Pome

Proposition 11.2.120009683E4137

Proposition 11.2.12.0009683E4137 If $n=2$, then every $u\in \mathcal H_\omega $ satisfies

\[ E_\omega (u)=\int _ Xu\omega ^2 +\int _ Xu\omega \wedge \mathrm{dd}^{\mathrm{c}}u +\frac13\int _ Xu(\mathrm{dd}^{\mathrm{c}}u)^2. \]

For every smooth real direction $v$, its differential satisfies

173E9E78941E 11.2.13
\begin{equation} DE_{\omega ,u}(v)-\int _ Xv\omega _ u^2 =\int _ Xv\bigl (u\mathrm{dd}^{\mathrm{c}}\omega +\mathrm{d}u\wedge \mathrm d^{\mathrm c}\omega -\mathrm d^{\mathrm c}u\wedge \mathrm{d}\omega \bigr ). \end{equation}

Proof. Expanding the radial integral gives the first identity. For $n=2$, one has $\Omega _ t=\omega +t\mathrm{dd}^{\mathrm{c}}u$, and since $\mathrm d$ and $\mathrm d^{\mathrm c}$ annihilate $\mathrm{dd}^{\mathrm{c}}u$, the derivatives $\mathrm d\Omega _ t$, $\mathrm d^{\mathrm c}\Omega _ t$ and $\mathrm{dd}^{\mathrm{c}}\Omega _ t$ are those of $\omega $. Formula (11.2.11) and $2\int _0^1t\, \mathrm dt=1$ give the second identity. $\square$

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