PomeLemma 12.16.1.744D35F9BE16 Suppose a smooth real $(n-1,n-1)$-form $T$ satisfies
\[ \int _ X \mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge T\geq 0 \quad \hbox{for every smooth real }f. \]
Then $T\geq 0$.
12.16 Curvature conditions for mass extremaA57EC3129CC8
A57EC3129CC8
We now ask when the lower and upper masses are attained by smooth potentials. At a smooth local extremum of the mass, the first variation forces the metric $\Omega =\omega +\mathrm{dd}^{\mathrm{c}}v$ to be Gauduchon, and the second variation gives a sign for $\mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})$. On a threefold these necessary conditions are also sufficient, because the mass is then a polynomial of degree two in the variation. Fix a Hermitian form $\omega $, and recall from section 11.1 the set $\mathcal H_\omega $ of smooth real functions $u$ with $\omega +\mathrm{dd}^{\mathrm{c}}u>0$ and the mass functional $\mathrm M_\omega (u)=\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u)^ n$.
For $n\geq 2$, theorem 11.1.7, proposition 12.15.2 give the differential conditions under which $\mathcal V_\mu (\omega )$ is independent of the smooth positive probability measure $\mu $. By proposition 11.38.3, the same conditions characterize when the difference $H_0-\operatorname {Ent}_{\mathrm{d}V}$ considered there is constant on probability measures of finite entropy, for a fixed smooth positive probability volume $\mathrm{d}V$; it suffices to test smooth positive probability measures. In dimension one the mass is always $\int _ X\omega $, so both conclusions hold automatically.
Recall that a smooth real $(n-1,n-1)$-form $T$ is nonnegative, written $T\geq 0$, if $\mathrm i\alpha \wedge \bar\alpha \wedge T\geq 0$ for every $(1,0)$-covector $\alpha $; in this bidegree this is the usual notion of positivity. The second variation of the mass produces integrals of the form $\int _ X\mathrm df\wedge \mathrm d^{\mathrm c}f\wedge T$, and the following elementary lemma shows that their signs detect the sign of $T$.
Proof. Suppose instead that $\mathrm i\alpha \wedge \bar\alpha \wedge T<0$ at a point $x_0$ for some $(1,0)$-covector $\alpha $. Every $(1,0)$-covector is the $(1,0)$-part of a real covector, so we can choose a real linear function $\ell $ in local real coordinates around $x_0$ whose differential has $(1,0)$-part $\alpha $ at $x_0$. Then $\mathrm{d}\ell \wedge \mathrm d^{\mathrm c}\ell $ is a positive multiple of $\mathrm i\alpha \wedge \bar\alpha $ at $x_0$, and after shrinking the coordinate ball, $\mathrm{d}\ell \wedge \mathrm d^{\mathrm c}\ell \wedge T$ is strictly negative on it. Choose a nonzero smooth cutoff function $\chi $ supported in this ball and, for positive integers $k$, set
\[ f_ k=k^{-1}\chi \sin (k\ell ), \]
extended by zero to $X$. Since $\mathrm{d}\ell $ is a constant nonzero covector in these coordinates,
\[ \mathrm{d}f_ k\wedge \mathrm d^{\mathrm c}f_ k =\chi ^2\cos ^2(k\ell )\, \mathrm{d}\ell \wedge \mathrm d^{\mathrm c}\ell +O(k^{-1}), \]
where $k$ times the remainder has uniformly bounded coefficients. In $\cos ^2(k\ell )=(1+\cos (2k\ell ))/2$, the oscillatory part contributes integrals tending to zero, as one sees by integrating by parts in a coordinate direction in which $\ell $ has nonzero derivative. Hence
\[ \lim _{k\to \infty }\int _ X\mathrm{d}f_ k\wedge \mathrm d^{\mathrm c}f_ k\wedge T =\frac12\int _ X\chi ^2\mathrm{d}\ell \wedge \mathrm d^{\mathrm c}\ell \wedge T<0, \]
which contradicts the hypothesis. $\square$
Proposition 12.16.2.CE7BE3992C5F Let $n\geq 3$, and let $v\in \mathcal H_\omega $ be a local minimum of $\mathrm M_\omega $ for the $C^2$ topology. Then $\Omega =\omega +\mathrm{dd}^{\mathrm{c}}v$ satisfies
\[ \mathrm{dd}^{\mathrm{c}}(\Omega ^{n-1})=0, \qquad \mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})\leq 0. \]
At a local maximum the first condition still holds and the second inequality is reversed.
Proof. For every smooth real $f$, the potentials $v+tf$ lie in $\mathcal H_\omega $ for all sufficiently small $|t|$. The vanishing of the first variation therefore gives the Gauduchon condition. The second variation is
\[ \begin{aligned} \left.\frac{\mathrm{d}^2}{\mathrm{d}t^2}\right|_{t=0}\mathrm M_\omega (v+tf) & =n(n-1)\int _ X(\mathrm{dd}^{\mathrm{c}}f)^2\wedge \Omega ^{n-2}\\ & =-n(n-1)\int _ X\mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge \mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2}), \end{aligned} \]
where the last equality follows by integrating by parts twice, using that $\mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})$ is both $\mathrm{d}$-closed and $\mathrm d^{\mathrm c}$-closed. At a local minimum this second variation is nonnegative for every $f$, and lemma 12.16.1, applied to $-\mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})$, gives $\mathrm{dd}^{\mathrm{c}}(\Omega ^{n-2})\leq 0$. The case of a maximum is identical, with the opposite sign. $\square$
In dimension three these necessary conditions are also sufficient, and every local extremum is global.
Theorem 12.16.3.26629526444F Assume $n=3$ and $v\in \mathcal H_\omega $. The following are equivalent:
$v$ is a local minimum of $\mathrm M_\omega $;
$v$ is a global minimum of $\mathrm M_\omega $ on $\mathcal H_\omega $;
$\Omega =\omega +\mathrm{dd}^{\mathrm{c}}v$ is Gauduchon and $\mathrm{dd}^{\mathrm{c}}\omega \leq 0$.
In this case $v_-(\omega )=\int _ X\Omega ^3$, and the infimum in (12.15.5) is attained. The corresponding statements for maxima hold with $\mathrm{dd}^{\mathrm{c}}\omega \geq 0$, and then $v_+(\omega )=\int _ X\Omega ^3<+\infty $.
Proof. Since $\mathrm{dd}^{\mathrm{c}}\Omega =\mathrm{dd}^{\mathrm{c}}\omega $, the preceding proposition shows that the first condition implies the third. Conversely, if $\Omega $ is Gauduchon, expanding the cube and integrating by parts gives, for every smooth real $f$,
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12.16.4
\begin{equation} \int _ X(\Omega +\mathrm{dd}^{\mathrm{c}}f)^3-\int _ X\Omega ^3 =-3\int _ X\mathrm{d}f\wedge \mathrm d^{\mathrm c}f\wedge \mathrm{dd}^{\mathrm{c}}\omega . \end{equation}
The form $\mathrm{d}f\wedge \mathrm d^{\mathrm c}f$ is nonnegative, so the right side is nonnegative if $\mathrm{dd}^{\mathrm{c}}\omega \leq 0$; then $v$ is a global minimum, and in particular a local one. By theorem 12.15.4, regularization identifies the infimum of the masses of smooth strictly $\omega $-plurisubharmonic potentials with $v_-(\omega )$, and proposition 12.15.6 proves attainment in (12.15.5). Reversing the sign in the same identity proves the statements for maxima, including the equality of $\int _ X\Omega ^3$ with the supremum over bounded potentials. $\square$
These conditions characterize smooth minimizers when they exist; neither argument produces a Gauduchon metric of the form $\omega +\mathrm{dd}^{\mathrm{c}}v$ in an arbitrary background.
For the Hopf metric $\omega $ and the pulled-back Fubini–Study form $\alpha $ of section 23.1, one has $\mathrm{dd}^{\mathrm{c}}\omega =-\alpha ^2\leq 0$, and $\omega $ is Gauduchon. Thus the minimum of the mass is attained by the constant potential and equals $6$, while the upper mass is infinite. On a surface, by contrast, the mass $\int _ X\omega ^2+2\int _ Xv\, \mathrm{dd}^{\mathrm{c}}\omega $ is affine in the potential, so a local extremum exists precisely when $\omega $ is Gauduchon, and then all masses are equal. In dimension $n\geq 4$ the sign condition of proposition 12.16.2 does not control the terms of order three and higher in the mass polynomial, and the argument above does not show that the conditions are sufficient.
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