A parabolic flow approach to generalized Gauduchon metrics
Abstract
We give a parabolic proof of the Fu-Wang-Wu theorem for semilinear elliptic equations associated with generalized Gauduchon metrics. Fixing a Hermitian metric ω, we introduce a flow ∂t v−Δv=ψ(|∇ v|2)+⟨ B,dv⟩−c(t)−f(x) for the conformal factor, whose stationary points solve the semilinear elliptic equation of Fu-Wang-Wu. Under the growth condition liminft→+∞ψ(t)/tμ>0 for some μ>1/2, we prove that the flow exists uniquely for all time and converges; the limit solves the elliptic equation and is unique up to addition of a constant. Our convergence argument does not rely on a gradient-flow structure or a monotone energy functional. In particular, this recovers, by a flow method rather than the continuity method, the existence and uniqueness theorem of Fu-Wang-Wu on k-th generalized Gauduchon metrics, providing a new parabolic approach to their elliptic theory.
Related
- Generalized Gauduchon metrics on compact complex spaces
- Existence and uniqueness of conformal Gauduchon metrics
- A note on the Chern-Yamabe flow on compact Hermitian manifolds with negative Gauduchon degree
- Parabolic Bogomolov--Gieseker Inequality and Pluriharmonic Metrics on Regular Parabolic Higgs Sheaves over Compact Kähler Manifolds
- Kähler-Einstein cone metrics and parabolic bundles
BibTeX
@misc{luo2026parabolic,
title = {{A parabolic flow approach to generalized Gauduchon metrics}},
author = {Han Luo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19887},
eprint = {2610.19887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19887},
}
RIS
TY - UNPB
TI - A parabolic flow approach to generalized Gauduchon metrics
AU - Han Luo
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19887
UR - https://arxiv.org/abs/2610.19887
AB - We give a parabolic proof of the Fu-Wang-Wu theorem for semilinear elliptic equations associated with generalized Gauduchon metrics. Fixing a Hermitian metric ω, we introduce a flow ∂_t v−Δv=ψ(|∇ v|^2)+⟨ B,dv⟩−c(t)−f(x) for the conformal factor, whose stationary points solve the semilinear elliptic equation of Fu-Wang-Wu. Under the growth condition liminf_t→+∞ψ(t)/t^μ>0 for some μ>1/2, we prove that the flow exists uniquely for all time and converges; the limit solves the elliptic equation and is unique up to addition of a constant. Our convergence argument does not rely on a gradient-flow structure or a monotone energy functional. In particular, this recovers, by a flow method rather than the continuity method, the existence and uniqueness theorem of Fu-Wang-Wu on k-th generalized Gauduchon metrics, providing a new parabolic approach to their elliptic theory.
ER -