{
  "abstract": "We prove that the energy functional associated with surfaces of prescribed mean curvature in $\\mathbb{R}^3$ exhibits a convexity property when restricted to maps from the unit $2$-disk having small Dirichlet energy and a fixed boundary value, provided that the $3$-form $H \\cdot \\text{vol}_{\\mathbb{R}^3}$ has a primitive satisfying certain bounds. Under milder assumptions on $H:\\mathbb{R}^3 \\to \\mathbb{R}$, we show that an analogous convexity estimate holds along the heat flow of the functional (the $H$-surface flow) when the initial Dirichlet energy is sufficiently small. This is done first for classical solutions, and then extended by approximation to weak solutions using a quantitative uniqueness result adapted from previous work on the harmonic map heat flow. As a consequence of the convexity estimate, we show that the $H$-surface flow with small-energy initial map of class $C^0 \\cap W^{1, 2}$ on the unit $2$-disk converges uniformly to a unique limit at infinite time, which solves the corresponding stationary problem (the $H$-surface system) with the same boundary value.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30740",
  "author_total": 3,
  "authors": [
    {
      "name": "Da Rong Cheng"
    },
    {
      "name": "Longzhi Lin"
    },
    {
      "name": "Xin Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/23byq64k9enbf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30740",
  "fulltext_url": null,
  "id": "23byq64k9enbf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30740",
  "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.AP"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214671200",
  "pdf_url": "https://arxiv.org/pdf/2609.30740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/23byq64k9enbf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.30740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Energy convexity and uniformity of the $H$-surface flow in $\\mathbb{R}^{3}$ with Dirichlet boundary condition",
  "title_text": "Energy convexity and uniformity of the H-surface flow in ℝ^3 with Dirichlet boundary condition",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/23byq64k9enbf",
  "venue": "arXiv",
  "versions": []
}