Example 23.2.45`FDF7812789EF`

Example 23.2.45.FDF7812789EF For

\[ \begin{gathered} 0<\varepsilon <\min \left\{ \frac12, \frac{(3/2)^{1-m}}{4\pi }\right\} , \\ C=1+\varepsilon \cos (\pi r),\qquad D=C'+C^{1-m}, \end{gathered} \]

the metric (23.2.36) is Gauduchon and non-Vaisman, with Lee form $\lambda =-C^{-m}\mathrm dr$ and density $F=C^ mC'+C$. The probability volume $\mathrm dV=\frac12\alpha ^ m\wedge \beta $ gives $\omega ^ n=2nF\, \mathrm dV$ and $\int _{H_ n}F\, \mathrm dV=1$.

Proof. Since $1/2<C<3/2$ and $m\geq 1$, we have $D\geq (3/2)^{1-m}-\pi \varepsilon >\frac34(3/2)^{1-m}>0$. The Gauduchon identity holds with $k=-1$, giving the Lee form and the positive density $F=C^ mD$. Its values $F(0)=1+\varepsilon $ and $F(1)=1-\varepsilon $ differ, so proposition 23.2.44 applies. The integral of $C^ mC'$ over a period is zero and the average of $C$ is one, so integration along the fibers of $q$ gives $\int _{H_ n}F\, \mathrm dV=1$. $\square$

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@misc{pome-tag-FDF7812789EF,
  author = {{Pome contributors}},
  title = {Example 23.2.45},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/FDF7812789EF/}
}

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Example 23.2.45FDF7812789EF

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Example 23.2.45`FDF7812789EF`