Theorem 12.9.18.AD62F7DA0E68 Let
\[ j_-=(1-\varepsilon )^ n\min _ X\frac{\omega ^ n}{\mathrm{d}V}, \qquad j_+=(1+\varepsilon )^ n\max _ X\frac{\omega ^ n}{\mathrm{d}V}, \]
and take $C_ H$ from proposition 12.9.15 with $\delta =1-\varepsilon $ and $\Lambda =1+\varepsilon $. For every $v\in \mathcal V_\varepsilon $, the exact maximizing current $T_ v\in \mathscr G_\omega $ for $F_ v$ is unique. Its determinant density $k_{T_ v}$ and the probability density $r_ v=\mathrm d\nu _{T_ v,F_ v}/\mathrm{d}V$ satisfy
FEAB45A97679
12.9.19
\begin{equation} \frac{j_-}{C_ Hj_+}\leq r_ v\leq \frac{C_ Hj_+}{j_-}, \qquad j_+^{-1/n}\frac{j_-}{C_ Hj_+} \leq k_{T_ v}\leq j_-^{-1/n}\frac{C_ Hj_+}{j_-}. \end{equation}
In particular both densities are uniformly bounded at the critical exponent $n/(n-1)$, without an assumption on the lower mass. Moreover,
\[ \| F_ v-F_0\| _\infty \leq n\log \frac{1+\varepsilon }{1-\varepsilon }, \]
and this normalized family contains infinitely many distinct nonconstant densities on the fixed manifold $X$.
Proof.
Use the normalization of proposition 12.9.15 and put
\[ N_ v=\int _ XJ_ v\rho _ v\, \mathrm{d}V, \qquad T_ v=\frac{\rho _ v}{N_ v}\Omega _ v^{n-1}. \]
The factor equation and $\mathrm{dd}^{\mathrm{c}}$-Stokes give
\[ \int _ X\omega \wedge \rho _ v\Omega _ v^{n-1} =\int _ X\Omega _ v\wedge \rho _ v\Omega _ v^{n-1}=N_ v. \]
Hence $T_ v\in \mathscr G_\omega $. It is the $(n-1)$st power of the Gauduchon metric $\gamma _ v=(\rho _ v/N_ v)^{1/(n-1)}\Omega _ v$. The equation $\Omega _ v^ n=c_ v\mathrm e^{F_ v}\mathrm{d}V$ gives $B_0(F_ v)=\log c_ v$. Thus theorem 12.7.3 identifies $T_ v$ as the unique exact maximizer, and $M(F_ v)=c_ v^{-1/n}$. The determinant and probability normalizations are therefore
2B0B40A18948
12.9.20
\begin{equation} k_{T_ v}=\frac{\rho _ v}{N_ v}J_ v^{(n-1)/n}, \qquad \Gamma _{F_ v}(T_ v)=c_ v^{-1/n}\frac{J_ v\rho _ v}{N_ v}\, \mathrm{d}V, \qquad r_ v=\frac{J_ v\rho _ v}{N_ v}. \end{equation}
Here $j_-\leq J_ v\leq j_+$ and, since $\int _ X\rho _ v\, \mathrm{d}V=1$, also $j_-\leq N_ v\leq j_+$. The factor bound gives the first pair of inequalities in (12.9.19); the identity $k_{T_ v}=J_ v^{-1/n}r_ v$ gives the second. The critical norms are bounded by the respective upper constants because $\mathrm{d}V$ is a probability volume.
The inequalities
\[ (1-\varepsilon )^ n\leq \frac{J_ v}{J_0}\leq (1+\varepsilon )^ n, \qquad (1-\varepsilon )^ n\leq \frac{c_ v}{c_0}\leq (1+\varepsilon )^ n \]
give the bound for $F_ v-F_0$ by subtracting logarithms. To prove nontriviality, choose a nonconstant smooth real $\phi $. For small $t$, the potential $t\phi -\sup _ X(t\phi )$ belongs to $\mathcal V_\varepsilon $ and gives the same density as $t\phi $. Differentiation at zero gives
\[ \left.\frac{d}{dt}F_{t\phi }\right|_{t=0} =L_\omega \phi - \frac{\int _ XJ_0L_\omega \phi \, \mathrm{d}V}{\int _ XJ_0\, \mathrm{d}V}. \]
This function is nonconstant: otherwise $L_\omega \phi $ would be constant, its values at a maximum and a minimum would force that constant to be zero, and the strong maximum principle would make $\phi $ constant. Choose two points where the displayed derivative differs. The difference of the densities at these points is strictly monotone for $t$ in a small interval. These give infinitely many distinct densities, with at most one parameter having equal values at the chosen points. Discarding that parameter gives the nonconstant family.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.