PomeA914D4D9C6CBProposition 11.23.4.A914D4D9C6CB Every form $\omega _ u$ with $u\in \mathcal H_\omega $ is Gauduchon if and only if
In that case the normalized multiplier is $\rho _ u=M^{-1}$ and $\mu _ u=M^{-1}\omega _ u^ n$ for every $u\in \mathcal H_\omega $.
Proof. If all the forms $\omega _ u$ are Gauduchon, the mass derivative (11.1.9) vanishes at every point of $\mathcal H_\omega $ and in every direction. Since $\mathcal H_\omega $ is convex, integration along the segment from $0$ to $u$ gives the mass identity. Conversely, proposition 11.1.8 shows that constant mass forces $\mathrm{dd}^{\mathrm{c}}(\omega _ u^{n-1})=0$ for every $u\in \mathcal H_\omega $. In that case the normalized multiplier is the constant $M^{-1}$, by theorem 11.4.2. $\square$
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