{
  "abstract": "We construct solutions to the Allen-Cahn equation $Δu+u-u^3=0$ in $\\mathbb{R}^{N+1}$, with $N\\ge 3$, whose zero level set is asymptotic (at infinity) to a given minimal cone. Our construction is quite general, since we only use a suitable nondegeneracy assumption; neither stability nor symmetry of the underlying cone is required. Moreover, we calculate the morse index of such solutions. In particular, our main result generalizes a result by Pacard and Wei (2013) dealing with stable solutions to the Allen-Cahn equation in high dimensions. More precisely, we give a positive answer to a remark raised in the paper by Pacard and Wei about the possibility to relax the assumption for the underlying cone is area-minimizing or at least stable. As a corollary, we also obtain solutions with infinite morse index and connected zero level set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01335",
  "author_total": 2,
  "authors": [
    {
      "name": "Oscar Ivan Agudelo Rico"
    },
    {
      "name": "Matteo Rizzi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5zps58ggdvcqf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01335",
  "fulltext_url": null,
  "id": "5zps58ggdvcqf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01335",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7215766611",
  "pdf_url": "https://arxiv.org/pdf/2610.01335",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5zps58ggdvcqf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01335",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01335"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Allen-Cahn equation and general minimal cones",
  "title_text": "The Allen-Cahn equation and general minimal cones",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/5zps58ggdvcqf",
  "venue": "arXiv",
  "versions": []
}