(2g+k-1)/3$ under a mild condition on the ends. The proofs use the harmonic one-form method of Ros and Chodosh--Máximo, with weighted $L^2$ spaces, adapted to the free boundary setting in the spirit of Ambrozio--Carlotto--Sharp. The main new ingredient is the computation of the dimension of the space of harmonic one-forms on a punctured compact Riemann surface with boundary which are tangential along the boundary and square integrable with respect to a weight. For a class of admissible weights $ρ$, we prove that this dimension is $2g+k-1+2\\sum_jN_j-\\varepsilon$, where $N_j$ is the maximal order of pole allowed by $ρ$ at the $j$-th puncture and $\\varepsilon\\in\\{0,1\\}$. For the weight of Chodosh--Máximo one has $N_j=d_j+1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27651",
  "author_total": 3,
  "authors": [
    {
      "name": "Marcos P. Cavalcante"
    },
    {
      "name": "Abraao Mendes"
    },
    {
      "name": "Ian R. dos Santos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/143x3wmvc9meq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27651",
  "fulltext_url": null,
  "id": "143x3wmvc9meq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27651",
  "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": "W7214237534",
  "pdf_url": "https://arxiv.org/pdf/2609.27651",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/143x3wmvc9meq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27651",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27651"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Index estimates for complete noncompact free boundary minimal surfaces",
  "title_text": "Index estimates for complete noncompact free boundary minimal surfaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/143x3wmvc9meq",
  "venue": "arXiv",
  "versions": []
}