SHARP CONVERGENCE RATES OF THE HERMITIAN-YANG-MILLS-HIGGS ITERATION ON RIEMANN SURFACES
Abstract
We study the iteration of Cao, Sun and Zhang for Hermitian--Einstein--Higgs metrics: starting from a metric h0 on a Higgs bundle over a compact Riemann surface, hm is the unique solution of K(hm)=hm−1hm−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 hm minimizes the Donaldson--Simpson functional plus the Bregman divergence of −logdet centred at hm−1, so the iteration is a Bregman proximal point method. Without stability, the spectral intervals of Km 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 L2 distance and all Ck norms. The asymptotic contraction factor of the functional is at most (1+λ)−2, where λ 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+τ)−1.
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BibTeX
@misc{alvaro2026sharp,
title = {{SHARP CONVERGENCE RATES OF THE HERMITIAN-YANG-MILLS-HIGGS ITERATION ON RIEMANN SURFACES}},
author = {Antón-Sancho Álvaro},
howpublished = {HAL},
year = {2026},
month = sep,
url = {https://hal.science/hal-05757643v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - SHARP CONVERGENCE RATES OF THE HERMITIAN-YANG-MILLS-HIGGS ITERATION ON RIEMANN SURFACES
AU - Antón-Sancho Álvaro
PY - 2026
DA - 2026/09/21
PB - HAL
UR - https://hal.science/hal-05757643v1
AB - 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−1h_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 −logdet 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+λ)^−2, where λ 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+τ)^−1.
ER -