Lemma 23.2.37

Lemma 23.2.37.AE38387ABD03 Every metric of the form (23.2.36) is strictly LCK. Its Lee form and Lee period are

\[ \lambda =l(r)\, \mathrm dr,\qquad l=\frac{C'-D}{C},\qquad L=-\int _0^2\frac{D}{C}\, \mathrm dr<0. \]

The metric is Gauduchon if and only if $C^{m-1}(C'-D)=k$ for a constant $k$. In terms of the positive smooth $2$-periodic function $F=C^ mD$, this condition is equivalent to

0452243D80CF 23.2.38

\begin{equation} C^ mC'=kC+F,\qquad D=\frac{F}{C^ m}. \end{equation}

Every positive periodic solution has $k<0$, and

9A128F70476F 23.2.39

\begin{equation} \begin{gathered} k\int _0^2C\, \mathrm dr+\int _0^2F\, \mathrm dr=0, \\ l=kC^{-m},\qquad L(C)=-\int _0^2\frac{F}{C^{m+1}}\, \mathrm dr. \end{gathered} \end{equation}

Proof. The differential identities are

\[ \begin{gathered} \alpha ^{m+1}=\beta ^2=0,\qquad \mathrm d\beta =-\mathrm dr\wedge \alpha ,\\ \mathrm d^{\mathrm c}\beta =-\alpha \wedge \mathrm d^{\mathrm c}r,\qquad \mathrm{dd}^{\mathrm{c}}\beta =-\alpha ^2. \end{gathered} \]

They give $\mathrm d\omega =(C'-D)\mathrm dr\wedge \alpha =\lambda \wedge \omega $. The closed form $\lambda $ is the Lee form, since wedging one-forms with $\omega $ is injective. The integral of $C'/C$ over a period is zero, which proves the formula for $L$; since $L\neq 0$, the Lee form is not exact. Two radial Lee forms have the same de Rham class exactly when their periods agree: a difference with zero period has a periodic primitive, and the radial loop detects the converse.

For a smooth function $G$ of $r$, the chain rule gives $\mathrm{dd}^{\mathrm{c}}G=G'\alpha +G''\beta $. Also $\mathrm d^{\mathrm c}(G\beta )=-G\alpha \wedge \mathrm d^{\mathrm c}r$, so

\[ \mathrm{dd}^{\mathrm{c}}(G\beta )=-G'\alpha \wedge \beta -G\alpha ^2. \]

Expanding the powers of $\omega $ therefore gives

\[ \begin{aligned} \omega ^ m& =C^ m\alpha ^ m+mC^{m-1}D\alpha ^{m-1}\wedge \beta ,\\ \omega ^ n& =nC^ mD\alpha ^ m\wedge \beta , \end{aligned} \]

and

\[ \mathrm{dd}^{\mathrm{c}}(\omega ^ m)= \bigl [(C^ m)''-m(C^{m-1}D)'\bigr ]\alpha ^ m\wedge \beta . \]

Since $(C^ m)''-m(C^{m-1}D)'=m\bigl (C^{m-1}(C'-D)\bigr )'$, this vanishes exactly when $C^{m-1}(C'-D)$ is constant. Multiplying the identity $C^{m-1}(C'-D)=k$ by $C$ gives (23.2.38). Since $C^ mC'$ has zero integral over a period, integrating (23.2.38) gives the first identity in (23.2.39), which forces $k<0$. The formulas for $l$ and $L$ follow from the formula for the Lee form. $\square$

Lemma 23.2.37`AE38387ABD03`

Comments

Lemma 23.2.37AE38387ABD03

Comments

Lemma 23.2.37`AE38387ABD03`