Existence and uniqueness of conformal Gauduchon metrics

Abstract

In this paper, we prove that every Hermitian conformal class on a connected noncompact complex manifold of dimension at least two contains a Gauduchon metric. We give a sufficient condition for uniqueness of such Gauduchon metrics up to scaling in terms of functions bounded from above and their Levi forms. This condition is necessary when the manifold has at least two ends. In that case, uniqueness in a fixed conformal class is equivalent to the assertion that every positive solution of the complex Laplace equation for a metric in that class is constant. For a connected noncompact complex Lie group of dimension n, uniqueness in every conformal class is equivalent to the assertion that, for every Hermitian metric, every positive solution of the complex Laplace equation is constant. These properties hold exactly when the complex tangent space of a maximal compact subgroup has dimension n−1; such a group is abelian. In a conformal class containing a balanced metric, the same equivalence holds without a restriction on the number of ends. We also prove uniqueness for manifolds with a suitable potential on a compactification with compact fibers over a point, the real line, or a parabolic Riemann surface. The examples include compact pluripolar deletions, partially deleted compact fibers, and semipositive line bundles. We construct conformal Gauduchon metrics which are not related by scaling on Stein manifolds and on the total space of 𝒪_PP¹(1)^⊕2.

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BibTeX RIS

BibTeX

@misc{yin2026existence,
  title = {{Existence and uniqueness of conformal Gauduchon metrics}},
  author = {Hanzhang Yin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17858},
  eprint = {2610.17858},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17858},
}

RIS

TY  - UNPB
TI  - Existence and uniqueness of conformal Gauduchon metrics
AU  - Hanzhang Yin
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.17858
UR  - https://arxiv.org/abs/2610.17858
AB  - In this paper, we prove that every Hermitian conformal class on a connected noncompact complex manifold of dimension at least two contains a Gauduchon metric. We give a sufficient condition for uniqueness of such Gauduchon metrics up to scaling in terms of functions bounded from above and their Levi forms. This condition is necessary when the manifold has at least two ends. In that case, uniqueness in a fixed conformal class is equivalent to the assertion that every positive solution of the complex Laplace equation for a metric in that class is constant. For a connected noncompact complex Lie group of dimension n, uniqueness in every conformal class is equivalent to the assertion that, for every Hermitian metric, every positive solution of the complex Laplace equation is constant. These properties hold exactly when the complex tangent space of a maximal compact subgroup has dimension n−1; such a group is abelian. In a conformal class containing a balanced metric, the same equivalence holds without a restriction on the number of ends. We also prove uniqueness for manifolds with a suitable potential on a compactification with compact fibers over a point, the real line, or a parabolic Riemann surface. The examples include compact pluripolar deletions, partially deleted compact fibers, and semipositive line bundles. We construct conformal Gauduchon metrics which are not related by scaling on Stein manifolds and on the total space of 𝒪_PP^1(1)^⊕2.
ER  -

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