23.2.25 Volume equations in Gauduchon gauge29CDAA9CF041

Let $X$ again be a compact LCK manifold of dimension $n\geq 2$. Prescribing the volume of an LCK metric in Gauduchon gauge couples two equations: one for the determinant and one for the conformal factor. To compare metrics with a fixed Lee class, we first identify the twisted groups for cohomologous Lee forms. For a closed real one-form $\lambda $, smooth real functions $a,u$ and a smooth differential form $\eta $, multiplication by $\mathrm{e}^{-a}$ gives

We use this identification when comparing classes; the additive constant in $a$ corresponds to a constant rescaling of the weight bundle.

Let $\omega _1$ be an LCK Gauduchon metric with Lee form $\lambda $, and let $\omega _2$ be an LCK metric with Lee form $\lambda +\mathrm{d}a$. If their twisted classes agree under the chosen identification, then

for a smooth real function $u$. Substituting $\omega _2=\mathrm{e}^ aA$ shows that $\omega _2$ has the same volume form as $\omega _1$ and is Gauduchon exactly when

The scalar Hermitian equation of section 14.1 does not impose the second equation, and it uses $\mathrm{dd}^{\mathrm{c}}u$ in place of $D_\lambda u$. The prescribed-volume Gauduchon theorem, theorem 14.10.1, does not assert that its solution is LCK or lies in the given twisted class.

The identification of classes makes this uniqueness problem precise. The question concerns metrics: uniqueness of a scalar potential $u$ would additionally require control of the kernel of $D_\lambda $. The counterexamples with degenerate densities in section 14.20 concern ordinary potentials and do not decide this question.