PomeExample 12.9.21.CAEF64468276 For every fixed $0<\varepsilon <1$, there are $v_ j\in \mathcal V_\varepsilon $ with $\| v_ j\| _{C^0}\to 0$ such that their normalized densities $F_{v_ j}$ are pairwise distinct, bounded in $C^0$, and unbounded in $C^1$. This family is not equicontinuous. Nevertheless, both densities of its exact maximizers satisfy the uniform bounds of theorem 12.9.18.
Example 12.9.21CAEF64468276
Proof. Choose holomorphic coordinates centered at a point $p$, put $x=\operatorname {Re}z_1$, and take a smooth real cutoff $\zeta $ supported in the chart and equal to one near $p$. We may assume $0\leq \zeta \leq 1$. On the chart set
\[ \beta =dx\wedge \mathrm d^{\mathrm c}x=\mathrm{dd}^{\mathrm{c}}(x^2/2) =\frac{\mathrm i}{8\pi }dz_1\wedge d\bar z_1. \]
This is a nonzero semipositive form of rank one. For a constant $a>0$ to be chosen, extend
\[ w_ j=\frac a{j^2}\zeta \sin (jx) \]
by zero outside the chart, and put $v_ j=w_ j-\sup _ Xw_ j$. The product rule gives
\begin{align*} \mathrm{dd}^{\mathrm{c}}w_ j={}& -a\zeta \sin (jx)\beta +\frac aj\cos (jx) (d\zeta \wedge \mathrm d^{\mathrm c}x+dx\wedge \mathrm d^{\mathrm c}\zeta )\\ & +\frac a{j^2}\sin (jx)\mathrm{dd}^{\mathrm{c}}\zeta . \end{align*}
All fixed forms in this expression have compact support after multiplication by the indicated cutoff or its derivatives. Thus a fixed $C>0$ bounds the operator norm of $\mathrm{dd}^{\mathrm{c}}w_ j$ relative to $\omega $ by $aC$ for every $j\geq 1$. Choose $a$ so small that $aC\leq \varepsilon /2$. Then $v_ j\in \mathcal V_\varepsilon $, and $\| v_ j\| _{C^0}\leq 2a/j^2$.
Write $s=\operatorname {tr}_{\omega (p)}\beta >0$. Since $x(p)=0$ and $\zeta =1$ near $p$, we have $\Omega _{v_ j}(p)=\omega (p)$ and
\[ \partial _ x\Omega _{v_ j}(p)=\partial _ x\omega (p)-aj\beta . \]
Differentiating the logarithmic determinant, and observing that its normalizing integral is constant in the spatial variable, gives the exact identity
\[ \partial _ xF_{v_ j}(p) =\partial _ x\log \frac{\omega ^ n}{\mathrm{d}V}(p)-ajs. \]
These derivatives are pairwise different and their absolute values tend to infinity. Hence the densities are pairwise distinct and their $C^1$ norms are unbounded. Their uniform $C^0$ bound is part of theorem 12.9.18.
To check failure of equicontinuity, take points $p_ j^\pm $ with $x=\pm \pi /(2j)$ and all other real coordinates zero. For large $j$ they lie where $\zeta =1$, and $\Omega _{v_ j}(p_ j^\pm )=\omega (p_ j^\pm )\mp a\beta $. The integral normalizing $F_{v_ j}$ cancels from the difference, so
\[ F_{v_ j}(p_ j^-)-F_{v_ j}(p_ j^+) \longrightarrow \log \frac{1+as}{1-as}>0. \]
Indeed, $\beta ^2=0$ gives $(\omega (p)\pm a\beta )^ n=(1\pm as)\omega (p)^ n$, and $as<1$ by the chosen form comparison. Both points tend to $p$, which proves the failure of equicontinuity. The exact-adjoint bounds still apply because the entire sequence lies in $\mathcal V_\varepsilon $. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.