{
  "abstract": "We study the iteration of Cao, Sun and Zhang for Hermitian--Einstein--Higgs metrics: starting from a metric $h_0$ on a Higgs bundle over a compact Riemann surface, $h_m$ is the unique solution of $K(h_m)=h_{m-1}h_m^{-1}$, where $K$ is the Hermitian--Yang--Mills--Higgs endomorphism. Well-posedness of each step is due to Fan, Wang, Yang and Yau. We prove that $h_m$ minimizes the Donaldson--Simpson functional plus the Bregman divergence of $-\\log\\det$ centred at $h_{m-1}$, so the iteration is a Bregman proximal point method. Without stability, the spectral intervals of $K_m$ are nested, so the supremum-norm defect is monotone, and they contain the Harder--Narasimhan slope range. A discrete parabolic estimate shows that the supremum-norm defect tends to zero, and the functional is $o(m)$, exactly for semistable bundles, while it diverges linearly otherwise. For stable bundles we prove uniform estimates, smooth convergence and geometric convergence of the functional, the $L^2$ distance and all $C^k$ norms. The asymptotic contraction factor of the functional is at most $(1+\\lambda)^{-2}$, where $\\lambda$ is the spectral gap of the Hessian at the limit, and this factor is attained. An estimate and a step-size version for Hitchin's equations in every degree follow. For the uniformizing Higgs bundle of a hyperbolic surface, Hitchin's equation reduces to a scalar equation solved with explicit bounds, and its implicit Euler scheme contracts in the supremum norm with factor $(1+\\tau)^{-1}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Antón-Sancho Álvaro"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/71wn18sggeqtn.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "71wn18sggeqtn",
  "issns": [],
  "landing_url": "https://hal.science/hal-05757643v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-dg"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05757643/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/71wn18sggeqtn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:37Z",
      "native_id": "hal-05757643",
      "source": "hal",
      "url": "https://hal.science/hal-05757643v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-dg",
      "subfield": "geometry-topology"
    }
  ],
  "title": "SHARP CONVERGENCE RATES OF THE HERMITIAN-YANG-MILLS-HIGGS ITERATION ON RIEMANN SURFACES",
  "title_text": "SHARP CONVERGENCE RATES OF THE HERMITIAN-YANG-MILLS-HIGGS ITERATION ON RIEMANN SURFACES",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:37Z",
  "url": "https://research.draconitic.com/paper/71wn18sggeqtn",
  "venue": "HAL",
  "versions": []
}