Complete harmonic metrics and subharmonic functions on the complex plane
Abstract
This paper continues our previous work \cite{Miy6} on an extension of the Li--Mochizuki theorem concerning the existence and uniqueness of complete diagonal harmonic metrics on cyclic Higgs bundles over open Riemann surfaces. We consider the extended Hitchin equation associated with general subharmonic weight functions on the canonical bundle, not necessarily induced by holomorphic r-differentials. We establish uniqueness of complete solutions with determinant $1$ for arbitrary subharmonic weight functions, without additional regularity assumptions on them. We also prove that, on the complex plane, such a complete solution exists whenever the weight function is not identically −∞. Furthermore, if (φε)ε>0 is a family of smooth subharmonic weight functions decreasing to such a weight function φ as εsearrow0, the associated family of smooth complete solutions (hε)ε>0 with det(hε)=1 converges monotonically to the unique complete solution h associated with φ and satisfying det(h)=1.
Related
- Minimal m-subharmonic functions with nonmaximal Hessian measures
- Maximal subextension of m-subharmonic functions
- Harmonic-Einstein maps and critical metrics for eigenvalues of the conformal Laplacian
- Nonexistence of complete metrics with uniformly positive scalar curvature
- Complete conformal metrics of constant positive scalar curvature with prescribed isolated singularities
BibTeX
@misc{miyatake2026complete,
title = {{Complete harmonic metrics and subharmonic functions on the complex plane}},
author = {Natsuo Miyatake},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23441},
eprint = {2610.23441},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23441},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Complete harmonic metrics and subharmonic functions on the complex plane
AU - Natsuo Miyatake
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23441
UR - https://arxiv.org/abs/2610.23441
AB - This paper continues our previous work \cite{Miy6} on an extension of the Li--Mochizuki theorem concerning the existence and uniqueness of complete diagonal harmonic metrics on cyclic Higgs bundles over open Riemann surfaces. We consider the extended Hitchin equation associated with general subharmonic weight functions on the canonical bundle, not necessarily induced by holomorphic r-differentials. We establish uniqueness of complete solutions with determinant $1$ for arbitrary subharmonic weight functions, without additional regularity assumptions on them. We also prove that, on the complex plane, such a complete solution exists whenever the weight function is not identically −∞. Furthermore, if (φ_ε)_ε>0 is a family of smooth subharmonic weight functions decreasing to such a weight function φ as εsearrow0, the associated family of smooth complete solutions (h_ε)_ε>0 with det(h_ε)=1 converges monotonically to the unique complete solution h associated with φ and satisfying det(h)=1.
ER -