PomeLemma 12.8.1.47843166E777 Suppose $T_ j\to T$ weakly in $\mathscr G_\omega $ and $\mathfrak D_ F(T_ j)\to M(F)$. Then $T\in \mathscr M(F)$ and $\Gamma _ F(T_ j)\rightharpoonup \Gamma _ F(T)$. On $\mathscr M(F)$ the map $T\mapsto \Gamma _ F(T)$ is affine and continuous.
Lemma 12.8.147843166E777
Proof. Upper semicontinuity gives $\mathfrak D_ F(T)\geq M(F)$, hence equality. The measures $\Gamma _ F(T_ j)$ have bounded mass; let $\Gamma '$ be a weak cluster point, whose mass is $M(F)$. For a continuous real function $h$, choose $\varepsilon >0$ with $1\pm \varepsilon h>0$. Since $\Gamma _{F+n\log (1\pm \varepsilon h)}=(1\pm \varepsilon h)\Gamma _ F$, upper semicontinuity of $\mathfrak D$ for these two continuous logarithmic densities gives
\[ \int _ X(1\pm \varepsilon h)\, \mathrm{d}\Gamma ' \leq \int _ X(1\pm \varepsilon h)\, \mathrm{d}\Gamma _ F(T). \]
Since both measures have mass $M(F)$, the two signs give equality of the integrals of $h$. Thus every cluster point equals $\Gamma _ F(T)$, which proves the convergence.
For affineness, take $T_1,T_2\in \mathscr G_\omega $ and $t\in [0,1]$, and express the coefficient measures of both currents through densities with respect to the sum of their trace measures. Concavity and homogeneity of the $n$th root of the determinant, applied to these matrix densities, give
\[ \Gamma _ F(tT_1+(1-t)T_2) \geq t\Gamma _ F(T_1)+(1-t)\Gamma _ F(T_2),\qquad 0\leq t\leq 1. \]
If $T_1$ and $T_2$ maximize, so does the current on the left, by concavity of $\mathfrak D_ F$. Both measures in the inequality then have mass $M(F)$, so equality holds. This proves affineness, and continuity is the special case of the first assertion in which every $T_ j$ is a maximizer. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.