{
  "abstract": "Let $(X,ω_0)$ be a compact Kähler manifold of complex dimension $n$, and let $U\\Subset X$ have $C^{3,1}$ uniformly strongly pseudoconvex boundary. Consider the signed parallel family $U_t=\\{ρ<t\\}$, where $ρ$ is a defining function agreeing near $\\partial U$ with the signed distance, negative in $U$. Let $u_t$ be the equilibrium envelope with zero obstacle on $X\\setminus U_t$, and write $Σ_t=\\partial U_t$, with outward unit normal $ν_t$. We prove that the normalized Aubin--Mabuchi energy $e(t)=E_{ω_0}(u_t)$ belongs to $C^{1,1/2}([-τ,τ])$ for some $τ>0$ and satisfies the Hadamard formula \\[ e'(t)=\\frac{1}{n+1}\\int_{Σ_t}\\left(-\\partial_{ν_t}u_t\\right)^2\\,dσ_{ρ,t}^{\\mathrm{tot}},\\qquad |t|<τ. \\] Here $dσ_{ρ,t}^{\\mathrm{tot}}$ is the interior outward Anzellotti trace of \\[ \\frac{1}{V}\\,d^cρ\\wedge \\sum_{p=0}^{n-1}(p+1)\\,ω_{u_t}^{p}\\wedgeω_0^{n-1-p},\\qquad V=\\int_Xω_0^n,\\quad ω_{u_t}=ω_0+dd^c u_t. \\] This measure is positive and uniformly comparable to the induced surface measure. The proof combines uniform $C^{1,1}$ estimates up to the boundary from the interior with $1/2$-Hölder stability of the normal derivatives under the normal-flow identifications. We identify the boundary part of the normalized Monge--Ampère measure with the negative outward trace of a divergence-measure flux current. The one-sided weak-* limits of the mixed Bedford--Taylor boundary measures yield the weighted total flux appearing in the variation formula.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.17187",
  "author_total": 1,
  "authors": [
    {
      "name": "Ziyu Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1qhrswcmhcxvw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.17187",
  "fulltext_url": null,
  "id": "1qhrswcmhcxvw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.17187",
  "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.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7169864646",
  "pdf_url": "https://arxiv.org/pdf/2607.17187",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1qhrswcmhcxvw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2607.17187",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.17187"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation",
  "title_text": "A Hadamard Formula for Equilibrium Envelopes under Parallel Deformation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/1qhrswcmhcxvw",
  "venue": "arXiv",
  "versions": []
}