PomeTheorem 23.1.35.D3E13E0C86B3 For every $n\geq 3$, the Gauduchon metric $\omega _ n$ admits smooth potentials $V_\eta $ with $\omega _ n+\mathrm{dd}^{\mathrm{c}}V_\eta >0$ whose total Monge–Ampère masses tend to infinity as $\eta \downarrow 0$.
Theorem 23.1.35D3E13E0C86B3
Proof. Use exactly the functions $V_\eta =W+g_\eta $ constructed in the proof of theorem 23.1.1, including the heat scale $\varepsilon =\eta ^{1/\eta ^2}$. Their endpoint properties give smooth potentials on every $H_ n$, and lemma 23.1.33 gives strict positivity. The slope bounds in lemma 23.1.25 and $-m_\eta \leq g_\eta '\leq 0$ imply
\[ \mu +(V_\eta )_ x\geq \tfrac 14 \qquad (x\geq 0) \]
for all sufficiently small $\eta $.
We must first localize the divergent threefold mass to this region. On $x<0$, the heat potential is $W=F_\eta (\varepsilon ,w)$ and has no $x$ derivatives. The determinant formula in the preceding proof and $\int _ E F_{w\bar w}\, \mathrm dA_ E=0$ give
\[ \int _ E J_3(V_\eta ;x,w)\, \mathrm dA_ E =12(\mu +g_\eta ') \bigl (g_\eta ''+\mu (1-\mu )\bigr ). \]
Here $0<\mu +g_\eta '\leq \mu $, and $g_\eta ''=r_\eta -m_\eta \psi $ with $\int r_\eta =m_\eta $ and $\int \psi =1$. Hence
\[ \int _{x<0}\! J_3(V_\eta )\, \mathrm dx\, \mathrm dA_ E \leq 12\int _{-\infty }^0\mu ^2(1-\mu )\, \mathrm dx +12\int _{\mathbb R}|g_\eta ''|\, \mathrm dx \leq \tfrac 32+24m_\eta . \]
No bound on the amplitude of the fiber Hessian is required; its mean has been integrated exactly. Since the total threefold mass tends to infinity, its mass on $x\geq 0$ tends to infinity. The dimensional comparison now gives
\[ \int _{H_ n}(\omega _ n+\mathrm{dd}^{\mathrm{c}}V_\eta )^ n \geq \frac{n(n-1)}6\, 4^{-(n-3)} \int _{x\geq 0}J_3(V_\eta )\, \mathrm dx\, \mathrm dA_ E \longrightarrow +\infty . \]
$\square$
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